LOCAL JORDAN ALGEBRAS

Thesis for the. Degree of PIL D.
MICHIGAN STATE UNIVERSITY
MARVIN EDWIN CW 7
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iichigan State
”a? University

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L WW3:- «3,

This is to certify that the

thesis entitled
"Local Jbrdan Algebras"
presented by

Marvin E. Camburn

has been accepted towards fulfillment
of the requirements for

Ph- D- degree infiLhQLfEQ-CS

KAT“. L_.. L/\’;-LK—‘

Major professor

Date 6-19-71

0-7639

 

ABSTRACT
LOCAL JORDAN AlGEBRAS

BY

Marvin Edwin Camburn

The purpose of this thesis is to study local Jordan algebras
and their completions. A local Jordan algebra J is a quadratic
Jordan algebra over a commutative associative ring o with identity
such that gKJ) is the unique maximal ideal of J, J/EKJ) satisfies

"° (k)
the minimum condition, and fl EKJ) = 0, where EKJ) is the

k=l
Jacobson radical of J and EKJ)(1) E 5K3), EKJ)(k+D

a U (k)(EKJ)(k)), k E;N. The main results of this thesis are:
) .

90
Theorem. If J is a local Jordan algebra, then the completion

of J is a local Jordan algebra.

Theorem. If J is a complete local Jordan algebra over a field

of characteristic not 2, then either

(1) J is a complete, completely primary local Jordan algebra,

(2) J is the vector space direct sum of two completely primary local
Jordan algebras and a subspace of J, or

(3) J is isomorphic to a Jordan matrix algebra bgbh,ja) of
order n 2 3, where (i) CD,j) is an alternative algebra
with involution and identity such that ©Q33j) C NCD), the
Smiley rldical ELD) is the unique maximal ideal of L&,j),

and ,D is complete in a topology induced by the topology

Marvin Edwin Camburn

on J, or LD,j) is an associative algebra with involution
such that ‘E¢&), the Jacobson radical of .8, is the unique

maximal ideal of g&,j) and .3 is.acomp1ete semi-local algebra-

If J is either a complete, completely primary Jordan
algebra, or J is one of the algebras in (3), then J is a

complete local Jordan algebra.

lDCAL JORDAN AlGEBRAS

BY

Marvin Edwin Camburn

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1971

ACKNOWLEDGEMENTS

I wish to express my appreciation to my research advisor
Dr. M. Tomber for the advice and encouragement which he provided
throughout the preparation of this thesis. I also wish to thank

Dr. E. Ingraham and Dr. C. Tsai for their helpful suggestions.

ii

Chapter

1

TABLE OF CONTENTS
INTRODUCTION

ALTERNATIVE AND ASSOCIATIVE AIEEBRAS

1.1 Algebras over Rings
1.2 Simple Artinian Algebras with Involution
1.3 Alternative Algebras

JORDAN AlGEBRAS

2 1 Quadratic Jordan Algebras

2 2 Inner and Outer Ideals

2 3 Invertible and Quasi-invertible Elements
2.4 Linear Jordan Algebras

2 5 Constructions of Jordan Algebras

2 6 Maximal Ideals

2 7 Linear Jordan Algebras over Fields

COMPLETIONS OF QUADRATIC JORDAN ALGEBRAS
3.1 Topological Modules

3.2 Topological.Quadratic Jordan Algebras
3.3 Local Jordan Algebras

STRUCTURE OF COMPLETIONS OF LOCAL JORDAN ALGEBRAS
1 Basic Concepts

2 Idempotent Lifting

.3 Structure of Completions

4 Complete Local Jordan Algebras

BIBLIOGRAPHY

iii

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95

INTRODUCTION

The importance of the theory of (commutative, associative)
local rings in algebraic geometry and the theory of commutative rings
is well-known (see [8], [12]). Cohen [2] provided the first structure
theory for complete Noetherian local rings in his now classic paper.
In 1957, Batho [l] in developing a theory of noncommutative, asso-
ciative, semi-local rings proved that any noncommutative complete
local ring is isomorphic to a full matrix ring over a complete, com-
pletely primary, local ring.

If A is any associative ring with identity, then A is an
algebra over the ring 2 of integers in the sense of §l.1. Thus A+
is a quadratic Jordan algebra, and when A is a local ring, much of
the structure of A can be seen to carry over to A+. Hence it
seems reasonable to ask the following question: is it possible under
a suitable definition to develop a structure theory for complete local
Jordan algebras? In this thesis, an affirmative answer is given for
a Jordan algebra over a field of characteristic not 2. This is
accomplished through the concept of a completion of a local Jordan
algebra.

Chapters 1 and 2 contain the basic concepts with which we will
be dealing throughout the thesis. In Chapter 1, algebras with involu-
tion are considered with emphasis on the simple ones. In Chapter 2,

both quadratic and linear Jordan algebras are defined as well as the

Jacobson radical, Jordan matrix algebras, and the concept of isotopy.
A sketch of the structure theory of certain simple Jordan algebras
over a field of characteristic not 2 is included. Many of the proofs
which are omitted in the first two chapters can be found in either
[5] or [6].

Chapter 3 deals with the completion of a local (quadratic)
Jordan algebra, beginning with an outline of completions of modules.
For a more extensive treatment of the material on modules, the reader
is referred to Zariski and Samuel [12]. The main result of this
chapter is the following: the completion of any local Jordan algebra
is a local Jordan algebra.

In Chapter 4, the main structure theory is developed through
the use of infinite series and an idempotent lifting property.
Completions of local Jordan algebras are classified according to the
capacity of their residue class algebras modulo the (Jacobson)
radical. When this capacity is greater than or equal to 3, a com-
pletion is shown to be a Jordan matrix algebra over a certain type
of alternative algebra or over a semi-local associative algebra with

involution.

CHAPTER I

ALTERNATIVE AND ASSOCIATIVE ALGEBRAS

1.1 Algebras over Rings

 

Let Q be an arbitrary commutative associative ring with
identity element 1, and let m and B be unital left Q-modules.
A mapping f : m X M a»% is bilinear, if f(x1 + x2,y) = f(x1,y) +
f(xz,y). f(X,y1 +y2) = f(X.y1) + f(x,y2). and f(a/X,,y) =
af(x,y) = f(x,ay), for all x1,x2,x,y1,y2,y E m, and a E Q. For

the special case B = Q, E is called a bilinear form on W. m is a

 

(unital) algebra over Q, if m is equipped with a bilinear multipli-
cation (x,y) a xy, x,y E m and M has an identity. The mapping
a X m X m a m defined by [x,y,z] = (xy)z - x(yz), x,y,z E m is

called the associator, and it is linear in each component. u is

 

alternative, if [x,x,y] = 0 = [y,x,x], for every x,y E m, and

m is associative, if [x,y,z] = O, for every x,y,z E m. For an

 

arbitrary algebra u we define the nucleus of N to be the set
N(m) = {n E m : [n,x,y] = [x,n,y] = [x,y,n] = O, for all x,y E m}.
The center of m is the set C(m) = {c E N(m) : ex = xc, for all
x E m}. N(m) is an associative algebra and COM) a commutative
associative algebra.

Let u be an algebra over Q. A Subalgebra K is a sub-

 

module of a such that for all x,y E K, xy 6 K. An ideal of u

is a submodule K such that xa and ax E K for all a E m,

x E K. Right and left ideals are defined in a similar fashion. The
concepts of residue class ring, homomorphism, kernel, etc., are taken
to be the standard ones (see [9]). (m,j) is an algebra with involu-
tion, if m is an algebra and j is an anti-automorphism of M

2
such that j = j. A subalgebra (ideal), K of (m,j) is a Sub-

 

algebra (ideal) of m such that KJ = K. If CB,k) is also an

algebra with involution, then a homomorphism from (m,j) into GB,k)
is a homomorphism e : u a E such that ej = k9. If a 6 m

satisfies aJ = a, then a is called a symmetric element of m

 

and the set of all symmetric elements of (m,j) is denoted by
©(m,j). If no confusion can arise, this set will simply be denoted
by ©(m). Now in general an algebra m is simple, if m has no

ideals other than 0 and m, and m2 : m fl # 0. An algebra with

involution (m,j) is a simple algebra with involution, if m

 

contains no j-invariant ideals other than 0, and N, and W2 ¥ 0.
If N and B are (unital) algebras over Q, the direct sum of W
and B is the set N x B together with componentwise addition,
scalar multiplication, and multiplication. The direct sum is denoted
by QIC)$, and fl and B are isomorphic to the ideals
{(a,o) E 91631) : a E u} and {(o,b) 6 91 (9’3 : b 6 fi}, reSpectively.
Hereafter we shall identify u and $ with these ideals. K is
an ideal of fl<3§8 if and only if K = K1 G>K2, where K is an

1

ideal of m and K2 is an ideal a. It is clear that if m and

B are associative, then so is fl(® B.

1, 32 be ideals

of M. It is well-known that, if 81 FIDZ = 0, and N = B1 +-$2 =

Let u be an associative algebra and let M

In this case we

{b1 +b2 : b1 6 531, b2 EVBZ}, then 91 3%199‘232.

suppress the ordered pair notation and write b1 + b2 for (b1,b2).
Also we will write 21 = 81 @332-

Let N be a (unital) associative algebra and define a new
product on m X m by x°y = yx, for all x,y E m. The opposite
algebra of m denoted by mo is the algebra determined by using
the Q-module structure of u together with the product 0. m° is
an associative algebra which is anti-isomorphic to m. Let
a =91 (+3210. Hence. if (x1.y1). (x2,y2) EB, then (x1.y1)(x2,y2) =
(x1x2,y3y2) = (x1x2,y2y1), and B is an associative algebra. If
1 is the identity of Qh,then l is the identity of Mo and (1,1) is
the identity for .B. Now define the mapping j : B -% by
(x,y)j = (y,x), for all (x,y) E B. Then j is an involution

called the exchange involution and so GB,j) is an algebra with

 

involution. It is clear that $68) = {(a,a) : a E m}. If K is

an ideal of ‘6, then K = K, C)K2, where K is an ideal of m

1 1

and K is an ideal of m°. Suppose (o,y) 6 K2. Then (o,y) E K

2
and KJ = K implies (y,o) = (o,y)J E K. Thus (y,o) E K1 and

(o,y) = (y,o)j, so (o,y) E Ki. Conversely, if (w,z) E Kj, then
(w,z) = (x,o)j, for some (x,o) 6 K1. Hence (w,z) = (o,x) and
(w,z) E K, so (w,z) 6 K2. Thus Ki c K2 and hence equality
holds; i.e., K = K1_C>Ki, K1 an ideal of m. On the other hand,
if K1 is an ideal of m, then K1<$ Ki is an ideal of GB,j).

Therefore there is a bijection from the set of ideals of GB,j)
onto the set of ideals of m.

Finally, note that, if Gu,j) is an algebra with involu-
tion such that 9; =15 @mj, m an ideal of 91, then fij 2&0 under

JbJ

j. This follows since (b1° b2)J = (b2b1)J = b1 2.

1.2 Simple Artinian Algebras with Involution

Let M be an associative algebra over a field Q. m

satisfies the minimum condition for right ideals, if m has no

 

infinite descending chain of right ideals. An algebra N which
satisfies the minimum condition for right ideals is called (right)
Artinian. By the Wedderburn-Artin theorem, an Artinian associative
algebra over a field Q is simple if and only if We: An, an algebra
of n X n matrices with entries from some division algebra A. A
simple Artinian algebra with involution Gu,j) is a simple asso-
ciative algebra with involution Such that m is Artinian. The
structure of these algebras will be determined next.

Let Gu,j) be a simple associative algebra with involution.
If m is not itself simple, then there exists an ideal. B of m
such that 4B # 0,N. Clearly fBj is also an ideal of u, so that
B 0%]. is an ideal of (m,j) such that I) fliij CB C91. By the
simplicity of (91,3'), we must have 23 [Haj = 0. Similarly, g3 +93j
is an ideal of (mg) and 0:33 cg} +29. Thus 9; =15 +g5j and

hence m ==$ QDEJ. If B is an ideal of 55, then B is an ideal

of m, and thus B +BJ is an ideal of Qu,j). Hence, B + BJ = O

or B +Bj ==m, so that either B = 0 or B =38. Therefore $
is a simple algebra.

Now assume Gu,j) is a simple Artinian algebra with involu-
tion. If m is simple, then ”:3 An, A an associative division
algebra with involution. If N is not simple, then m ==% C)%j,
where ‘B is a simple ideal of m. Since N is Artinian, 53 is
Artinian, and thus E a Ah’ A an associative division algebra. It
is clear that SBj’% (A°)n, so that MEE An(® (A°)n, with j the

exchange involution.

1.3 Alternative Algebras

 

Let m be an alternative algebra over a field Q. Then
[x,x,y] = O, for all x,y E m, and [y,x,x] = O, for all x,y E m.
These statements are equivalent to the left and right alternative
laws: x2y = x(xy), yx2 = (yx)x, for all x,y E u. By the theorem
of Artin, these laws are equivalent to the statement that every
subalgebra of m generated by two elements is associative. The
associator in an alternative algebra has the following prOperty:
if a is any permutation of {1,2,3}, then [xlo,x20,x3o] =
(sgn o)[x1,x2,x3], for all x1,x2,x3 E m. The Moufang identities

also hold in any alternative algebra m:

(1) (XYX)Z X[y(XZ)]
(2) Z(XYX) = [(ZX))']X
(3) (Xy) (ZX) = X(YZ)X.
for every x,y,z E m [9].
Assume that m is alternative with identity 1. x E N is

invertible, if there exists y E m such that xy = yx = 1. If x

 

is invertible with inverse y, then [x,y,z] = O, for all z E m
and y is unique, (see [9], p. 38). If x is invertible, the
unique inverse of x will be denoted by x-l. Suppose x,y,z E m
are such that yx = 1 = xz. Then by the second Moufang identity,

1 = yx = y(xzx) = [(yx)z]x = zx. Hence x is invertible with

-1 =

z y. Suppose x,y E m are invertible. Then

0 = [x'1.xy.y-1] [XY.y-1.x'1] = [(xy)y'1]x“1 - (xy)(y'1x'1) = 1 -

(xy)(y-1x-1). Thus (xy)(y-lx-1) = 1. Similarly, (y-lx-1)(xy)

so xy is invertible, and (xy)-1 = y-lx-l. Next suppose x,y E N

inverse x

ll
I'd
U

are such that x and xy are invertible. Then

y = (X-IX)y = x-1(xy), is invertible.

Definition 1.3.1. Let u be a (unital) alternative algebra. x E M
is quasi-invertible with quasi-inverse y, if 1 - x is invertible

with inverse 1 - y. An ideal K of m is a qgasi-invertible

 

ideal, if every element of K is quasi-invertible.

Proposition 1.3.1. Let K be a quasi-invertible ideal of u. If
u E N is invertible and x E K, then u - x is invertible.

Proof. u-1(u - x) = l - u x is invertible, since u x E K

. . -1 . . . . . . .
implies u x is quaSi-invertible. Thus u - x 18 invertible.

Proposition 1.3.2. If A and B are quasi-invertible ideals of
m, then A +'B is a quasi-invertible ideal of m.
Proof. If a E A, b E B, then 1 - (a +tb) = (l - a) - b is

invertible by Proposition 1.3.1.

Theorem 1.3.1. Let u be an alternative algebra with 1 over a
field Q. Then m contains a unique maximal quasi-invertible

ideal EKM) which contains every quasi-invertible ideal of u.
Moreover, M/EKN) contains no nonzero quasi-invertible ideals.
Proof. Since 0 is a quasi-invertible ideal, the set of all quasi-
invertible ideals is nonempty, and Zorn's lemma may be applied to
find a maximal quasi-invertible ideal Efiu). If A is any quasi-
invertible ideal of m, then by Proposition 1.3.2, A +-EKM) is a
quasi-invertible ideal of m. Hence A +-EKm) = Exm), and A C 51%).
Thus the uniqueness of EKM) is clear. Now if a +-EKM) is
invertible in m/EKM), then there exists b 6 N such that

ab - 1 E EKM). Hence ab = 1 - z, z E EKm) and, so ab is

invertible. Similarly ha is invertible in m. Thus

abza = (ab)(ba) is invertible in m. Let c E m be the inverse

2 2 2 2 2
of ab a. Then a[b (ac)] = (ab a)c = l = C(ab a) = [(ca)b ja, and
a is invertible in m. Therefore, if x +-gxm) is quasi-invertible
in m/Exu), x is quasi-invertible in m, and if K is a quasi-
invertible ideal of fl/EKM), the complete inverse image of K in
m under the natural homomorphism is a quasi-invertible ideal of m

and thus is contained in RKu). Hence K = 0.

Definition 1.3.2. The unique quasi-invertible ideal ETD) of m

 

is called the Smilely radical of M [10].

thanow turn our attention to alternative algebras with involu-

tion, and first consider the defining conditions for a quaternion

 

algebra Q over a field Q of characteristic not 2. Recall that

Q is a noncommutative associative algebra with identity 1, and

2

generators i,j satisfying i2 = XI, j ul, and ij = -ji, where

x,u E Q are nonzero. The set {l,i,j,k ij} constitutes a basis

2

for Q, and the following are also satisfied: k = 'lul;

. = _ . = _ ., . = _. = _ .. = . . +

Jk k] p1, k1 1k k] If a all + (121 + 013] 04k,
then a a a = all - azi - a3j - 04k is an involution called the

standard involution, and the norm of a E Q, n(a), is defined by

 

a 3 = n(a)l = a a. The norm is a quadratic form on Q whose
associated symmetric bilinear form, n(a,b) = n(a + b) - n(a) -
n(b) = a5 + b3, is nondegenerate.

Let 0 = Q @Q, where the sum is a vector Space direct sum,

and define multiplication in (3 by

(a,b)(c,d) = (ac + vdb, da +-bE), v # 0 e Q

10

O' is a (not associative) alternative algebra over Q called the
algebra of octonions defined by O» and v. 0' has identity (1,0)
and the subset {(a,o) : a E Q} is an isomorphic copy of Q. If

we write a for (a,o), and let L = (0,1), then every element of
0' can be written uniquely as a +~bL, a,b E Q. If x = a + bL E(3,
the mapping x a fi = a - bL is an involution in Or again called
the standard involution. As with Q, the quadratic form n defined
by x i = n(x)1 = x x, x E O» is called the norm, and the associated
bilinear form is nondegenerate. Q and 0, are examples of simple

alternative algebras.

Definition 1.3.3. An algebra with involution LB,j) is called a
composition algebra, if .B is alternative and for all x E.D,

xxj = Q(x)1 = xjx, where Q is a quadratic form whose associated
bilinear fornl Q(x,y) E Q(x +-y) - Q(x) - Q(y) is nondegenerate.

CD,j) is a split composition algebra, if .D is not a division

algebra.

Now any composition algebra c&,j) is finite dimensional
and is isomorphic to a two-dimensional commutative associative algebra
with basis {1,q] where qj = -q, a quaternion algebra with standard
involution, or an octonian algebra with standard involution [5].
In addition, we state a result in which the determination of certain
simple alternative algebras with involution is given. This result

as well as a more detailed discussion of the topics in this section

can be found in [5].

Proposition 1.3.3. let Cb,j) be a simple alternative algebra

with involution and identity element Such that every nonzero symmetric

ll

element is invertible in the nucleus, and let P be the subset of
.3 of symmetric elements in C69). Then F is a subfield of .B
and the following possibilities occur for CD,j): I. .8 = A<D A0,
A an associative division algebra, j the exchange involution;
II. CD,j) an associative division algebra with involution;

III. a Split quaternion algebra over F isomorphic to F2;
IV. an algegra of octonions over F with standard involution.

Conversely, any algebra in one of the classes I-IV satisfies the

given conditions.

CHAPTER 2

JORDAN AICEB RAS

2.1 Quadratic Jordan Algebras

 

Quadratic Jordan algebras were first introduced by McCrimmon
[7 , althou h the presentation iven in this section follows that of
8 8

Jacobson [6].

Definition 2.1.1. Let Q be an arbitrary connmtative associative
ring with l and let M,N be left Q-modules. Then a mapping

Q : M a N is called quadratic, if (1) for all a E Q, a E M,
Q(aa) = 0.2%»). and (2) Q(a.b> .5 cm +b) - Q(a) - MM is a

Q-bilinear mapping from M X M into N.

If P is a commutative associative ring extension of Q

in the sense that P is a(unital) algebra over Q, let MP = P 85M, M

a unital Q-module. If Q : M a EndQM is a quadratic mapping,

then there exists a unique quadratic mapping 6 : MP 4 EndPMP

such that the following diagram:

_9_.
EndQM

.1 1,

MP ——-9 EndPMP

~

Q

is commutative, where v(a) = 1 ®>a, Q(A) = 1 @>A, and l<x A(p E>a)

p Q9 A(a) [6].

12

13

We are now ready to define a quadratic Jordan algebra.

Definition 2.1.2. A (unitalquuadratic Jordan algebra over a

 

commutative associative ring with identity is a triple (J,U,1),
where J is a unital left Q-module, 1 a distinguished element of

J, and U : J a End J (a a Ua) is a quadratic mapping such that

QJ1 U1 = IdJ
U =
QJ2 for every x,y E J, Ux yUx UUX(Y)
J3 If U = U - U - U is the associated s mmetric
Q x,y X+y X y y

bilinear mapping, and VX y E End J is defined by

V 2 = U x , for all z E J, then
X,y( ) z,y( )

V U = U V = U
x,y y y Yax Uy(x)sy
QJ4 If P is any commutative associative algebra over Q

and U is the extension of U to Jp, then U satisfies

QJl.- QJ3

By a linearization of QJ2 and QJ3, the following identities

are obtained:

J5 U U U + U U U = U
Q x y x,z x,z y x UX(Y):UX 2(Y)

3

6 U U U + U U U +‘U U U = U + U
QJ X y 2 Z y X x,z y x,z UX(Y),UZ(Y) UX(Y):Z
QJ7 U U U + U U U +‘U U U +‘U U U
x y z,w x,z y x,w z,w y x x,w y x,z
=U +U
Ux,z(Y)’Ux,w(Y) UX(Y)’Uz,w(Y)
QJ8 V U +'V U = U V +'U V

x,y Ysz x,z y Yaz YSX y 23X

14

In the presence of QJ1 - QJ3, QJ4 is equivalent to QJS - QJ8.
Hence QJ1 - QJ3 together with QJS - QJ8 constitute an intrinsic set

of defining conditions for a quadratic Jordan algebra [6].

Definition 2.1.3. For x,y E J, (i) x2 E UX(1), (ii) x ° y E UX y(l),

(iii) VX E Vl,x°
It follows immediately from Definition 2.1.3 that VX(y)
= V1,x(y) = Ux,y(1) = Uy,x(1) = y o x, and o is a symmetric bilinear

composition. We now list identities which are necessary for this

thesis [6].

QJ9 VX = Vx,1 = Ux,l
l = . o =
_ 2
QJIO U 2 - U
x
x
J11 U U +‘U U = U +'U - V U V
Q x x Ux(y),y x°y x y x
QJ12 U V +'V U =
x x x,Xoy
_ 2
x
QJ14 U +v = v v
x,y x,y y X
J14' U z +U = z o x 0
Q x,y( ) z,y(><) ( ) y
. . . 0 1 2
Definition 2.1.4. If x E J, let x = l, x = x, x = UX(1) and

for n 2 2, xn = Ux(xn-2).

From Definition 2.1.4, we have
QJ15 U = U
QJ16 (x ) = x

QJ17 X o X = 2X [6]

15

. . n n+2
For any nonnegative integer n, Ux(x ) = x . If

n n+2m
U m(x ) — x
x

n
, m a nonnegative integer, then U m+1(x ) =

m+1 n _ m n _ n _ n+2m _ n+2nr-l-2 _ n+2(m+l)
Ux (x ) - Uxe(x ) — Uxem(x ) Ux(x ) — x x .

Therefore by induction on m we have

+2
QJ18 U m(xn) = xn m, m,n nonnegative integers.
x

16

2.2 Inner and Outer Ideals

 

Let (J,U,1) be a quadratic Jordan algebra. If A,B C J,
let UB(A) be the set of all Q-linear combinations of elements of

the form Ub(a), b E B, a E A.

Definition 2.2.1. Let (J,U,1) be a quadratic Jordan algebra. A

subset K of J is an inner ideal of J, if K is a submodule of

 

J and UK(J) C K. A subset K is an outer ideal, if K is a sub-

 

module of J and UJ(K) C K. A subset K is an ideal of J, if
K is both an inner and an outer ideal of J. A subset K is a

subalgebra of J, if K is a submodule, l E K and UK(K) C K.

If K is an inner ideal of J and x E J, then Ux(K)
is an inner ideal of J, so in particular UX(J) is an inner ideal
called the principal inner ideal determined by x. UX(J) need not
contain x, so the inner ideal generated by x is Qx +-UX(J) [11].

For outer ideals we have the following:

Proposition 2.2.1. Let (J,U,1) be a quadratic Jordan algebra and
K an outer ideal of J. If x,y E J and k E K, then

. k = _ ' ' = = o
(1) Ux,y( ) Vk,y(x) E K, (11) Vk(x) Vx(k) x k E K, so

J ° K C K, and (iii) Ux k(y) = Uk X(y) = Vy k(X) = Vy X(k) E K.

Proof. (i) ”x,y(k) = UX (k) - Ux(k) - Uy(k) E UJ(K) C K.

.{ny
(ii) By QJ9, x o k =Vx(k) = Ux,1(k) e K by (i).

(iii) By QJ14', Uk,x(y) = (y ° k) 0 x - UX y(k) E K by (i) and (ii).

3

Definition 2.2.2. Let (J,U,1) and (J,U,1) be quadratic Jordan

algebras. A mapping 9 : J ~ J is a (Jordan) homomorphism, if e

 

is a linear mapping and

l7

(1) 9(1) = 1

(2) e<ux(y>> = 69(x)<e<y)>. for all x,y e J.

Isomorphism, endomorphism and automorphism all have the usual mean-
ings.

It is clear from Definition 2.2.2, that the kernel of any
homomorphism 9 is an ideal of J. If K is any ideal of J,

then (J/K, U, I) is a quadratic Jordan algebra, where

Ux+K(y + K) = Ux(y) +'K and l = 1 +-K. The natural Q-module
homomorphism 9 : J a J/K is a Jordan homomorphism and if
(J,U,1) is any homomorphic image of (J,U,1), then 3 asJ/ker @-
If K is any inner (outer) ideal of J, then 9(K) is an inner
(outer) ideal of J, for if x E J, k E K, Ue(k)(e(x)) =

Q(Uk(x)) E Q(K) (5 (9(k)) = 6(Ux(k)) E e(K)). Conversely, if

6(X)
K is any inner (outer) ideal of J, then K = 9-1(K) is an inner
(outer) ideal of J and ker 9 C K, for x E J, k E K implies
U9(k)(e(X)) 6 K, so 9(Uk(x)) e K, and Uk(x) e K(e(Ux(k)) =
69(k)(G(X)) E K implies Ux(k) E K). Thus the (inner, outer)

ideals of J are in one-to-one correSpondence with the (inner,

outer) ideals of J which contain ker 9.

Proposition 2.2.2. If (J,U,1) is a quadratic Jordan algebra and
K is an outer ideal of J, then for all x,y E J Such that

n n
x - y E K, x - y E K, for all n E,N.

Proof. The case = 1 holds by hypothesis. Let x - y = w E K,

n
k k
and assume x - y E K for k < n. Then

l8

n-2 n-2
x - y = UX(X ) ‘ Uy(y )

n-2 n-2
Uy+w(x ) - Uy(y )

ll

-2 -2 -2 -2
(xn ) +-Uy(xn ) + Uw(xn ) - Uy(yn )

II
C.‘

st

ll

n-2 n-2 n-2 n-2
Uy,w(x ) + Uy(x - y ) + UW(X ) E K

by Proposition 2.2.1 and the induction hypothesis.

Definition 2.2.3. Let (J,U,1) be a quadratic Jordan algebra and
(1)

jK be an ideal of.J.Then K(O) E J, K E K, and for n 2 l,

+
Km 1) =U
K

(n)
(n) 0‘ )°

Proposition 2.2.3. If (J,U,1) is a quadratic Jordan algebra and
K is an ideal of J, then K(n) is an ideal of J for n = 0,1,2,...
Proof. Clearly K(o), K(1) are ideals of J. The proof follows
by induction on n, if K(2) = UK(K) is an ideal. Let x E J,

k ,k e K. Then U (x) = U U U (x) = U [U (U (x))]
1 2 Uk1(k2) k1 k2 k1 k1 k2 k1

E UK(UK(J)) C UK(K)’ since K is an inner ideal. Thus K(2) is

an inner ideal. To show that UK(K) is an outer ideal first note

that, if k e K, then U (k) = U (k) - U (k) U (k)
k1,k2 k1+k2 k1 k2
E UK(K). Thus by QJ11 and QJ12,

UxUk1(k2) = UUx(k1),k1(k2) +-Uxok1(k2) - UklUX(k2) VxUk Vx(k2)

l

UU (k ).k (k2) +'Uxok (k2) ' Uk Ux(k2) ' (Uk ,X0k
x l l 1 l

1 l

- Ulex)Vx(k2)
= u (k ) +‘U o (k ) - U [U (k ) -
Ux(k1).k1 2 x k1 2 k1 x 2 3

Ukl x°k1(x ° k2) +'Uk1(x ° (X ° k2))

3

E UK(K).

l9

(2)

Thus K is an ideal, and the proposition follows as indicated.

Proposition 2.2.4. Let (J,U,1) be a quadratic Jordan algebra and

m E K(k) k

K an ideal of J. If x E K, then x for all m 2 3 ,

k Eli.

3‘"1 (k)

Proof. We first Show that x E K , k E N by induction on

k. If k = 1, then x E K(1) and if k = 2, then

3

2
x3 = U (x) E UK(K) = K( ). Suppose the property holds for r < k,
X k-l 3k-2 (k_1)
where k 2 2. Then by QJ18, X = U (x ) E U (K )
3k-2 K(k-l)
x
k-l
= K(k). Now, if m 2 3k, then xm = U k_1(xm-2°3 ) E K(k), since
3
(k) X

K an ideal.

20

2.3 Invertible and Quasi-invertible Elements

 

Definition 2.3.1. Let (J,U,1) be a quadratic Jordan algebra.
x E J is invertible, if there exists y E J Such that
(i) Ux(y) = x, and (ii) Ux(y2) = l. y is called the inverse of

. -1
x and we write y = X .

Proposition 2.3.1 (Theorem on Inverses). l. The following are
equivalent: (i) x is invertible; (ii) Ux is invertible in
End J; (iii) 1 e UX(J).

2. If x is invertible, then (i) x.1 is unique and

x”1 = U;1(x), (ii) U _1 = U;1, (iii) x-1 is invertible and
x

- - - 2 _
(X1)1=X,(1V)XOX1=2,(V)XOX1=2X,(V1)V_1=

-1 _ x
v U = U 1V .
X x x x

3. Ux(y) is invertible if and only if x and y are

. . -1 -l
invertible, and [Ux(y)] = U _1(y ). [6]

X

By 1(ii) and 1(iii), if UX is onto for x E J, then UX

is one-to-one. However, the converse need not be true. If UX

is not onto, then x is called a zero divisor of J. Thus x E J

 

is a zero divisor if and only if there exists y # O E J Such that
Ux(y) = O. A quadratic Jordan algebra which has no zero divisors

other than zero is called a Jordan integral domain. Similarly, a

 

quadratic Jordan algebra in which every nonzero element is invertible

is called a Jordan division algebra. z E J is an absolute zero

 

divisor, if Uz = 0. J is (strongly) nondegenerate, if J has

 

no nonzero absolute zero divisors. The ideal, zer J, generated

by the set of all absolute zero divisors of J is a nil ideal in the uSual

n
sense that z E zer J implies z = 0, for some n th, and hence,

21

if K is an ideal of J such that zer J c K, then J/K is non-

degenerate [6].

Definition 2.3.2. Iet (J,U,1) be a quadratic Jordan algebra.
z E J is called quasi-invertible, if 1 - z is invertible. If
the inverse of 1 - z is denoted by 1 - w, then w is called

the quasi-inverse of z.

 

Let 2 be quasi-invertible with quasi-inverse w. Then

the following properties are immediate from the definition (see
[6]).
(1) u (l-w)=l-z u [(l-w)2]=1
l-z ’ 1-2
2
(2) W+z-Z-WoZ+Uz(W)=O
2
(3) 2w - z2 + UZ(2w) - Uz(w ) + 22 - 2w 0 z - w2 + w2 o z = O
-1 2.
(5) zow=2(z+w)

(6) w + z + Uz(1 - w) = 0.

From (6), it is clear that any ideal which contains 2 must also
contain w. Also note that w is quasi-invertible with quasi-
inverse 2.

An (inner, outer) ideal is quasi-invertible, if every

 

element is quasi-invertible. The following useful lemma is stated

without proof (see [6]).

Lemma 2.3.1. Let (J,U,1) be a quadratic Jordan algebra and let
K be a quasi-invertible ideal of J. If u E J is invertible and

z E K, then u + z is invertible.

22

If K1 and K2 are quasi-invertible ideals of J, and
x E K1, y E K2, then 1 - x is invertible, and thus, by Lemma 2.3.1,
1 - (x +»y) = (1 - x) +-y is invertible. Hence K1 + K2 is a
quasi-invertible ideal. This fact together with Zorn's lemma implies
the existance of a unique maximal quasi-invertible ideal EKJ), which

contains every quasi-invertible ideal of J. EKJ) is called the

Jacobson radical of J and J is (Jacobson) semisimple, if

 

EKJ) = 0. Since the homomorphic image and the complete inverse
image of a quasi-invertible ideal is a quasi-invertible ideal,
J/EKJ) contains no nonzero quasi-invertible ideals. Finally, since
EKJ) contains every nil ideal of J, in particular zer J C EKJ),
and so J/EKJ) is nondegenerate [6].

We conclude this section with a brief discussion of isotopes.

For more details, see [6].

Definition 2.3.3. Let (J,U,1) be a quadratic Jordan algebra and
(C)

X

c E J invertible. Define U(C)

1(C) = c-1. Then the quadratic Jordan algebra

:JaEndJ by U =UXUC,

x E J, and let

J(C) = (J, U(C), 1(C)) is called the c-isotope of (J,U,1).
(U (d))
If d E J is also invertible, then (J(C))(d) = J C .
-2
_ _ -2 (U (C ))
Thus if c 2 a (c 1)2, (J(C))(C ) = J C = J(1) = J. Now

if K is an ideal of J and x E J, k E K, then Uic)(x)

= UkUC(X) E K and U(C)(k) = Uch(k) E K. Thus K is an ideal

x
of J(C). Similarly, if K is an ideal of J(C)

, then K is an
( ('2
ideal of (J C)) C ) = J. Thus the ideals of J and any isotope

coincide. In particular, EKJ(C)) = RKJ) (see [6]).

23

2.4 Linear Jordan Algebras

Linear Jordan algebras play an important role in Chapter 4,
where the only algebras considered will be linear Jordan algebras

over fields of characteristic not 2.

Definition 2.4.1. A (unital) linear Jordan algebra over a com-

 

mutative associative ring Q (with 1) containing'% is a triple
(J,R,1) such that J is a (unital) left Q-module and R is a
mapping from J a End J such that R is a Q-homomorphism, and

if the image of x E J is denoted by Rx’ then

J1 R1 = idJ
J2 R R — R R
RX(X) x x RX(X)
' d ' d = = .
J3 If Ex 18 efine by Lk(y) Ry(x), then EX Rx

Let (J,R,l) be a linear Jordan algebra and for x,y E J
define x.y = Ry(x). Then . is a bilinear composition, by J1
2 2
x.l = x, by J3 x.y = y.x, and by J2, (y.x).x = (y.x ).x, where

2

x = x.x. Conversely, let J be a unital left Q—module -l E Q,

’ 2
equipped with a symmetric bilinear composition . Such that l

is an identity for . and (y.x).x2 = (y.xz).x for all x,y E J.
By defining R by Rx(y) = y.x, R satisfies J1 - J3 and (J,R,l)

is a linear Jordan algebra. Thus we have the more familiar alternate

definition which is equivalent to Definition 2.4.1.

Definition 2.4.1'. A (unital) linear Jordan algebra J over the

 

. . . . . . . 1 .
commutative assoc1ative ring Q (With 1) containing 3' is a
unital left Q-module with a bilinear composition . and l E J

such that (i) 1.x = x = x.l, x E J, (ii) x.y = y.x, for x,y E J,

24

2
(iii) (Jordan identity) (y.x).x = (y.xz).x, for all x,y E J.

If (J,R,l) is any linear Jordan algebra over Q (% E Q)
and P is any commutative associative algebra over Q, then there
exists a unique extension R of R to Jp = P ®§.J So that
(JP, ii, 1 a 1) is a linear Jordan algebra [6].

Finally, if (J,R,l) is a linear Jordan algebra, define
U : J a End J by UX = 2R: - R 2. Then U is quadratic and
(J,U,1) is a quadratic Jordan :lgebra. Conversely, if (J,U,1)

is a quadratic Jordan algebra over Q, where % E Q, then define

1
: .4 d = ——
R J En J by Rx 2 VX

. In this case (J,R,l) is a linear
Jordan algebra. In fact there is a category isomorphism between
the category of linear Jordan algebras with homomorphisms as
morphisms and quadratic Jordan algebras over rings Q such that
E'E Q with homomorphisms as morphisms [6].

For the remainder of this section let (J,U,1) be a
quadratic Jordan algebra over the unita1.conmutative associative

ring Q and aSSume %-E Q. Then we will write J for the triple

(J,U,1) and the triple (J,R,l), Since J is also a linear Jordan

algebra in the sense of Definition 2.4.1' with x.y -'x o y.

2

First note that K is an ideal of J if and only if K is an

outer ideal of J. Since one direction is clear, assume K is
2

an outer ideal of J. Then for k E K, x E J, Uk(x) = 2Rk(x)

- R 2(x) = 2(x.k).k.- x.k2 = l-(x o k) c k - é-x 0 k2 E K by

2
k
Proposition 2.2.1. Hence K is also an inner ideal and therefore

an ideal. Also it is clear that K is an ideal of J if and only

if K is a submodule of J and x.k E K for all x E J, k E K.

25

It is also possible to state the concepts of inverse and

quasi-inverse in terms of ° . Suppose x E J is invertible with

l
inverse y. By Proposition 2.3.1, x.y = %-x ° y = — 2 = 1 and

2
2 l 2 l . .
x .y = E-x 0 y = E'Zx = X. Conversely, if for X E J there eXiSts
2
y such that x.y = l and x .y = x, then we Show first that
X2.y = l and x.y = y. Since %’E Q, the linearized form of the

Jordan identity holds; i.e., for a,b,c,d, E J, [(b.c).a].d

+ [(c.d).a].b + [(d.b).a].c = (a.b).(c.d) + (c.b).(a.d) + (d.b).(a.c).

X and a

2 2
(x .y ).y = y. Hence ux(y> = 2R§<y> — R 2(y)

X
2
y)

2
By taking c = d b = y, we get x .y2 = 1. Then

2 2 2

x.y = (x -y)-y
2 2 2 2
= 2(y.x).x - y.x = 2x - X = x, and Ux(y ) = 2Rx(y ) - R 2(
2 2 2 .X. .

= 2(y .x).X - y .X = 2y.x - l = 2 - l = l, and so X is invertible
with inverse y. By a straightforward computation it can be Shown
that x E J is quasi-invertible with quasi-inverse y if and only

if X +-y - x.y = O and x2.y - X.(x.y) = O.

. _ _ . 0 _ l _
By QJ3 With y — l, UXVx — Vxe° Thus Since x — l, x — X,
2 2 _ 2 2
X = Ux(1) = 2Rx(1) - R 2(1) — 2X.x - x , so that X = x.X, we have
n+1 n-l l X n-2 l n-2 n
X = UX(X ) = E'UXVX(X ) = E'VXUX(X ) = X .x by induction.

Hence when ‘% E Q, powers can be defined in terms of -, and

n m l n m n+m , , , _ ,
X .x = E'X ° X = x implies that (J,',l) is power-assoc1ative.

We finish this section with a special result for linear

(n)

Jordan algebras concerning K , n E N.

Proposition 2.4.1. Let J be a linear Jordan algebra (%-E Q)

and K an ideal of J. Then K(n+1) = (K(n))'3, n.E N, where
A'3 a A.(A.A).
Proof. Since K<n+1> = U (n)(K(n)), the result will follow by

induction, if UK(K) = K° . If k1,k2 E K1 then

26

2 2 .3
Uk1(k2) 2Rk1(k2) - Rk2(k2) — 2(k2.k1).k1 - k2.k1 e K . Con-

l t
versely, suppose k1,k2,k3 E K. Then by QJ14 , 4(k1.k2).k3
=(kok)°k =U (k)+U (k)=U (k)-U (k)
l 2 3 k1,k;.3 2 k2,k3 l k1+k3 2 k1 2

-U (k)+U (k)-U (k)-U (k)EU(K).
R3 2 k2+k3 1 k2 1 k3 1 K

27

2.5 Constructions gf Jordan Algebras

 

 

I. Let m be an associative algebra with 1 over the com-
mutative associative ring Q> with identity. Define U : m 4 End m
by x a Ux’ where Ux(y) = xyX, y E N. Then U is a quadratic
mapping and m+ = (M,U,l) is a quadratic Jordan algebra (see [7]).
The associated bilinear map is determined by Ux,y(z) = xzy + yzx.

1

1 —l =— =1 =1
If 262.318“ X-y-zxoy 2VXO') 2Ux,1(y) 2(XY+YX)-

1
Conversely, if m is an associative algebra and 2 E Q, then m

Nlp—I

together with - defined by x.y = (xy + yx) is a linear Jordan
algebra.

Clearly every ideal of U is also an ideal of m+. The
following theorem due to Herstein has an interesting consequence.
Let (x) denote the associative ideal of m generated by X.
Theorem 2.5.1 (Herstein). Let u be an associative algebra (with
1) and 0 ¥ K an ideal of M+U Then either (1) b2 = 0 for every
b E K, and if b # 0, then (b) ¢ 0 but (b)3 = O, or (2) there exists
b e K, b2 # 0, and o # (b2) : K.
Proof. First note that X2 = xlx = UX(1) for all x E J and
x 0y = Ux,y(1) = xy + yx, for all x,y E m. Secondly, note that
bxby and szy E K for all b E K, x,y E m, since bxby = (bx)(by)
+ (by)(Xb) - b(yx)b = Ub,by(x) - Ub(yx), and szy = (Xb)(by)
+ (bX)(by) + (by)(xb) + (by)(bX) - bxby - b(yX)b - bybx = (x ° b) ° by
- Ub(yx) - bxby - bbe.

(1) Assume b2 = O for every b E K. Then for all a,c E K,
ac + ca = a 0c = Ua,c(1) = Ua+c(1) - Ua(l) - UC(l) = (a+c)2 - a2 - c2

= 0, so ac = -ca. Now, if b # 0, then b E (b) = mbm # O. For

28

2
x,y E m, bxbyb = (bxby)b = -b(bxby) =-b be = 0, so
(b)3 = o.
2 2 2 2
(2) If b E K and b # 0, then b E (b ), so 0 # (b )

= szfl C K.

Definition 2.5.1. A quadratic Jordan algebra (J,U,1) is simple, if

J # O and the only ideals of J are 0 and J.

We now derive a corollary to Herstein's theorem which pro-
vides many examples of simple Jordan algebras. Recall that an
associative algebra m is simple, if m2 ¢ 0, and O and m are
the only ideals of m. Since m2 is an ideal of m, m2 = U when

m is Simple.

Corollary 2.5.1. m is a simple associative algebra with identity
if and only if m+ is a Simple quadratic Jordan algebra.

Proof. If m+ is simple, then m is simple since every ideal

of u is an ideal of U+. Hence assume m is simple. Then

m+l# 0, so let K be a nonzero ideal of U+. If b E K and

b2 ¥ 0, then 0 # (b2) C K C m. Since U is simple (b2) = m and

hence K = fl+. 0n the other hand, if b2 = 0 for all b E K,

then (b) # 0, so (b) = M. But by Theorem 2.5.1, 0 = (b)3 = m3,
and thus m2 = O, a contradiction. Thus the only nonzero ideal of

U+D is m+' itself, and W+ is simple.

Proposition 2.5.1. x E m is invertible with inverse y if and

. . . . . + . . . .
only if x is invertible in m With inverse y. x is quaSi-
invertible in m with quasi-inverse y if and only if x is quasi-
invertible in m+' with quasi-inverse y. Let x’n denote powers

.n n .
of X in m+. Then x = X“, where x denotes powers in m.

29

2 2
Proof. If xy = 1 = yx, then Ux(y) = xyx = x and Ux(y ) = xy x
2
= (xy)(yx) = l = 1. Conversely, if Ux(y) = x and Ux(y2) = l,
2 2
then x(y x) = l = (xy )x, so x is invertible in m, and

2 2 -1
y X = xy = X . Since X = Ux(y) = xyx, and x invertible,

-1 -1 -1 -1_ -1
y = x xyxx = XX — X .

x If X is quasi-invertible in m,
. . . . . . . +
then 1 - x is invertible in m and hence invertible in U .

+
Thus X is quasi-invertible in U and conversely. If x has

3

quasi-inverse y in m, then 1 - x and l - y are inverses in
+ .

m , and conversely. Thus the second statement is clear.

Clearly x.1 = X1 and x"2 = Ux(l) = XlX = x2, so for

n > 2, xon = Ux(Xn-2) = xxn-ZX = X“.

It can be shown that the nil radicals and Jacobson radicals
of m and fl+ coincide. Since only the result concerning the
Jacobson radicals is needed for this thesis, it is the only one
which will be proved. However, the proofs in the two cases are

nearly identical, both results being corollaries to Herstein's

Theorem (2.5.1).

Corollary 2.5.2. Let U be an associative algebra (with l), and
let EKM) be the Jacobson radical of m. Then m is (Jacobson)
+

semi-simple if and only if m is (Jacobson) semi-simple.

Proof. By Proposition 2.5.1, EKU) is a quasi-invertible ideal of
+ + + . , .

m so that EKM) C EKm ). Now suppose m is semi-Simple. Then

Q(y) C Ralf.) = 0 implies E021) = 0; i.e., 91 is semi-simple.
Conversely, assume m is semi-simple. If x2 = 0 for all

X E EKm+), then (x)3 = O, and (X) C EKm) = 0. Thus X = O and

m+ is semi-simple. If there exists x E EKm+) such that X2 # 0,

30

then 0 # (X2) C EKU+). But then (X2) is a nonzero quasi-invertible

+
ideal of m, a contradiction. Thus U is semi-Simple.

.8

Corollary 2.5.3. emu) = EKm+), m as in Corollary 2.5.2.
Proof. Since EKm) C EKM+), it remains only to Show the reverse
inclusion. let 9 : U a fl/EKM). Then clearly 9 is a Jordan
homomorphism from M+'a (m/EKM))+I with kernel EKM). Hence
9147,90,!) =3: (21/,Q(9,[))+, where the latter is semi-s imple by Corollary

2.5.2. Since EKm+)/Eflu) is a quasi-invertible ideal of m+/EKM),

R’(91+)/R(91) = 0 so nab : mi).

Definition 2.5.2. A quadratic Jordan algebra (J,U,1) is Special,
+

if there exists a (Jordan) monomorphism from J into m , where m

is an associative algebra with identity. Otherwise (J,U,1) is

exceptional.

 

+
Clearly, if m associative, then m is Special.

11. The construction in (I) can be generalized to m an
alternative algebra with identity. Again Ux(y) E xyx, and the

resulting quadratic Jordan algebra is denoted by N+. If E Q,

CD NIp—a

1 +
x.y E §(xy + yx) yields W as a linear Jordan algebra. ince

in this case m+'3:Hom¢ (M,W)+, W+ is clearly Special. Finally, it
is again clear that every ideal of m is also an ideal of m+.

111. Let (u,j) be an associative algebra with identity and
involution j, and let ©(M,j) = {X E m : xj = x] be the set of
j-symmetric elements of (W,j). Since ©(W,j) is clearly a sub-
module of m, 1j = 1, and Ux(y)j = (xyx)j = ijij = xyX = Ux(y)

for all x,y E ©(m,j), ©(M,j) is a subalgebra of M+. If

31

9 : (M,j) * (U,3) is a homomorphism of algebras with involution,

then e‘bCflsj) is a (Jordan) homomorphism from ©(Waj) * ©(fiaJ)-

Now let mo be the opposite algebra of m, E ==m $>m°
and j the exchange involution. Then we have seen that E is
an associative algebra with involution and identity (1,1). Hence
$36.3) = {(a,a) ; a E 9,1] is a Jordan subalgebra of 13+. Since
fl+.a:©06,j) under the mapping a a (a,a), we have a second equi-

valent construction of fl+, W associative.

IV. Let V be a vector Space over a field Q which is
equipped with a symmetric bilinear form f, and let J = Q1 €>V
be the vector Space direct sum of V with the one-dimensional

vector Space Ql with basis {1]. We define a product on J by

(011 + X)-(81 + y) = (018 + f(x,y))l + (Bx +ozy)~

Then if Q does not have characteristic 2, (J,~,l) is a linear

Jordan algebra. Let V‘L = [X E V : f(x,y) = O for all y E V}.

..L

Then f is nondegenerate, if V‘L = O. V is a subSpace of J

 

which is properly contained in J, and if a1 + x E J, y E Vi,
then (0.1 + x).(01 + y) = f(x,y)l + ay = ay 6 vi. Thus vL is
an ideal of J, and so if J is simple, V‘L = O, and f is non-
degenerate. Conversely, if f is nondegenerate and dim V > 1,

then J is simple.

32

2.6 Maximal Ideals

 

Let J be a quadratic Jordan algebra. Since 1 is quasi-
invertible if and only if J = O, we assume throughout this section

that J # 0. Hence 1 E EKJ) so that EKJ) CiJ.

Definition 2.6.1. An ideal M of a quadratic Jordan algebra J

is a maximal ideal, if M ¥ J and M Q N, N an ideal of J,

 

implies N = M or N = J. An ideal P is prime, if A and B
ideals of J such that UB(A) C P implies A C P or B C P

(for equivalent conditions see [11]).

Before considering properties of maximal ideals, note that
K an ideal of J implies K O {x E J : x is invertible] = ¢ or
K = J. For if x E K is invertible, then 1 = UX(x-2) E K so

that y = U1(y) E K for all y E J.

Proposition 2.6.1. Let J be a quadratic Jordan algebra with

Jacobson radical EKJ). Then:

(1) Every maximal ideal of J is prime.

(2) M is a maximal ideal of J if and only if J/M is simple.

(3) Every ideal properly contained in J is contained in a maximal
ideal.

(4) EKJ) is contained in every maximal ideal of J, and if EKJ)
is itself a maximal ideal, then EKJ) is the only maximal
ideal of J.

(5) If M is the unique maximal ideal of J, then every ideal
properly contained in J is contained in M, and conversely.

(6) J/E(J) is a division algebra if and only if 80) = {x e J

x is not invertible]. In this case EKJ) is the unique

33

maximal ideal of J.
Proof. (1) Let M be a maximal ideal of J, and assume A and
B are ideals of J such that AstM and BCM. Then MCA+M
and 'M<: B +-M, so A +“M = B + M = J. Hence J = U1(J) C UJ(J)
C UB+M(A +-M). Let a E A, b E B, and x,y E M. Then Ub+x(a + y)
= Ub,x(a + y) + Ub(a + y) + Ux(a + y) = Ub,x(a + y) + Ub(a) + Ub(y)
+Ux(a +y) E UB(A) +M. Thus J ; UBm(A +M) c UB(A) +M, which
implies that UB(A) C M, Since M # J. Hence M is prime.
(2) K is an ideal of J such that M<: K if and only if K/M
is a nonzero ideal of J/M. Thus the result is immediate.
(3) Let K be an ideal of J Such that K # J. The set of all
ideals containing K and properly contained in J is nonempty,
since this set contains K. Thus Zorn's lemma can be used to imply
the existence of a maximal element M of the set. That is M # J
is an ideal, K C M, and if K C N, N an ideal of J, then N C M
or N = J. But any ideal of J containing M must necessarily
contain K, and so M is actually a maximal ideal of J.
(4) Suppose M is a maximal ideal of J and EKJ) C M. Then
M<: EKJ) +-M, so that EKJ) + M = J. Hence there exists 2 E EKJ)
and X E M Such that z + X = l, and so x = 1 - z is invertible.
This is a contradiction, and so EKJ) C M. If EKJ) is a maximal
ideal, then it is clear that EKJ) is the only maximal ideal of J.
(5) If M is the only maximal ideal of J and K is any ideal
different from J, then K C M by (3).
(6) One direction is clear. Hence assume J/EKJ) is a Jordan
division algebra. If x E EKJ), then Q(x) # 0, where 9 is the

natural homomorphism from J onto J/RKJ). Thus Q(x) is invertible

34

and there exists y E J such that Ux(y) +-EKJ) = 1. Hence there
exists 2 E EKJ) such that Ux(y) = l - 2, which implies that
Ux(y) is invertible. By Proposition 2.3.1, X is invertible.

The maximality of EKJ) follows immediately.

We next turn to the relationship between maximal ideals

of an associative algebra m and the maximal ideals of m+.

Proposition 2.6.2. Let u be an associative algebra with l. M
is a maximal ideal of m if and only if M is a maximal ideal

of M+l If M is the unique maximal ideal of m+, then M is
the unique maximal ideal of N. If M = EKM) is a maximal ideal
of m, then it is the unique maximal ideal of both m and U+.
Proof. Recall that U+YM E:(u/M)+l By Corollary 2.5.1, U/M is
simple if and only if (ELI/M)+ is simple. Thus N/M is simple

if and only if M+YM is Simple and the first result follows from
Proposition 2.6.1 (2). If M is the unique maximal ideal of fl+,
then by Proposition 2.6.1 (5), M contains every ideal of m+
which is properly contained in fl+l In particular maximal ideals
of m are such ideals, so M is the only maximal ideal of m.

If M = EKM) is maximal in m, then M is the unique maximal
ideal of m, since EMU) is contained in every maximal ideal of
m. Since EKM) = EKM+) by Corollary 2.5.3, M is also the unique

maximal ideal of m+l by Proposition 2.6.1 (4).

Proposition 2.6.2 allows one to construct many examples of
quadratic Jordan algebras with unique maximal ideals. Let
91‘ {a/b E Q : b ¢O(mod p), p a fixed odd prime in 2]. Then 91

is a unital algebra over 2 with unique maximal ideal

35

ecu) = [a/b E 9.1 : a E 0(mod p)]. Now 91 itself is a quadratic

Jordan algebra, but we can derive a less trivial one by forming
+ + . . . +

(91“) , n 2 2. Then (91m) has unique max1mal ideal Rain)

= Exam) = Emu)“. Of course any associative algebra m with

identity such that EKW) is a maximal ideal would do as well.

36

2.7. Linear Jordan Algebras over Fields

 

In this final section of Chapter 2, it is assumed that J
is a linear Jordan algebra (with 1) over a field Q of charac-
teristic not 2. None of the proofs will be given for results
whose proofs can be found in [5].

We first consider the associativity of certain finitely
generated subalgebras of J. Recall that J is special, if there
exists a monomorphism o : J u.m+; m an associative algebra.
Whenever this occurs in what follows, J will be identified with
JO and 1 will be assumed to be the identity of M (see [5‘],

p. 10). For the sake of completeness, the fact that J is power-

associative is restated in another form. Note: 13 a Subalgebra

of J implies 1E8.

Proposition 2.7.1. For every x E J, the Subalgebra generated by

{1,x} is associative.

Proposition 2.7.2. Let RJQB) [Rb E Hom§(J,J) : b E.B]. If .8
is a subalgebra of J and X a set of generators of .6 containing

1, then the subalgebra of Hom¢(J,J) generated by RJOS) and

IdJ is generated by {Ux,y : x,y E X}.

Proposition 2.7.3. (Shirshov-Cohn). Any Jordan algebra (with 1)

generated by two elements (and l) is Special.

Proposition 2.7.4. Let K be the subalgebra of J generated by
{1,x,y}, where J is a subalgebra of m+; m associative . If
xy = yx, then K is associative.

Proof. Note that J is a subalgebra of m+, m associative, by

Proposition 2.7.3. We first show that xy = yx implies Rx’R

and R commutes. For
x.y

37

l
Rny(z) = (z.y).x = 2(zyx + yzx +-xzy +-xyz)
1
= Z(zxy +-yzx + xzy + yxz)
= (z.x).y = Rny(z),
and
l 2
Rx ny(z) = (z.x).(x.y) = §(zx y + zxyx +~xzxy +-xzyx +-xyzx
2
+ xyxz + yxzx + yX z)
1 2 2
= §(zxyx + zyx +-xyzx + yxzx +-xzxy + xzyx + x yz + xyxz)
= [z.(x.y)].x = RxRx.y(z) .
Similarl R = , ' = -
)IRx.y y Rny.y Now Since Ua,b RaRb + RbRa Ra.b’ the

set {Ua,b : a,b E {l,x,y]] is a commutative subset of HomQ(J,J).
Since the subalgebra of Hom¢(J,J) generated by RJ(K) is gen-
erated by {Ua,b : a,b E [l,x,y]], this associative Subalgebra is
commutative, call it 6.

Suppose a,b,c E K. Then Rb,
= RCRb(a) = RCRa(b) = RaRc(b) = (b.c).a = a.(b.c). Thus K is

RC E B, and so (a.b).c

associative.

Proposition 2.7.5. Let K be the subalgebra of J generated by
{l,x,y]. Then K is associative, if (i) x is invertible and

y = X- , or (ii) X is quasi-invertible and y is the quasi-
inverse of x.

Proof. Let m be the associative algebra such that J is a

-subalgebra of fl+b By Proposition 2.7.4, it is sufficient to

Show that xy = yx in each case. In case of (i) or (ii), x

38
and y are inverses (quasi-inverses) in m by Proposition 2.5.1.
Thus xy = 1 = yx, if (i) holds or xy = X + y = yx, if (ii) holds.
We next consider idempotents in J.

Definition 2.7.1. Let (J,U,1) be a quadratic Jordan algebra (no

. . . . . 2
restriction on Q). An element e E J is idempotent, if e = e.

 

Two idempotents e and f are orthogonal (e l f), if e 0 f

 

= Ue(f) = Uf(e) = O. A set of orthogonal idempotents
n

{e1,e2,...,en] is supplementary, if ‘21 ei = l.
1:

It is immediate from the definition that e an idempotent
implies en = e for all n E N. Now if %-E Q, then e and f
orthogonal implies e.f = % e o f = 0. Conversely, if e.f = 0,
then 9 o f = 2e.f = 0 and Ue(f) = %-[V:(f) - V 2(f)]
= %((f 0 e) 0 e - f o e) = O; and similarly Uf(e: = 0. Thus
e l f.

If e is an idempotent of J, then Ue(J) is an inner

ideal of J, since x,y 6 J implies UUe(x)(y) = UeUXUe(y)

N

= UeEUer(Y)] E Ue (J). Moreover, if Ua = Ua‘Ue(J)a

a

E Ue(J),
then (Ue(J), U, e) is a quadratic Jordan algebra. Since QJ2-
QJ3 and QJS-QJ8 are inherited by U from U, all that needs to

be shown is that Ua is an endomorphism of Ue(J) and

” = Id .
Ue Ue(J)

BC = =
u UUe(x)[Ue(y)] UeUerUe(Y) Ue(Uer(y))
E Ue(J) and UeUe(X) = Ue(X) proves this is so. Thus if

é-E Q, Ue(J) is a linear Jordan algebra under ° . In partic-

ular, this also implies X2 = Ux(e) for x E Ue(J).

39

Let {e1,e2,...,en] be a supplementary set of idempotents

in J. Then for i ¥ j, Ue ,U e form a set of pairwise orthogonal
i ei’ j
n
. - = + . =

pr0jections and IdJ .2 Ue, ,2,Ue_,e_ If Jii Ue,(J) and
i=1 i i<j i j i

Jij = Ue.,e (J), 1 < 3, then we have J = .2. C>Jij, which is

i j 15]

called the Pierce decomposition of J relative to [e1,e2,...,en].

 

The inner ideals Jii’ i = 1,2,...,n are called the Pierce inner
ideals determined by the ei. AS before we will be interested in the
case where Q is a field of characteristic not 2, and since the
decomposition is a vector Space direct sum, any Jij’ i s j will

be called a Pierce space of the decomposition. We have the follow-

ing characterizations of the Pierce Spaces for this case:

(1) J..

11 [x E J : x.e. = X}

1

(2) J {XEJ :x.e.

l = }
l 2 x x.ej .

ii
Proposition 2.7.6. Let J be a linear Jordan algebra over a field
Q of characteristic not 2, and let J = Z C>Jij be the Pierce
isj
decomposition of J relative to the supplementary set of idempotents

[e1,e2,...,en}. Then:

(i) Jii°Jii ; Jii, i = 1,2,...,n,

(11) JijoJiichj’ 1f lUj)

... . c + .f . .
(iii) Jij Jij Jii Jjj’ 1 1 E J,

(1V) Jiionj = 0: 1f 1 U J:
(v) Jij'ij C Jik’ if i,j,k are distinct,
(vi) J .J = 0, if i,j,k are distinct.

ij kk

4O

(vii) Jij°JkL = 0, if i,j,k,L are distinct.

Proposition 2.7.7. Let J be a linear Jordan algebra with Pierce

decomposition J 2 C>Ji, with reSpect to the Supplementary

isj
set of idempotents {e1,e2,...,en], and let K be an ideal of J.

If e : J a J/K is the natural homomorphism and for S C J we

write 6(3) = 5, then [51,52,...,én] is a Supplementary set of

 

idempotents of J. and JI= Z (3 J], is the Pierce decomposition
isj
of J relative to this set of idempotents.
-2 —_ - - - _
Proof. Since e. = e? = e, and 6,-6. = €.-€. = 0, for i # j,
i i i i j i

{€1,52,...,6n] is a set of orthogonal idempotents of J. This
n n
set is Supplementary, since 2 6, = z e, = 1. If X E (J) .
. i . i ii
i=1 i=1
= U; (J), then § = 5; (F): Y E J. Hence X U (Y) + K E Eli.
i i ei
Conversely, if {c e J.., then a? = U (y) + K 5- (y) e U- (3)
ii e. e e.
i i 1
(J)ii’ y E J. Hence (J)ii = Jii' Now if X E (J)ij

Eéaé (J), i 3‘ j, then )2 = U; ,5 (y)
i j i 3
Thus X E Uei,ej(J) = Jij
X =U . e.(y) +K=U‘.:;
1 J 1 3

U8 ,8 (y) + K, y E J.
1 J
. Conversely, if Q E Eij’ then

(i) e (3)”. Hence (3),]. = 31,"

Definition 2.7.2. Two nonzero orthogonal idempotents ei,e2 E J

u12 E Ue1,82(J) which is

e1+e2(J). e1 and e2 are strongly connected,

if there exists an element u E U (J) such that
12 e ,e2

2 = e + e .

1
u12 1 2

are called connected, if there exists

invertible in U

 

The relation of connectedness is transitive.

41

Proposition 2.7.8. Let e1,82,e3 be nonzero orthogonal idempotents
such that e1 and e2 are connected (strongly connected) and e2
and e are connected (strongly connected). Then e and e

3

are connected (strongly connected).

1 3

The usefulness of connected idempotents will become apparent
after Jordan matrix algebras have been defined. Let .8 be an
algebra over a field Q of characteristic not 2 with identity
element 1 and involution j : d a dj = 3. Let '&n be the algebra
of n X n matrices with entries in .8. let 31 E N(.8) (1 $68),

i = 1,2,...,n (NED) is the nucleus of .D and 33(8) = {d E .8:

ll

8 d]), and assume ai is invertible in N68). If

a = diag [a1,az,...,an], then we define an involution on 'fih

at
called a canonical involution by : X ~ a X a, x E fig, where

 

ja
at
X is the conjugate tranSpose of X under j. If a = In’ then
the involution is called standard. Let ©(8n,ja) denote the set
+
of symmetric elements of '3h° If 5%. is the algebra which has
the same underlying vector Space structure as 8%, but multiplica-
tion defined by X.Y = %(XY +-YX), then $(85,ja) is a subalgebra
+

of ‘fih° We W111 write egbh) for $(8h,j1n).

The following notation will be used for elements of 8%

and Q(Dh,ja) wherever these algebras are encountered. Let

eij’ i,j = 1,2,...,n be the element of 8%. with l in the (i,j)
n
position and 0 elsewhere, and if x E.B, identify x with 2 x eii
i=1

= diag {X,X,...,X}. Then xeij is the matrix with x in the
(i,j) position and O elsewhere, and for x E .8 we put

3' -
(JMAO) x[ij] = xeij + (xei.) a = xe,. + (ajlfi ai)e

J l] i,j = 1,2,...,n.

Ji’

42

3
Since the characteristic of Q is not 2, awn’ja) = {X + X a :

X E .Dn], and hence every element of Q(fin’ja) is a Sum of elements

x[ij], x 6.8, i,j = 1,2,...,n. If fiij = [x[ij] : X E.8], then

n
53” = bji and Q(fin’ja) = is§=1 E) bij' The follOWing multiplica-
tion rules hold in ©C8n,ja):

(JMAl) 2 x[ij].y[jk] xy[ik], if i,j,k are distinct;

(m2) 2 mil-yin] (xy + (a;1 s ai>y>[111. if i e j;

(JMA3) 2 x[ij].y[ji] xy[ii] + yx[jj], if i # j;

(JMA4) 2 x[ii].y[ii] (x + a;1 52 ai).(y + ail 3‘: ai)[ii].

.. -1 x . ..
Also we have x[ij] = (a1 x ai)[ji], and x[ij].y[k{,] = 0,
if {i,j} O {k,{,] = <25:
The next proposition determines when bwn’ja) is a

(linear) Jordan algebra for n 2 3.

Proposition 2.7.9. awn’ja) for n 2 3 is Jordan if and only
if (.8,j) is associative or n = 3 and (.D,j) is alternative

with symmetric elements in the nucleus.

For n = 2, Swn’ja) is clearly Jordan if (.8,j) is
associative (see Section 2.5), and for n = 1, nothing more can
be said than to state the Jordan conditions for S)(.8n,ja). In
any case, whenever @wn’ja) is Jordan, Q(fin’ja) is called a

Jordan matrix algebra of order n.

Proposition 2.7.10. Let b = Q(finda) be a Jordan matrix algebra
of order n 2 3 defined by the canonical involution ja in 8n
such that a1 = 1. Then the mapping /3 -' 5n n S; is a lattice

isomorphism of the lattice of subalgebras [3 of (.D,j) containing

43

a a.1 '

= 1,2,...,n onto the lattice of subalgebras of b
containing the elements l[i,j], i,j = 1,2,...,n. Also the mapping
[.3 «an O b is a lattice isomorphism of the lattice of ideals B
of C8,j) onto the lattice of ideals of b, and

B = {d E .8: d[ij] E 8n (1.2; for every (i,j) E [1,2,...,n]

X [l,2,...,n]]. In either the ideal case or Subalgebra case,.6

is characterized by .6 = {d E.8 : d is an entry of a matrix in

an n 3;}.

Note that there is no loss of generality in assuming

a1 = l in the preceeding prOposition (see [5], p. 128).

Proposition 2.7.11. Let ©C8n,ja) and ©(6n,kb) be Jordan matrix
algebras of order n 2 3 determined by canonical involutions ja
and kb reSpectively where the first is defined by an involution
j in .8 and a diagonal matrix a with a1 = l and the second
by an involution ltzhics and a diagonal matrix b with b1 = 1.

If n is a homomorphism of E8,j) into (6,k) such that
a? = bi’ i = 1,...,n, then the restriction o to ©(8h,ja) of
the mapping (dij) a (dyj) of 8%. is a homomorphism of @cfih’ja)
into 53(55ka such that 1[ij]° = 1{ij}, where d[ij] and
e[ij} are defined by (JMAO) in ©E8n,ja) and ©(6n,kb)
reSpectively.

Conversely, if c is a homomorphism of ©E8n,ja) into
9(6n,kb) such that 1[ij]O = 1{ij}, i,j = 1,...,n, then there
exists a homomorphism n of C8,j) into (6,k) such that
a? = bi, i = 1,2,...,n and a is the restriction to swnga)

. a N
of the mapping (dij) (dij) of Inf

44

Corollary 2.7.1. Let 9 = ©63h’ja) be a Jordan matrix algebra

of order n 2 3 determined by a canonical involution ja with

a1 = 1, and let K = 5n n D be an ideal of b. Then b/K
saga/6)“, ~15), where (,8/6,j) is the image of (.8,j) under the
natural homomorphism H and 5 = diag {a¥,a2,...,a2].

Proof. By Proposition 2.7.11, there exists a homomorphism o
from ©(8h,ja) into b((8/6)n,ja) which is the restriction of

the homomorphism (di ) a (dgj)' Since n onto implies that (dij)

.1
—0 d1), . t , ' , ° —9 a

( 1]) 13 on 0 o is clearly onto Let 8 ..8h fib/an (.8/6)n
be the natural map. Then ker o = ker 9 O b = 6n O 9 = K, and

hence J/K e: 83((8/6)n,ja)-

Proposition 2.7.12. (Coorinatization Theorem). let J be a linear
Jordan algebra over a field Q of characterization not 2, let
[e1,e2,...,en] be a supplementary set of nonzero idempotents of

J, n 2 3, and let J = 2 <3 Jij be the correSponding Pierce
isj
decomposition. If for j = 2,3,...,n there exists ulj E J1j

which is invertible in J + J, +-J , then there exists a
11 Jj 13
Jordan matrix algebra bcbn’ja) and an isomorphism g of J
C _ 1 . C _ .
onto ©(8n,ja) such that ei - §[ll] and ulj — l[lj],

. . 2 _ .
j = 2,3,...,n. If in addition ulj - 81 + ej, j = 2,3,...,n,

then the involution ja is standard and J 259E8n).

Since Supplementary connected idempotents satisfy the
first hypothesis and supplementary strongly connected idempotents
the latter in addition, it follows immediately that if J has a
Supplementary set of connected (strongly connected) idempotents

{e1,e2,...,en], n 2 3, then J is isomorphic to a Jordan matrix

45

algebra 9(8h,ja) (963m))-

Lamma 2.7.1. let (J,U,1) be a quadratic Jordan algebra and
e # 0 an idempotent in J. If K = Ue(J), then EKJ) O K C KKK).

Proof. If Ue(x) E EKJ) O K, then for any Ue(Y) E K, U (Ue(Y))

Ue(X)
= UeUXUe(y) E EKJ) O K, and UUe(y)(Ue(X)) = UeUyUe(X) E EKJ) O K,
Since EKJ) is an ideal and K = Ue(J). Thus EKJ) n K is an
ideal of K. Now Ue(x) E EKJ) 0 K implies Ue(x) is quasi-

invertible in J with quasi-inverse y. Then y = U1_y[(Ue(X))2

- Ue(x)] = U1_yUe(Ux(e) - x). since (U800)2 = ”u (x,(e> = UeUXUe<e>
= UeUx(€)o Thus Ue(y) = UeU1_yUe(Ux(e) - x) = UeU1_yU:(Ux(e) - x)

(UeUx(e) - Ue(x)) = U [(Ue(x))2 - Ue(x)]. There-

= U
Ue(1'Y) e-Ue(y)
fore Ue(x) has quasi-inverse Ue(y) E K, and EKJ) m K is a

quasi-invertible ideal of K. Thus EKJ) O K C EKK).

The next proposition plays a fundamental role in obtaining

the results of Chapter 4.

Proposition 2.7.13. let J be a linear Jordan algebra, K a quasi-
invertible ideal of J such that 3.5 J/K.2:©(6n,jb), a Jordan
matrix algebra of order n 2 3 determined by a canonical involu-
tion, and let {51,52,...,5n] be a supplementary set of connected
idempotents such that éi ~ %[ii] under the isomorphism of J

onto ©(§H,jb) for i = 1,2,...,n. If J contains a supplementary
set of connected idempotents [f1,f2,...,fn] satisfying fi = 51
for i = 1,2,...,n, then

(1) J 3 gwn’ja)’ a Jordan matrix algebra,

(2) (6,1) as (.8/3‘,j), where a is the ideal of .5 such that

K a 3n O ©(8h,ja) under the isomorphism of J onto

46

agenda), and b = diag {a1 +3, a2 +a,..., an +3}.
Proof. Without loss of generality we may let ei = fi’ i = 1,2,...,n.
Thus, if.J = 2 (O'Ji' is the Pierce decomposition of J with
isj
reSpect to the set {e1,e2,...,en], then by Proposition 2.7.6,

3': 2 C)J;, is the Pierce decomposition of J. with reSpect
iSj

to {él,é2,...,gn]. Let T be the given isomorphism of J. onto

©(6n,jb). Then T(éi) = % [ii] (as defined in (JMAO)), i = 1,2,...,n.

lj E Jlj such that

Also for every j = 2,3,...,n, there exists u

—1j is invertible in Ell + Elj + 3jj

£1 + éj' Then there exists v E J such

.1
A
c:

...:
V
II
H I
r-‘\
f—J
“I
W
m
:3
D.
C

I
C
I

$’

A
“I
v
E'
I"?
(D

II

that Uu (v) = e + z, z E EKJ) (since K C EKJ)). Hence by
11

. . . . . . + =
Lemma 2 3 l, Uu1j(v) is invertible in J11 +Jlj Jjj Ue(J),

d o o o a + + 0
an so ulj is invertible in J11 J1j Jjj’ by Proposition
2.3.1. For j = 2,3,...,n, let uj1 be the inverse of ulj in

J +J +J... Then r5, =i'1.
JJ ( J1) l3 1

ll lj
Now by hypothesis the ei are connected and supplementary,
so by Proposition 2.7.12, there exists a Jordan matrix algebra

$(8n,ja) and an isomorphism g of J onto ©(8h,ja) Such that

g(ei) ='% [ii], i = 1,2,...,n, and Q(ulj) = l[lj], j 2,3,...,n.

let v : J a J. be the natural homomorphism. Then o T o v o g-

is a homorphism of QCDn’ja) onto ©(6n,jb) such that

(i) O(%[ii]) T 0 V o g-1(%[ii]) T 0 V(ei) = T(ai) = %lii}’

(ii) 0(1[11]> T . v . g'1(1[111> = r . v<u1j> = r(fi1j) = Tilj}.

(iii) o(l[jl]) i{jl}, as in (ii).

47

Hence by Proposition 2.7.11, there exists a homomorphism
n of C8,j) into (6,j) satisfying “(ai) = bi’ i = 1,2,...,n
and g is the restriction to ©C8h,ja) of (dij) a (dgj). Since
0 is onto, n is onto, and (6,j) a (8,j)/ker n. Let 3 = ker fi.
Hence Q(K) = ker o = (ker H)“ O QCfih’ja) = ?n O 9(8h,ja), Since

0 is the restriction of the homomorphism (dij) a (dgj)'

We next identify the Jacobson radical of a Jordan matrix

algebra for n 2 3, and .8 associative.

Proposition 2.7.14. Let E8,j) be an associative algebra with
involution and identity over a field Q of characteristic not 2.
Then a = ©(8h,ja), ja a canonical involution and n 2 3, satisfies
as) = m9)n n s.

Proof. By Proposition 2.7.10, there exists an ideal 6 of E8,j)
such that EKb) = 6n O 9. Now .8 associative with 1 implies

19(3)“ = Ewn)’ and hence emu O 1?) = Ewn) O S) is a quasi-invert-

ible ideal of S). Thus ,€(.8)n O b C E63). Suppose x E 6. Then
n
2
x[12] E 6n O b = EKQ). Also u = l[12] + Z eii E b and u = I ,

i=3 n

so u is invertible in .8H and hence in 9. Since x[12] is

an element of the quasi-invertible ideal EKb), u - x[12]

= (l - x)[12] +.;3eii is invertible in 9. One verifies by
direct computati6E that the inverse of u - x[12] has the form
(1 ’ Y)[21] +.;38ii’ where 1 ~ y = (1 - X)-1. Therefore x
is quasi-inverfible in .8 and 6 is a quasi-invertible ideal.

Hence 6 : 8(8) and 9(3)) = 6n n S.) : PCB)In n 3;.

Finally, we consider linear Jordan algebras which satisfy

minimum conditions on inner ideals.

48

Definition 2.7.3. Let J be a linear Jordan algebra over a field
Q of characteristic not 2. An inner ideal B of J is called

a minimal inner ideal, if B # 0 and C an inner ideal of J
such that O C C C B implies C = 0 or C = B. J satisfies

the minimum condition (for inner ideals), if (i) there exist no
properly decreasing sequence Ue (J):D Ue (J):D ..., e = e , and

1 2 i i
(ii) every inner ideal U (J), e2 = e # 0, contains a minimal
e

inner ideal.

Jordan algebras will now be considered which satisfy the

following axioms:

(i) J has an identity;

(ii) J is nondegenerate; i.e., J has no nonzero absolute zero
divisors;

(iii) J satisfies the minimum condition .

Recall that J/EKJ) is nondegenerate, and therefore, if
J is a linear Jordan algebra such that J/EKJ) satisfies the
minimum condition , then J/EKJ) satisfies the axioms (i)-(iii).

This case becomes of prime importance in Chapter 4.

Definition 2.7.4. Let J be a linear Jordan algebra. An idempotent

e E J is primitive, if e # 0 and e cannot be written as

e = e1 + e2, where e1 and e2 are nonzero orthogonal idempotents.

An idempotent e E J is completely ppimitive, if e # 0 and

Ue(J) is a division algebra. J has (finite) capacity n, if

J contains a supplementary set {e1,e2,...,en} of completely primitive

orthogonal idempotents and n is minimal with reSpect to this

property.

49

Note that a nonzero idempotent e is primitive if and
only if Ue(J) contains no idempotents # 0, e, and J has

capacity 1 if and only if J is a division algebra.

Proposition 2.7.15. Any Jordan algebra J satisfying axioms (i)-

(iii) has a finite capacity.

Proposition 2.7.16. (Second Structure Theorem). The following
conditions on a Jordan algebra J are equivalent: (1) J is a
simple algebra satisfying axioms (1)-(iii); (2) J is either a
division algebra, a Jordan algebra of a nondegenerate symmetric
bilinear form in a vector Space V over an extension field F

with dim'V/F > 1 (only if the capacity of J = 2), a Jordan
matrix algebra bcbh’ja) where n 2 2 and (8,j) is either

A 0A0 with A an associative division algebra and j the
exchange involution, an associative division algebra with involu-
tion, a split quaternion algebra over an extension field (n = 2
only), an algebra of octonians over an extension field with standard
involution (only if n = 3); (3) J is either a division algebra,
a Jordan algebra of a nondegenerate synuetric bilinear form in

a vector Space V over an extension field F with dim V/F > 1,

a Jordan matrix algebra ©(83,jy) where .8 is an octonian algebra
over an extension field with standard involution and j is a
canonical involution, or an algebra $(fl,j) where (m,j) is

simple Artinian with involution.

CHAPTER 3

COMPLETIONS OF QUADRATIC JORDAN AIGEBRAS

3.1 Topological Modules

 

Let Q be a commutative associative ring with identity

and J a unital Q-module. If J = KO 2 K 2 K 2

1 2 ...

decreasing sequence of Submodules of J, then it is well-known

is any

that the set {x + Ki : i = 0,1,2,... and x E J] forms a basis

for a topology on J. If S is any subset of J, then

so

c1.J S = O (S + Ki) so that S is closed if and only if

1:0

8:
i

"38

(S +-Ki). In particular every Open subset of J is closed.
0

Clearly the topology is l0 countable, and it can be Shown that

so
the topology is Hausdorff if and only if O Ki = 0 (see [12]).
i=0
Let I be the topology induced by a decreasing sequence of
co
submodules of J as above, and assume O Ki = 0. Since X,y E J
an i=0
are distinct if and only if x-y E O Ki’ X # y implies there
i=0
exists k E {0,1,2,...} such that x - y E K —-K Let e

k k+1'
be any fixed real number greater than 1, and define d : J X J a R

-k
by (i) d(x,y) = 0, if X = y, (ii) d(x,y) = e , if X # y and
x - y E Kk _'Kk+l' Then it is well-known that (J,d) is a metric
space and f(d) = I. We now establish some useful inequalities

for d.

50

51

Proposition 3.1.1. (i) For every x,y,z E J, d(x,z)
s max [d(x,y), d(y,z)].
(ii) For every x,w,y,z E J, d(x +vy, w + z)

s max [d(x,w), d(y,z)].
(iii) For every x,y E J and o E Q, d(ax, ay) s d(x,y).
Proof. (1) If x = z, then d(x,z) = O s d(x,y) = d(z,y) = d(y,z).
If x = y, then d(x,z) = d(y,z) and d(x,y) = O s d(y,z).

Similarly if y = z, and (1) holds for these special cases. Now
-k -k

assume x,y,z are distinct. Then d(x,y) = e 1, d(y,z) = e 2,
-k
_ 3
and d(x,z) - e , where X - y E Kk Kk +1, y — 2 E Kk Kk +1
1 1 2 2
and X - z E Kk- Kk +1. Thus x - z = (x - y) + (y - z)
3 3
C .
E Kmin{k ,k 1, so that Kk Kmin[k ,k } Hence R3 2 k1 or
1 2 3 _ 1 2 -k -k
k3 2 k2 and thus d(x,z) = e 3 s max [e 1,e 2]

= max {d(x,Y)s d(y,z)}:
(ii) If x = w, then (x +-y ) - (w + z) = y - z and so
d(x +-y, w + z) = d(y,z) s max [d(x,w), d(y,z)]. Similarly, if

y = z, the result holds. Hence assume x ¢ w and y # 2. Then
-k -k

d(xsw) = e 1, d(y,Z) = e 2, where x - w E K -— K and
k k +1
1 l
y - z E Kkfi— Kk2+1° So (x + y) - (w + z) = (x - w) + (y -z)
E Kmin{k1,k2] and the result follows as in (1).

(iii) If x = y, then d(ax,ay) = 0 = d(x,y). If x # y, then

d(x,y) = e-k, where x - y E Kk- K Hence ox - HY = Q(x ' Y)

k+l°
-k
E Kk’ and d(ozxm') S e = d(x,y).

From Preposition 3.1.1 (ii), it follows easily that the

mapping (x,y) a x +'y from J X J 4 J is (uniformly) continuous.

52

The mapping (a,X) » ax from Q X J a J is also continuous when
Q has a suitable topology: in particular, if Q is given one

of the trivial topologies. Since we are primarily concerned with
a ring theoretic structure, the continuity of the module operation
plays no role. Thus a module with a continuous addition will be

called a topological module in this thesis.

 

The notions of sequence, Cauchy sequence, series, and con-
vergence will be taken to be the uSual ones associated with metric
spaces. Since f(d) = I it is convenient to have these ideas

stated in terms of the topology T.

Proposition 3.1.2. Let J = KO 2 K1 2 K2 2... be a decreasing

00
sequence of submodules of J such that O Ki = 0. Then:
i=0
(1) If {Xn] is a sequence in J, then 1im xn = x if and only
n

if for every nonnegative integer k there exists N E N
Such that n 2 N implies xn E x (mod Kk)°

(2) {Xn} is a Cauchy sequence if and only if for every
k E [0,1,2,...] there exists N E,N such that n,m 2 N
' ‘ E d .
implies xn Xm (mo Kk)

(3) If every Cauchy sequence in J converges in J, then
m n
2 xi E lim ( 2 xi) converges if and only if 1im xi = 0.

i=0 n i=0 i

Proof. (1) lim Xn = x if and only if for every 6 > 0 there
exists N E N nsuch that if n 2 N, then d(xn,x) < 3. Suppose

1im xn = x and let k be given. Choose a > 0 so that e S e-k.
TUen there exists N E,N Such that if n 2 N, then d(xn,x) < e,

which implies d(Xn,X) < e-k. Hence for n 2 N, xn E x (mod Kk).

Conversely, let 6 > 0 be given. Then for some k, e-k < e-

53

Hence for this k there exists N E N so that xn E x (mod Kk),

if n 2 N; i.e., if n 2 N, d(xn,x) S e-k'< 6- Hence lim xn = X.
n
(2) Proof is similar to (l).

(3) All that remains to be shown is that lim xi = 0 implies

an i

2 xi converges. This will be accomplished by Showing that the
i=0

sequence of partial Sums is a Cauchy sequence. Thus let k be

any nonnegative integer. Since limxi = 0, there exists N E N

III

1
such that i 2 N implies xi 0 (mod Kk). Thus for n 2 m 2 N,

n m n

x. - Z X.= 2 x. E 0 (mod K ), and therefore the sequence of
. 1 . i. i k
i=0 i=0 i=m+l

partial Sums is Cauchy by (2).

Using the standard metric Space arguments, there exists
* *
a metric Space (J ,d ) which is the completion of (J,d) in
* * a
that (i) (J ,d ) is a complete metric Space, (ii) d coincides

*
with d on J X J, (iii) CL *J = J ; i.e., J is a dense Subset
J

* a *
of J , and (iv) (J ,d ) is unique to within isometry. Addition
* * a
and scalar multiplication are defined on J by x + y =
o * o o *
= lim (xn + yn) and ax = lim ax“, where o E Q, 11m xn = x ,

n n n

*
lim yn = y and [Xn], [yn] are sequences on J. Of course

*
every element of J is the limit of a sequence on J, and more-
* * *
over, if we let S = CL *S for S C J, then x E S if and
J

only if there exists a sequence {xn] in S Such that lim Xn =

ll

*
We now establish some important properties of J

* *
Proposition 3.1.3. Let (J ,d ) be the completion of (J,d)
as above. Then:

* *
(1) If l;m Xn = X , lgm yn = y , {xn], [yn] sequences on J,

54

* 71‘ ‘k
then d (X ,y ) = lim d(Xn,yn).

* n
(2) J is a unital Q-module.

* * * *
(3) J = KO 2 K1 2 K2 2... is a decreasing sequence of submodules

* a *
of J with O Ki = O and thus induces a metric p on J

1:0

*

*
(4) f(d ) = 7(9)
(5) There is a one-to-one correSpondence between the open Sub-
*
modules A of J and the open Submodules of J given
* * *
by A e A , and A = J O A . In particular, Ki = J O Ki
for i = 0,1,2,...
* *
(6) If A is any open submodule of J, then J/A=3 J /A as
Q-modules.
*
Proof. (1) Since 1im xn = x and lim yn = y , for every
n n

* *
s > 0, there exists N EEN Such that n 2 N implies d (xn,x ) < 3/2

* *
and d (yn,y ) < 3/2. Thus for every n 2 N,
* * * * * * * * * * *
d(xn.yn) - d (X ,y ) = d (Xn.yn) - d (x ,y ) s d (Xn,x ) + d (x .yn)
* * *
' d (X 3y )
a * a * * a * * * *
sd (xn,X)+d (X .y)+d (y .yn) 'd (X .y)
< 6/2 + 6/2 = 6:
and
d* d* * x d* d* * d* *
(xnsyn) ‘ (X sy ) 2 (anyn) ' (X syn) ' (yn’y )
* * x * * *
2 d (anyn) ' d (X :Xn) ' d (anyn) ‘ d (ynry )
> -e/2 - 6/2 = 'e .

* * a * s *
Therefore, [d(xn,yn) - d (X ,y }] < e, and lim d(xn,yn) = d (X ,y ).
n
* *
(2) It will be Shown first that the definitions X + y = lim (Xn +-yn)
n

55

*
and ax = lim (axn), where a E Q and {Xn], {yn] are sequences

n * * * *
in J such that lim xn = X E J , 1im yn = y E J , are independent
n * * n *
of representations of x , y . First Suppose x = lim xn,
n
* . 3 * . * . y . .
X = 11m Km, and y = lim yn, y = lim yn. Then by Prop081tion
n n n

8 I U I
3.1.1, d(xn + yn, xn +-yn) s max [d(xn,xn), d(yn,yn)] and
d(axn,ox;) S d(Xn,X;), for every n E;N. But it is clear that if
[Wu] is a sequence in J and {wé} is a sequence in J, then

* *
lim Wn - w , lim w; = w if and only if for every 6 > 0 there
n n
exists N E,N so that d(wn,w;) < e, for all n 2 N. Hence
. _ . g 3 . = . I
lim (xn + yn) — 11m (xn +-yn) and lim (axn) lim (axn) follows
n n n n
immediately from the above inequalities. Note that the fact that
[xn + yn], and {axn} are Cauchy sequences also follows from these
inequalities. Thus the operations are well-defined and clearly
*

closed. That J is an Abelian semigroup follows easily from the

corresponding properties in J. Also it is clear that the constant

*
seQuence [0} is an identity Since X +'0 = lim (xn + 0) =

* * . . 2
= lim Xn = x . Finally -X = 11!" (‘anslnce x + 1im (-xn)
n n n
= lim (xn + (-xn)) = lim 0 = 0, and d(-Xn,-Xm) = d(xn,xm) so

n n
*
that {-xn] is Cauchy. Thus (J,+) is an Abelian group. Now,
if 01,3 E Q, we have
. * * . , * *
i) a(x +~y ) = lim [n(xn + yn)] = 11m (axn +-ayn) = dx +-oy )
n n

11> (e + w" = lim (o + 8)er = lim (can + ex“) = dx* + ex",

n n

iii) (a8)x* = lim (oB)xn = lim d(BXn) = Q(BX*),
- *—-n1 -1'n-*

iv) 1X — l;m ( Xn) — :m xn — x ,

*
Since J is a unital Q-module. Hence J is a unital Q-module.

* * * * .
2..., J = KO 2 K 3..., and Ki is a

(3) Since J = KO 2 K 1 _

l
* * m *
Q-submodule of J exactly as in (2). Hence let x E n Ki'
i=0
* * * .
Then for each i = 0,1,2,..., x E Ki’ and since K1 is closed,
*
there is a sequence (x, } on K. such that lim x, = x . Now
in i n in

let 6 > 0 be arbitrary, and choose i so that e.1 < 6/2.

. * __ -i
Since xin E Ki’ for every n E,N, d (Xin’o) d(xin’o) < e < 6/2.

*
Also since lim X'n = x , there exists N E N Such that n 2 N
i

n
. . * * , . * *
implies d (x ’xin) < 6/2. Thus, in particular d (x ,0)

* * * * *
Sd (x XiN) +d (XiN,O) < e/2 + 9/2 = g, and SO (1 (x ,0) = 0.

*
Hence x = 0.

The second statement about the existence of p follows
immediately from the definition of the metric determined by the

* * * *
Submodules J 2 KO 2 K1 2 K2 2... .

*
(4) To show T(d ) = 3(p), it is sufficient to show that the two
* *
metrics are equivalent. If e > 0 is given and x = y , then
* * * * * * * *
d (x ,y ) = O = p(x ,y ), and d (x ,y ) < e if and only if

* * * *
Q(x ,y ) < 6- Hence we may assume x # y . First assume

* * * * -k * * * *
p(x ,y ) < 5/2. Then p(X ,Y ) = e : where X ' y E Kk _'Kk+1

-k 9:, .
and e < 5/2. Since Kk is closed, there eXists a sequence

* *

{Zn} in Kk such that lim 2n = x - y . Thus there exists
n

* * *
N EEN Such that n 2 N implies d (x - y ,zn) < e/Z. Also

* -
z E K for all n Elfl implies d (z ,0) = d(z ,0) S e k < 3/2.
n k n n

Hence

1- d —1' d —d* * *
rim (Xn,yn) ‘ :‘m (Xn " ynao) — (X ‘ y :0)

* * * *
s d (x - y ,2“) +'d (Zn,0)

*
d(XaY)

57

**-k ***
Conversely, assume d (x ,y ) < e/Z. By (1) d (x ,y )

1im d(xn,yn), and hence there exists N E N such that when

n * * * . * * *
2 N, ‘d(xn,yn) - d (x .y )\ < e/Z; 1-e-, d(xn,yn) < d (X :Y )

:1

+ 6/2 < e/2 + 6/2 = e for all n 2 N. Thus if n 2 N,
-k 00

._ n _ = .
xn - yn E Kkn Kkn+1, where e < e or xn yn 0 E .H Ki

Let k be the smallest positive integer kn’ n 2 N. Then

Kkn C Kk for all n 2 N and thus xn - yn E Kk for all n 2 N,

- * * * *
where e < 6- Since Kk is closed, x - y = lim (Xn - yn) E Kk’

* * - .
and p(x , y ) s e k < 6- Therefore the two metrics are equivalent.
* *
(5) By (4) convergence, etc. in (J ,d ) is characterized by

*
Proposition 3.1.2. Also since d \ = d, the topology on (J,d)

JXJ
* *

coincides with the subSpace topology inherited from (J ,d ).

Let A be an open submodule of J and define m on the set of

*
open submodules of J by m(A) = A . First, A open and 0 E A

*
implies that there exists k E,N such that K C A c A . Thus

k
* * * * * * * *
Kk C A and for any x E A , x +Kk C A . Hence A is open
*
and m is a map into the set of open ideals of J . Since
I o * - * o 0
A1 — A2 implies A1 — A2, m is well-defined. Next let B be

*
any open submodule of J . Then A = B n J is an open submodule
*
of J and A C B. Now B Open implies B closed in J so
* * *
that A C B = B. Suppose x E B. Then there exists a sequence
*
{xn} in J such that lim xn = X . Hence for every k E_N
n * *
there exists N(k) EEN such that if n 2 N(k), then Xn - x E Kko
*
Also B is open in J and 0 E B so there exists m E N such
* *
that Km C B, which implies Ki C B for all i 2 m. Define a

*
new sequence Clearly 1im y = x , and since
m

yn = xN(m)+n-l° n

58

* * *
X E B and x - X E Km C B. Thus yn E B 0 J = A for

N(m)+n-l
* * *
all n, and so x E A . Thus B C A , and equality holds. From
this the remainder of (S) can be deduced. First, this certainly
*
implies that m is onto the set of open submodules Of J
Secondly, if A is open in J, then A = B H J where B is open
* *
in J which implies that A = A n J by what was just shown.
* *

Finally m is one-to-one, for if A = B where A and B are

* *
Open in J, then A = J 0 A = J H B

B.
* *
(6) Let A be any open submodule of J and define m : J 4 J /A
*

by Q(X) = X + A for x E A. If 0:8 E Q, X.Y E J, then

* * * * *
cp(cvx+ay)=ax+ay+A =ax+A +By+A =a(x+A)+B(y+A)
= a¢(x) + Bm(y). Thus m is a @-module homomorphism. Suppose

* * * *
x +'A E J /A . Then there exists a sequence {xn} in J such

* *
that 1im xn = x . Also A is open, so A is Open by (5) and
n * *
there exists k.E N Such that Kk C A . Hence there exists
* *
N E,N such that xn E X (mod Kk) for all n 2 N. Hence

_ * * * _ * * .

XN = X (mod A ), and ¢(xN) = xN +-A — x +-A . Thus m is
* ‘k *

onto, and J /A :3 J/ker m. But X E A implies x E A so that
* *

¢(X) = X + A = A and X E ker m, and X E ker m implies

* ‘k *

A = m(x) = X +'A so that X E J H A = A. Hence ker m = A

and (6) follows.

59

3.2. TOpological.Quadratic Jordan_Algebras

Let (J,U,1) be a quadratic Jordan algebra, and consider

the ideals J = K(O),K(1),K(2),... as defined in Definition 2.2.3.

(n+1) =

Since for n 2 l, K (n)), clearly K(O) Q K(1)

UK(n)(K
2 K(2)

Q 0.. 0

Definition 3.2.1. An ideal K Of J is called a nucleus for J,

G)
if n x(“) = 0.
n=o
Since J and K(n), n = 0,1,2,..., are @-modules, the con-

siderations in §3.l apply. Hence if K is a nucleus for J, K

induces a topology in J called the K-topology. We have the

 

metric d as defined in §3.1, and of course (J,d) is a metric
Space whose topology coincides with the K-topology.

As we have previously seen,addition and scalar multiplica-
tion are continuous (the latter under suitable aSSumptions about
é). Next consider the mapping (x,y) » Ux(y) from J X J a J.

We shall now prove that this operation is also continuous. Let

Ux(y) +'K(k) be any basic open set of f(d) which contains

Ux(y). Clearly (X + K(k)) X (y +-K(k)) is Open in the product
k

topology on J X J and (x,y) E (x +~K( )) X (y + K(k)). If

(k)
(k)

(k), then x + a E x + K(k) and y + b

(k) (k)

a E K and b E K

E y +-K so that (x + a, y + b) E (x +-K ) X (y +~K ).

Also Ux+a(y +'b) - Ux(y +~b) + Ua(y +-b) +-Ux,a(y + b)

= Ux(y) + Ux(b) +-Ua(y + b) +-Ux,a(y +'b) E Ux(y) + K(k), since

(k) (k) (k)

K is an ideal and a,b E K implies U

X“310' + b) E K

by
Proposition 2.2.1. Thus (x,y) a Ux(y) is continuous and J

is a topological Jordan algebra according to

60

Definition 3.2.2. A quadratic (linear) Jordan algebra J is

called a tOpological Jordan algebra, if J is a topological Space

 

and the mappings (x,y) a X +'y, (x,y) « Ux(y) are continuous.

Before considering completions of topological Jordan

algebras with reSpect to the metric topology determined by a nucleus

K, the following useful inequality is established.

Lemma 3.2.1. Let K be a nucleus for J and d the metric of
the K-topology. Then d(Ux(y),Uw(z)) s max {d(x,w),d(y,z)} for
all x,y,w,z E J.

Proof. Suppose X = w so that d(x,w) = 0. If y = 2, then
Ux(y) - Uw(z) = UX(y - z) = UX(0) = O, and the inequality holds.

If y # 2, then d(y,z) = e.k where y - z E K(k)- K<k+1).

Hence Ux(y) - Uw(z) = Ux(y - z) E K(k) implies d(Ux(y),Uw(z)) s e-

d(y,z). Now assume x # w so that d(x,w) = e'k where

a E K(k) _ K(k+1).

x - w = If y = 2, then Ux(y) - Uw(z)

(”x - ”x-a)<y) = (”x - ”x - ua +'Ux,a>(Y> = ”x,a<Y> - ua<y>

E K(k). Thus in this instance d(UX(y),UW(z)) s e-k = d(x,w)

K(m+l).

where y - z = b E K(m)-— Then we have

Ux(y) - UW(Z) Ux(y) - Ux_a(y - b) = UX(y) - Ux(y - b)

- Ua( y - b) + UX a(y - b)

ux(y) - ux(y> + Ux<b) - Ua(y - b) + ”X a(y - b)

3

(L)
Ux(b) - ua<y - b) +‘”x a(y - b) e K ,

(k) (L) (m) (L)

where L = min {k,m}, Since K C K and K C K .

max {d(x,w),d(y,z)}. Finally assume y # 2 so that d(y,z) = e

-m

k

61

Thus d(Ux(y),Uw(z)) s e-L = max {d(x,w),d(y,z)}.

Definition 3.2.3. Let J be a quadratic Jordan algebra with
nucleus K and let d be the metric of the K-topology. Then a

* *
quadratic Jordan algebra (J ,U ,l) is a completion of (J,U,1)

 

with reSpect to d, if
* *
i) (J ,d ) is a complete topological Space,
* *
ii) (J,d) is a dense subSpace of (J ,d ),
*
iii) J is a topological Jordan algebra,

*
iv) J is a subalgebra of J

Theorem 3.2.1. If (J,U,1) has nucleus K with metric d, then
* *
there exists a completion (J ,U ,1) which is unique to within

isomorphism.
Proof. By the considerations in §3.1 there exists a unital @-

* *
module J with metric d such that among other things, J is

*

* *
a dense subSpace of J , d | = d, and J is complete with

JXJ
*
respect to the metric d . Recall that addition and scalar multi-
* * *
plication were defined by x +-y = lim (xn + y ), ax = lim axn,
n
n n

* ‘k *
a E Q and x = 13m Xn, y = lim yn E J where {xn}, {yn] are

*
sequences in J. In a similar fashion define U by

* * _ .
U *(y ) - 11m UX (yn) -
X n n

_ . , * . * ' *
Suppose 11m x = X , 1im X = x , lim y = y , 1im yn = y . Then
n n n n n n n

for any 3 > 0 there exists N E N such that n 2 N implies

d(xn,x;) < e and d(yn,y;) < 6. Hence by Lemma 3.2.1,

d(UX (yn),UX;(y;)) s max {d(xn,x;),d(yn,y;)} < e for all n 2 N.
n

62

Thus 1im Ux (yn) = 1im Ux,(y;) and so the Operation is independent
n n n n

* k *
of representation. Similarly, it follows that U *(y ) E J . Hence

X
* * * * .

U * E Endé J . Also for every a E Q, U *(y ) = lim U01x (yn)

X ax n n
, 2 2 * * * . .

= 11m a U (y ) = a U *(y ). Therefore U is a quadratlc mapping
n xn n x

* x *
from J into EndQ J . If U * * is the associated bilinear
x,y

mapping, then

* * * * * * * *
X :Y X *7 X y
=lim[U (2)-U (2)-U (2)]
ti Xn+yn I1 Xn n yn n
= lim UX , (zn) .
n n n
* * * * _
Similarly V * *(Z ) = lim VX (Zn) and V *(y ) = 11m V (yn)’
X ,y n n’ n X n xn
* * * * .
so that x 0 y = V *(y ) = lim VX (yn) = lim xno yn, where the
X n n n

definitions of the Operators are as in §2.1. It can easily be

seen that the fact that (J,U,1) satisfies QJ1 - QJ3 and QJS -
* *

QJ8 implies that these hold for (J ,U ,1) also. For example,

by QJ3 in (J,U,1),

* ~k * .
U *V * *(z ) 11m UK v

y ,x (2“)
X y ,X n n n

it * *-
v * *U *(z ),
X .y x

* * * x
and thus QJ3 holds for (J ,U ,1). Therefore (J ,U ,1) is a

quadratic Jordan algebra with (J,U,1) as a subalgebra.

* x x .
By Proposition 3.1.3 (1), d (x ,y ) = lim d(xn,yn), if
n
, * * * * * * *
11m Xn = X and lim yn = y . Thus d (X +-y ,w + z )
n n

63

* * * * * * * * * * *
5 max {d (x ,W )sd (3' ’z )} and d (U *(y ),U *(2 ))
* * * * * * * x w
s max {d (x ,w ),d (y ,z )}. Let p be the metric induced on

* * * * * * *
the product J X J where p ((x ,y ),(w ,z ))
* * * * * * * * * * *
= max {d (x ,w ),d (y ,z )}. Then for any 6 > O, p ((x ,y ),(w ,z ))

, * * * * * * * * * *
< 6 implies d (x +Iy ,w + z ) < e and d (U *(y ),U *(z )) < e-
x

w
* *
Hence the Operations are uniformly continuous, and (J U

3 3

1)

is a topological Jordan algebra.

AA

If (J,U,1) is another completion of J, then there

* .

exists an isometry m : J a J Such that m is onto and m(x) = x
* * * * *

for all x E J. Since d (x ,y ) = d(¢Cx ),m(y )) for all

* * * * * *

X ,Y E J , if x = lim X“, y = lim yn, then ¢(x ) = lim Xn’

n n n

¢(Y*) = lim yn. Hence m(qx* + By*) = ¢[lim (axn + Syn)]
* n * * * n x *
= a¢(x ) + BmCy ), and mCU *(y )) = m(lim Ux (Yn)) = U * (@(Y ))-
x n n ¢(X )

Thus m is a Jordan homomorphism and J* 2:3.
Since a completion of J* of J is unique to within
isomorphism, J* will be called the completion Of J. Also since
f(d*) = f(p), where p is the metric induced by the decreasing
(0)* 2 K(1)* 2 K(2)* 2

*
sequence of modules J = K ..., p will

*
be used to denote the metric for J . Some important properties

*
of J are considered next.

*
Lemma 3.2.4. If J has nucleus K and J is the completion
of J, then:
* *
(i) K (n) s: (K(n)) , n = 0,1,2,... .

m) (opcxum

m *
(ii) fl (K )

n=o

= 0, and if (K for every

* *
n = 0,1,2,..., then K is a nucleus for (J ,p).

64

* *
(iii) If A is an ideal of J, then A is an ideal of J
*
(iv) If A is an open ideal of J, then A = A n J and
k
A a A is a one-to-One correSpondence between the open
*
ideals of J and the Open ideals of J .
* *
(v) For any open ideal A of J, J/A a J /A as quadratic
Jordan algebras.

* * * *
Proof. (i) Certainly K (0) .—.J = (KM) and K (1)

= (x(1))*.

*
= K

. . *(n)
Proceed by induction. For n 2 I assume K

*. * K*(n+1)

*
If z E , then 2 is a o-linear combina-

*(n)

: (K(n))
* * * *
tion of elements of the form U *(y ), X ,y E K Since

X

+1 *
K(n )) is closed under such Sums, it suffices to Show

+ * * * *
(n 1)) for every x ,y E K (n). By the induction

(

* *
U*(y)€(K
X

* *(n)

. * . . * * (n) *
hypotheSis, X ,y E K implies x ,y E (K ) . Hence there

*
exist sequences {Xk}, {yk} in K(n) such that x = lim Xk,
k
* * *
y = lim y SO that U *(y ) = lim U (y )‘ NOW U (y )
k k x k Xk k xk k
+1 * *
E U (n)(K(n)) = K(n ) for every k = 1,2,..., so U *(y )
K x
+1 *
e (1<(n )) .

.. ”° *(n) °° (n) * . .
(ii) 0 K C n (K ) = O, by PropOSition 3.1.3 (3). Thus
n=o n=o
* k * *
K is a nucleus for a topology for J . If (K(n)) ; K (n),

x *
then (K(n)) = K (n), and the topology coincides with the 9-
metric topology.
... , * * * x
(111) Let A be an ideal of J and x E A , y E J . Then

there exist sequences {xn}, {yn} in A and J reSpectively

* *
such that lim xn = X , 1im yn = y . Thus for all n E N,
n n * * .
U (y ),U (X ) E A. Hence U *(y ) = lim U (y ),
X n y n X n
n n x n n

6S

* * * * *
U *(X ) = lim U (Xn) E A . Since A is a submodule of J by
y n yn * *
Proposition 3.1.3 (2), A is an ideal Of J .
(iv) Let A be any open ideal in J. By Proposition 3.1.3 (5),
* * x *
A = A n J and A is open in J . Thus the map ¢(A) = A is
a one-to-one map of the set of open ideals of J into the set Of

* *
Open ideals of J . If B is any open ideal of J then A = B H J

*
is an open ideal of A and B = A . Thus m is onto.
(v) From the proof of Proposition 3.1.3 (6) we have, if A is
* * *
an Open ideal of J, then m : J a J /A defined by Q(x) = x + A ,
*
X E J, is a module epimorphism with kernel A. Now Q(l) = l +’A
* * * ~*
and l + A is the unit of J /A . Let U denote the quadratic
* * * *
mapping for J /A induced by U . Then m(Ux(y)) = Ux(y) +~A
*

* * ~* * ~
=Ux(y)+A =U *(y+A)=U

l .
“A qmum» for a 1 x.y e J

* *
Hence W is a Jordan homomorphism so that J/A a J /A as

quadratic Jordan algebras.

66

3.3. Local Jordan Algebras

If J is a quadratic Jordan algebra, then EKJ) denotes

the Jacobson radical of J (see §2.3).

Definition 3.3.1. A quadratic Jordan algebra J is a local
Jordan algebra, if (i) EKJ) is the unique maximal ideal of J,
(ii) EKJ) is a nucleus for J, and (iii) J/EKJ) satisfies

the minimum condition.

Let m be a commutative associative ring with 1. If M

has a unique maximal ideal, then this ideal is RKM). Since
k-l

m
90') (k) C £011) is clear by induction on k, if H Q(y)“ =
n=o
then H ”EKM)(R) = O and EKM) is a nucleus for m as a

k 0
quadratic Jordan algebra over Z. m/Eflu) is a field and hence

contains no idempotents other than 0,1. Thus m/EKW) trivially

satisfies the minimum condition. Therefore any not-necessarily

Noetherian,<xmmmtative, associative, local ring is a local Jordan algebra.
We now proceed to consider the completion of a local

Jordan algebra.

Lemma 3.3.1. If K is an ideal of J and J/K is semi-simple,
then Q(J) C K.

Proof. Let Q : J a J/K be the natural homomorphism. Then
m(EKJ)) is a quasi-invertible ideal of J/K, and thus

cp(,€(J)) = 0. Hence Q(J) s: K.

Lemma 3.3.2. Let J be a local Jordan algebra with completion

* * on n

J , 18t an E Q, n E N, and 2 E EKJ) . Then 2 a z converges
* n=l n

in J .

67

Proof. Let k,E N and N = 3k. By Proposition 2.2.4, n 2 N implies

2n 6 emf“) : emu” (10*

, and hence anzn E EKJ) for n 2 N.

G
Thus lim anzn = O, and Z anzn converges by Proposition 3.1.2 (3).
n n=1

*
Lemma 3.3.3. Let J be a local Jordan algebra with completion J

* * *
e an idempotent in J , and K = Ue(J ) closed in J . If
* co
2 E EKJ) H K has quasi-inverse w E K, then -w = z z

Proof. Recall that K is a quadratic Jordan algebra with quadratic

~* * ~* *
mapping U = U ‘K and unit e (§2.7, p. 38). Since U = U on

*
K we Shall use U to denote both mappings. Note that for all

2 * * *
Xay E K, X - Ux(€) and x 0 y - Ux,y(e) - Ux e(y), the latter

9

*
following Since UX y(e) = (x +~y) - x2 - y . Thus the operations
,

*
in J coincide with those in K. Hence we have for any m EIQ,

* mn * mn * mn * mn
U (e+zz)=U(e+2z)+U(e+£z)-U (8+ 2)
e-z e z e,z
n=1 n=1 n=1 n=1
m m m
* * * * *
=U(e)+ ZU(Zn)+U(e)+ ZU(zn)-U (e>‘ EU (Zn)
e e z z e,z e,z
n=1 n=1 n=1
m m m
= e +- 2 2n + 22 + 2 zn+2 - e 0 z - 2 z 0 2n
n=1 n=1 n=1
m m+2 m
+
= e - 2z +' 2 zn +- 2 2n - 2 2 2n 1
n=1 n=2 n=1
= e _ z _ zm+l + zm+2
k m+2 +1 * k *
Givenany kEN, if m23,then z -zm EEO) (k)c:,?(J)(),
m
*
and thus it is clear that lim Ue-z(e +' Z zn) = e - z. By Lemma
m n m * n=1
3.3.2, 2 z = u exists in J , and hence, Since K is closed,

n=1
u E K. Now

68
* .. 1 * + m n *
Ue_z(e +'u) — gm Ue_z(e “:12 ) = e - z — Ue_z(e ‘ W):

since w is the quasi-inverse of z in K. By the invertibility

*
of U (on K), -w = u.

co
Lemma 3.3.4. If J is a local Jordan algebra and z x. = x,
oo 00 i=1 1
'2 21 = 2, then .2 (xi + 21) = x + z .
i=1 i=1
Proof. For any nonnegative integer k, there exists N E,N Such
m n (k)
that when m 2 N, x - 2 Xi and z - 2 z, are in EKJ) . Thus
i=1 i=1 1
m m m
for all m 2 N (x + z) - E (x, + z.) = (x - 2 x.) + (z - 2 2,)
'=1 1 1 i=1 1 i=1 1
(k) 1

E Q(J) .

*
Lemma 3.3.5. If J is a local Jordan algebra with completion J ,
* *
then Q(J) = Q(J ).
* * * *
Proof. By Lemma 3.2.4, J/E(J) .2: J /,Q(J) , and hence J /R(J) is
* *
semi-simple. By Lemma 3.3.1, EKJ ) c EKJ) . Also by Lemma 3.2.4,
* *
RKJ) is an ideal of J , and thus the result will follow, if
* * *
every element of EKJ) is quasi-invertible. If x E EKJ) ,

*
then there exists a sequence on EKJ) Such that 1im xn = x

n
Thus for each n EEN there exists yn E EKJ) which is the quasi-

inverse of xn. We will first Show that the sequence {yn} is

Cauchy. Now for any k = 0,1,2,..., there exists N E N such

that n,m 2 N implies xn - Xm E EKJ)(k). By Proposition 2.2.2,
(xn)1 - (Xm)1 E EKJ)(k) for every 1 E N. By Lemma 3.3.3,
” i w i ” i
“ya = .3 (xn) and -ym = .2 (xm) , and so yn - ym = .2 (xm>
i=1 m i=1 i=1
a - i ' *
- z (x )l = 2 [(x) - (x )1] 6 em“) for all n,m 2N.
i=1 n i=1 m n '

Therefore {yn} is a Cauchy sequence on EKJ), so that there

* * *
exists y E EKJ) such that y = lim yn.
n

69

* *
Since U *(1 - y ) = lim U1_ (1 - yn) = lim (1 - x )
l-X n xn n n
* * * 2 , 2 . _

= 1 - x . and u ,[<1 - y > 1 = 11m ”1-x [(1 - yn> 1 = 11m 1 — 1.
l-x n n n

* * *

y is the quasi-inverse of X in J

Theorem 3.3.1. If J is a local Jordan algebra with completion
* *
J , then J is a local Jordan algebra.
* * * *
Proof. By Lemma 3.3.5, Q(J) = Q(J ), and so J/mj) a J /,Q(J)
* *
= J /&KJ ). Since EKJ) is the unique maximal ideal of J, J/EKJ)
* *

is simple by Proposition 2.6.1 (2). Hence J /EKJ ) is simple

* *
and so EKJ ) is a maximal ideal of J . By Proposition 2.6.1 (4),

* * m * (n)
EKJ ) is the unique maximal ideal of J . Finally fl EKJ ) = 0
n=0

* *
by Lemma 3.2.4, and J /EKJ ) satisfies the minimum condition

*
since J/EKJ) does. Thus J is a local Jordan algebra.

The last result of this section is concerned with properties

*
of power series in J which will be useful in Chapter 4.

PrOposition 3.3.1. Let J be a local Jordan algebra with comple-

* *
tion J , z E EKJ ), an’en E o, n = 0,1,2,... . Then:

a:

(i) Any regrouping or rearrangement of Z ann gives a con-
n=0
vergent series (not necessarily a power series), which con-
m n
verges to 2 a z .
n
n=0
m n m n w n
(ii) If y = 2 a U , and u = 2 B Z , then y = 2 y 2 , where
n n n
n=0 n=1 n=0

2
Y0 = 0‘01’ Y1 = 0’1551’ Y2 = C'182 + 0’281’ Y3 = 0’153 + 20’25152
2 _ 2 2 4

Proof. (i) First note that by Lemma 3.3.2, every series of the
m

* *
form 2 a z“, z E gKJ ), an E Q, converges in J . Trivially
n=1 n
00 n on
then, so does any series of the form 2 a z = aol + 2 a 2n
n
n=0 n=1

70

G)
n
Now any series formed from Z anz by regrouping (adding paren-
n=0
theses) will have its sequence of partial sums as a subsequence of
Q
n
the sequence of partial sums of z anz , and will therefore con-
oo n=0
n
verge to z a z .
n
n=0

n
Next let an = anz and let n be any permutation of

m
N U {0}. Let bn = afi(n) and Bm = 2 bn. Then letting
n=0
co m m
a = 2 a 2n = 2 a , and A = 2 a , we have a = lim A . Now
n n m n m
n=0 n=0 n=0 m
given m E N, let r(m) = max {n(i) : 0 s i s m}. Then for any
m m
HIE N B = b = a = A - 2 a . Let
’ m n20 n “:0 n(n) r(m) n(n)$r(m) n(n)

n>m
k E N be given and let h = max {n : n(n) 3 3k}. Then for every

n 2 h + 1, n(n) > 3k so that an(n) = o[1_r(n)zrr(n) E ETJ*)(k)
(10* 0°

Since 2 3n = a, there exists Nl.€ N such that
n=0

j 2 N, implies a - Aj E EKJ*)(k) c EKJ)(k)*. Now let p E;N

C EU)

satisfy n(p) = N1, and let N2 = max {p,h}. If m 2 N2, then

(i) r(m) 2 n(n), for all n s m implies N1 = n(p) s r(m), so

* *
that a - Ar(m) E EKJ )(k) C EKJ)(k) , and (ii) m 2 N2 2 h
*
implies for n > m, n 2 h + 1 so an(n) E EKJ )(k) C EKJ)(k)*.
Thus for m 2 N , a - B = a - A + Z 8 E EKJ*)(k)
co n>m
(10* . ..
C EKJ) . Hence 2 bn = lim Bm — a.
m n=0 m m
(iii) let w = Z y zn be the series derived from Z q un by substituting
n n
n=O n=0
m n m n
u = 2 an for u and collecting terms. Now 2 ynz is derived
n=1 n=0

from the series a0 + “1812 + 018222 +~a28izz +-918323 + ... by
no

regrouping and if this series is denoted by 2 bi’ then 1im bi = 0,
i=0 1

since there are only finitely many terms of a fixed power of z.

71

a) a:
Thus 2 b. converges by (i) and w = 2 b..

. 1 . 1

i=0 i=0

Next consider the following matrix in which the entry in

the mEh-row and nEh-column is denoted by Cmn'

a 0 O 0 O .
0 C’1312 0’1327‘2 “18323 0’16424 ° °"
0 0 azfiizz 202315223 (2028163+UZB§)24 . ...
0 0 O a38i23 3036i8224 . ...
0 0 0 0 aaBiz4
0 0 O 0 0 . ...

*
Since each row constitutes a power series in z E EKJ ), with the
00

first m terms 0, each row series converges. Let cm = 2
n=0
r
m = 0,1,2,... . Also let Br = E b, and let k E_N be given.
i=0 1
Then there exists N1 E_N such that for all r 2 N1, w a B
(k) (k)*

C EKJ) . Now Br is a finite sum of entries from

C :
mn

r

*
E EKJ )
the above matrix, SO let m'(r) = max {row subscripts Of terms of Br}

and n'(r) = max {column subscripts of terms of B }. Then w - B
We n'<r) r r

= w - 2 ( 2 c ) E EKJ*)(k) C EKJ)(R)*, for every r 2 Nlo
m=0 n=0 m:
Also, since cm = nEoCmn, there exists N2 E;N such that p 2 N2
P
implies cm - z Cmn = x(p) E EKJ*)(k) C EKJ)(k)*. Hence for
n=0 m'(r) m'(r)
n'(r) 2 N2, W ‘ Br = W ' Z [Cm ‘ X(P)] = W ‘ X Cm + m'(r)x(p),
m=0 m=0
m'(r)
and if r 2 N1,w - Br E EKJ*)(k) C EKJ)(k)* implies w - 2 cm
=0
m' (r) m
+ m'(r)x(p) e EKJ*)(k) : ng)(k)*, so that w - z c e g«J*)(k)
(10* "“0 m

C EKJ) . Hence if Q = m'(N1), for any q 2 Q, there exists

72

q=m' (r) “'03)

r 2 N Such that B = 2 ( z c ) and therefore the above
1 r mn
m=0 n=0
. . q * (k) (10*
computations give w - 2 cm E EKJ ) C EKJ) . Hence
q CD [n=o co n 00
w = lim 2 c = Z c . Finally, this gives 2 Y z = w = Z c
_ m _ m _ n _ m
q m-O m—O n—O m—O
m n

CHAPTER 4

STRUCTURE OF COMPLETIONS OF lOCAL JORDAN ALGEBRAS

4.1. Basic Concepts

 

Throughout.Chapter 4 all Jordan algebras will be assumed
to be linear Jordan algebras over a field of characteristic dif-
ferent from 2 unless it is stated otherwise. Recall that such an
algebra is equipped with a commutative bilinear product which
satisfies the Jordan identity. If g is a local Jordan algebra
with completion J, then, in the notation Of Chapter 3, U:‘fl = UX
for all x E g, and so we will write U for both the quadratic
mapping on g and the quadratic mapping on J. Likewise the stars
will be omitted from the remaining operators, but we will still
denote cLJS by 8*. In particular, we have J = 3*. Let
{xn}, {yn} be Cauchy sequences on 3 such that lim Xn = x and

n

° — = .1'. o - ' l
1;m yn — y. Then x,y E J, and Xoy — 2 X y lgm (2 Xn Y“)

= lim Xn.yn. Thus the product . satisfies the usual defining
congition for a product, and the continuity of U clearly implies
the continuity of ° .

In a linear Jordan algebra J, the subalgebra generated by
l and an arbitrary element X E J is associative; i.e., J is
power-associative. If g is a local Jordan algebra with completion

J, then the set of convergent power series in x E J with coefficients

in Q is an associative subalgebra of J. We now prove this in

73

74

a somewhat limited form.

Proposition 4.1.1. Let g be a local Jordan algebra with comple-

CD

tion J, let 2 E EKJ) be fixed, and M2 = { B an zn : an E Q,
n=0

n = 0,1,2,...}. Then m2 is an associative subalgebra of J.

Proof. Note that by Lemma 3.3.2, all elements of $2 are con-

00

vergent series. Suppose 2 a z“, 2 an zn E m and y E Q.
n=0 “ n=0 2
co co m n m
Then 2 anzn +- 2 Sn 2n = lim ( 2 an 2 +' 2 an 2 n) = lim ( Z (a M+fi )zn )
m n=0 n=0 m n=0 n=0 m m n=0
= 2 (an + an )zn E mz, and y z anzn = lim (y 2 an 2 n)
n=0 n=0 m n=0n
= 1im ( 2 (Van )zn )= 2 (Van )zn Hence m2 is a vector
m n=0 n=0
subspace of J. Also
co m m m m n+j
(Barf-z)(28nn=z) 1im[(Zann.Z)(ZBnnZ)]=lim2 2%sz
n=0 n=0 m n=0 =0 m n=0 j=0
m n n 2m 2m-n+l n
=lim[Z(EO’jBn_-Z) + 2 < z ajBn_.)Z]
m n=0 j=0 j n=m+l j=n-m J
m n n 2m 2m-n+l n
=11m 22(20)an )2 +1im 2 ( z OI.Bn_.)Z
m Fflj4)J j m nqfilfmmiJ J
on n
n
=Z_3(2cern_J )2 E912
n=0 j=0
Finally, the associativity of $2 follows from the
associativity of Q(z); for if 2 gnzn, 2 ann: 2 Yn zn E m2, then
n=0 n=0 n=0
m m m
[(EanIIL)(EBnn.Z)] Zvnnz =lim[(2ar,r12).(rlr,rl-nnZBZ)]EYZ
F0 n=0 n=0 m n=0 =0 n=0
m
=11m Zannz ,[(28nn.nz)(zonnyz)]
m n=0 n=0
=2 n~1G};(Bnn-nz)E_onniznfl-
n=Oa n=0

75

00

Clearly I = 11 +- 2 0 2n E flz and the proposition follows.
n=1

For the last prOposition of this section recall that a
set Of idempotents {e1,e2,...,en} is Supplementary, if they are

n
pair-wise orthogonal and z ei = 1.
i=1

Proposition 4.1.2. Let J be a quadratic Jordan algebra over a
commutative, associative ring Q with identity, and let K be
a nucleus for J. If J = E €>J,. is the Pierce decomposition

isj
Of J relative to a set of supplementary idempotents {e1,e2,...,en),

then Jij is closed for i s j.
Proof. We have that Jii = {X E J : Ue (x) = x},

1
j = {x E J : Uei’ej(x) = x], 1 < 3. Thus if x E CLJJii’ then
there exists a sequence {Xk} on Jii such that x = lim xk.
Hence Ue (x) = lim Ue.(xk) = lim xk = x, and so x E Jii' Thus
1 k i k
Jii is closed. Since Ue,,e, = Ue,+e, - Ue. - Ue.’ Jij’ i < J,
1 J 1 J 1 J

is closed in a similar manner.

J.
1

In particular, if g is a local Jordan algebra, then the
* *
Pierce Spaces of g and g are closed in the f(g) and 5K; )-
*
topologies, respectively, and therefore those of g are closed

in the completion topology.

76

4.2. Idempotent Lifting

If g is a quadratic Jordan algebra over a ring Q and
l
K is an inner ideal of y, then for any x E K, X E K and

xn = UX(Xn-2) E K for all n > 1.

Lemma 4.2.1. Let g be a local Jordan algebra with completion J,
and let K be any closed inner ideal of J. Then for every

Y E EKJ) n K, there exists x E EKJ) n K Such that x2 - X = y.
Proof. Define a sequence {an} as follows: let 01 = 1 E Q

n

and an+1 = iglaian-i+1 for all n 2 1. Clearly, an E Q for

every n E,N. Since y E EKJ) n K, -y E EKJ) D K, and so

G (D
rowa wmaaa H x=2ogwf.wm xemnnx.
n=1 n=1

since each is closed in J. Now from the proof of Proposition
m n-l a

2
4.1.1, x = z ( 2 aian_i)('Y)n = 2 an('Y)n- Thus
n=2 i=1 n=2
2 m n m n
x - x = z: an(-y) - 2‘. an(‘)') = y -
n=2 ' n=1

Lemma 4.2.2. Let g be a local Jordan algebra with completion

J, and let K be a closed inner ideal of J. If v E EKJ) n K,

as
w = 2 v v“, y E Q, and u2 - u = v, then
n n
n=1
2 2
(i) (u.w) = u .w2
.. 2
(11) u.(u.w) = u .w
(iii) (u.w).w = u.w2
m n 2
Proof. Let W = 2 v v , m E N. Since v = u - u, W E Q(u)
m n m
n=1
for all m EN, and Q(u) associative yields that for all m E N,

2 2 2 _ 2 _ 2
(u.wm) — u .Wm, u.(u.wm) - u .Wm, (u.wm).Wm — u.Wm. Hence

77

liniW , implies (u.w)2
m m
lim u2.W
m
m

m
.(u.w)

(2

lim u.(u.wug

m 2
-.w.

1im u.W
m

U

Theorem 4.2.1. Let

J, and let K be a closed inner ideal

be the natural homomorphism and let K

idempotent, then there exists

?2

e E K

Proof. -?=0 and feE

Now 1 + 42

~42 E Q(J) n K
1

implies
m n
z (-4z) .
n=0

Q(J) O K

3.3.3, (1 +42)” Thus

is closed.

E EKJ) D K, since

m

E a [-z.(l + 42)
n

n=1

= -l n

exists x

-z.(l + 4z)-1.

and x - =

f2

Since 2

eneo.

2

2
f.(f.x) = f .X, and (f.X).x = f.x .

h

l - 2x

Next let and g

= (1 - 2,02 + 2x.(1

2 2
.z x +

(l - 2X)2.z

2
and g +-h

x + z - z.[(l + 4z).(l +~4z)-1]

lim (u.W
m

2
= u

such that

implies that

-z.(1 + 42)-

J ’ 01 =

By Proposition 3.3.1,

- f, Lemma 4.2.2 implies

2x) = 1 - 4x + 4X2
x - z.[(l

x + z - z

2

U

2
.w ,

)2

lim u2.W2
m

m

.w, and (u.w).w

lim (u.Wm).Wm
m

g be a local Jordan algebra with completion

of J. Let : J a J-

J//?(J)

=9(K). If (9E is

and A; = f.

e e

2
f - f =

z E Q(J) OK.

is invertible, SO by Lemma

1 n-l n
z

{-4)

By Lemma 4.2.

1
E Q, an+l

(f.

2

Then h

X. 2g.h

4X2

1 - 2X

+2)!“ 11,

4z)-1 1

- 1 +4z.(l +42)"]

+

x Now let

= g.

e = f.h + g. Since f,x E K, e = f.h + g = f - 2f.x + x E K, and

since x E EKJ), -2f.x +*x E EKJ), so 2 = f. It remains to Show
2 2

that e is idempotent. Since (f.h) = f2 - 4f.(f.x) + 4(f.x)

2

2 2.x +’4f .x

= f - 4f

2_22

f .h and (f

= f.x - 2f.x2 = f.(h.g), we have

.h).g = f.x - 2(f.x).x

78

2 2
(f.h)2 + 2(f.h).g + g = (f +-z).h2 + 2f.(h.g) + g

(D
II

2 2
f.h2 + z.h2 + 2f.(h.g) + g = f.(h2 + 2h.g) + g2 + h 02

f.h + g = e .

For the next theorem, recall that if J is a linear Jordan

2
algebra over Q and % E Q, then for all x E J, Ux = 2R - R

Theorem 4.2.2. Let g be a local Jordan algebra with completion
J. If {fi,fé,...,fg} is a set of nonzero orthogonal idempotents
in J.= J/EKJ), then there exists a set {e1,e2,...,en} Of nonzero

idempotents in J such that ei = fi, 1 = 1,2,...,n. Furthermore
if {f1,...,fn} is Supplementary, then {e1,...,en} is a supple-
mentary set.

Proof. First note that if {e1,...,en] is a set of orthogonal
idempotents such that E, = fi, 1 = 1,2,...,n, then the ei are

nonzero. The proof is by induction on n. The case n = 1 follows

from Theorem 4.2.1, where we take K = J. Hence assume for

l s r < n, orthogonal idempotents e1,e2,...,er exist with ej = fj,
r
j = 1,2,...,r. Let u1 = .2 ej and u2 = l - ul. Then {u1,u2}
J=1 \
is a supplementary set of idempotents. Let J = J11 OJ12 OJ22
be the Pierce decomposition of J relative to {u1,u2}. Since
r
for j = 1,2,...,r, u1.ej = (iilei).ej = ej, ej E J11 for all
j = 1,2,...,r. Now put f = Uu2(fr+1)’ so that f E J22. Since

2
f - Uu2(fr+l) - 2Ru2(fr+l) - Ru2(fr+l) — 2(fr+1.u2).u2 - fr+1Ou2
2

= 2[fr+1.(1 - u1)].(1 - ul) - fr+1.(l - u1)

= 2fr+1.(1 - ul) - 2(fr+1.u1).(l - ul) - f (l - ul)

r+l°

= fr+1 ' fr+1‘111 ' 2(1 +1 111)“1 ' ”1)’
. = = .f = = .
and r+1 u1 .2 fr+1 1 § fr+1 i 0’ 1 fr+1 # 0
i=1 i-l

Now by Proposition 4.1.2, J22 is closed, and J22 is
an inner ideal of J. Also f2 =1f E 322’ so by Theorem 4.2.1,

. 2 -' - _
there eXiStS e E J22 Such that e — e and e — f. Let er+1 €-
Then er+1 is an idempotent in J and er+1 = e = f = fr+l° Now

for any i = 1,2,...,r, ei E so ei.er

J11,
{e1,e2,...,er+1} is a nonzero set Of orthogonal idempotents of

J such that ei = fi, 1 = 1,2,...,n.

n n
Finally, assume 2 f, = l, and let e = 2 e,. Then
i=1 1 i=1 1
e = 2 e1 = 2 fi = l, and hence e = l + 2, z E EKJ). Now
i=1 i=1

(1 + 2)2 = e2 = e l + 2, so 1 + 22 + z2 = (l + z)2 = l + 2.

Thus z.(1 + z) = 0, and [z.(l + 2)].(1 + z).1 = O.

N
+

N
H

But 2, l + 2, (l + z) E m2, so 2 = 0.

Lemma 4.2.3. Let g be a local Jordan algebra with completion
J, e a nonzero idempotent of J, and K = Ue(J). Then;

(i) If 31E K is invertible in K, then u is invertible in K.

(ii) If u = e - 42, z E EKJ) H K, then there exists w = g a 2
such that (e - 2w)2 = u = e - 42. =

(iii) If u E K and u2 - e E EKJ) O K, then there exists v E K
such that v2 = e and ;'= 5.

Proof. (i) Recall that K is a linear Jordan algebra and

EKJ) H K C KKK). If B1 is invertible in K, then there exists

X'E K1 such that 6;};5 = E, i.e., Uu(X) - e = z E EKJ) O K.

u
Hence Uu(X) = e + 2, and since -2 E EKJ) H K C EKK), e + z is

80

invertible in K. Thus Uu(x) is invertible in K and so u

is invertible in K by PrOpOSition 2.3.1.

(ii) Since K = Ue(J) is a Pierce Space from the Pierce decomposi-
tion of J relative to {e, l - e}, K is closed. Hence by Lemma
4.2.1, there exists w E RKJ) O K such that w2 a w = z, where

°° n 2 2 2 2
w = z anz , an E Q. Now (e - 2w) = e - 4e.w + 4w = e - 4w + 4w

n=1
=e-42.

(iii) Since u2 - e E EKJ) O K, let u2 = e - 42, z E EKJ) H K.

Then by (ii) there exists w = ; anzn such that (e - 2W)2 = e - 42.
Since EKJ) 0 K is closed, 2w 2i;(1) D K C KKK), so 2w is quasi-
invertible in K. Hence e - 2w is invertible in K. Let

v = u.(: - 2w)-1. Clearly v E K. Now by lemmam3.3.3, (e - 2w)

= e +. g (2w)n, so v = u.(e - 2w)“1 = u.e +'u. Z (2w)n = u + u. E (2w)n.

n=1 n=1 n=1
a n
Since 2w E EKJ) and EKJ) is closed in J, 2 (2w) E EKJ), and
m ‘_ ‘_ n=1 m n
so u. 2 (2w)n E EKJ). Thus v = u. Finally, since w = Z qnz ,
n=1 n=1
‘1 °° n .. .
(e ' 2W) = e + 2 B Z , by Prop031tion 3.3.1. Thus, Since
n=1 n
m m a
-l , n , n 2 n
(e - 2w) = lim(e +' 2 B 2 ) = 11m [e + 2 ‘-— (e - u ) 1, and
n n
m n=1 m n=1 4
m 8n n
e + 2 —E'(e - u ) is in the subalgebra of K generated by
n=1 4

- - 2 -
e and u, v2 = [u.(e - 2w) 112 = u2.(e - 2w) 2 = (e - 2w) .(e - 2w) 2

= e

Theorem 4.2.3. Let g be a local Jordan algebra with completion

J and let be nonzero orthogonal idempotents in J. Then

81,92

are (strongly) connected if and only if E, and e2

e1 and e2

are strongly connected in J = J/7?(J).

81

Proof. If e and e are (strongly) connected, then there exists

1 2
2
. . . . = + .
u E Ue ,e (J) such that u is invertible 1n Ue +9 (J) (u e1 e2)
1 2 l 2
Hence 5' is invertible in U;, __(J) (52 = E, + 32), and
e +e
._ ,_ ,_ l 2
u E U_, _,(J).
e1,e2 _ _ _
Hence assume e1, e2 are connected in J, Then there
exists 5,2 E U;, __(3) such that :12 iS inVEYtible in 5:133,
81,82 e
where e = e1 + e2. Let v12 = Ue ,e (ulz). Then v12 E Ue ,e (J),
l 2 l 2
and 5,2 = U;_ E (:12) = 312’ so 3,2 is invertible in U;(J).
e1, 2 e
By Lemma 4.2.3, v12 is invertible in Ue(J). Thus e1 and e2

are connected.

If, moreover e and e are strongly connected, then

1 2
we may assume that :12 = El Let v12 be as above. Then
V E U (J), and G2 = E: By Lemma 4.2.3 (iii), there exists
12 e1,e2 12
v E Ue(J) Such that v2 = e and ;'= 3,2. Even more can be

said from the proof of the lemma, since

v - lim [v +-v ; EB (e - 2 )n B t +'V ; -2- 2 )n
12 12‘ n V12 3" ‘1 v12 12' II“ v12
n=1 4 n=1 4
E U (J), for all m E N, and U (J) is closed. Therefore
e ,e e ’e

l 2 l 2
v E U (J), and e ,e are strongly connected.

el)e2 1 2

The importance of Theorem 4.2.3 lies in the relationship
between (Strongly) connected idempotents and Jordan matrix algebras.
In essence, it allows us to infer the structure of a completion from

the known structure of J/RKJ).

82

4.3. Structure of Completions

 

If m is any algebra, A an ideal of m and 9 : m a.m/A
the natural homomorphism, then for S C m and x E S we will write

Q(x) = X. and 9(8) = 81 throughout this section.

Definition 4.3.1. Let J be a Jordan algebra with radical EKJ).

J is said to have (finite) radical capacity, n, if J/EKJ) has

 

finite capacity n. J is completely primary, if J has radical

 

capacity 1.

Lemma 4.3.1. Let J be a Jordan algebra. Then the following are
equivalent:

(1) J is completely primary.

(2) 3 = J/RU) is a division algebra.

(3) EKJ) is the set of noninvertible elements of J.

Proof. See p. 49 and Proposition 2.6.1.

If J is the completion Of the local Jordan algebra y,
then J is a local Jordan algebra (Theorem 3.3.1). Consequently,
J1= J/EKJ) is simple (PrOpOSition 2.6.1), 31 satisfies axioms (1)-
(iii), and there exists n E N such that J has radical capacity
n (Proposition 2.7.15). Thus, the structure Of J. is completely
determined by the Second Structure Theorem (Proposition 2.7.16). The
several possibilities are listed according to the radical capacity
n of J.
I (n = l). J. is a Jordan division algebra.
II (n = 2). J. is isomorphic to a Jordan algebra of a nondegenerate

symmetric bilinear form in a vector Space V over an extension

field F/Q with dim V/F > 1, or a Jordan matrix algebra

83

©c32,ja), where (D,j) is an associative division algebra
with involution or .D:2 A C>A°, A an associative division
and j the exchange involution.
III (n = 3). J. is isomorphic to a Jordan matrix algebra
©(&3,ja), where (B,j) is an associative division algebra
with involution,.D a A(g A0, A an associative division algebra r
and j the exchange involution, or an Octonian algebra with E
standard involution over an extension field f/Q. 5
IV (n 2 4). J is isomorphic to a Jordan matrix algebra Egbh,ja), 1
where (D,j) is an associative division algebra with involution L
or .D a A 69 A0 , A an associative division algebra and j the

exchange involution.

Since J has finite radical capacity n, J1 contains a
supplementary set of (strongly) connected idempotents (2,,gé,...,gg}.
By Theorem 4.2.2 and Theorem 4.2.3, we may aSSume that
{e1,e2,...,en} is a Supplementary set of (strongly) connected
idempotents in J. If n = 1, then J is a completely primary
local Jordan algebra (lemma 4.3.1). If n = 2, then
J

OJ22 C>J (Pierce decomposition relative to {e1,e2}),

= J11 12

where J11 and J22 are completely primary local Jordan algebras
and J12 is a subSpace of J. This follows, Since {$11321 a

set of completely primitive idempotents implies that Jii is a

div181on algebra, i = 1,2, and thus, Jii = Jii/Jii n EKJ) is
semi-simple, which in turn implies Jii fl EKJ) = EKJii) (Lemma
2.7.1 and lemma 3.3.1). There does not seem to be more that can
be said for this case, because of the lack of a coordinatization

theorem for n = 2.

84

If n 2 3, then J asQCDn,ja), a Jordan matrix algebra,
and if 3 = «9/771: where Q(ewnqan 1:" 771“ n ’QCB‘nLia), then
Ja- 326111.12) with E = diag 121’32"°"3n1 for
a = diag {a1,a2,...,an} (Proposition 2.7.13). Of course C§,j)
is characterized by III and IV. Also, Since J is a local Jordan
algebra, Q(awn’ja» is the unique maximal ideal Of Sgwn,ja).
Therefore, by Proposition 2.7.10, W2 must be the unique maximal
ideal of CD,j).

Since Q(bn,ja) is Jordan and n 2 3, LD,j) is associative
or n = 3 and w,j) is alternative with S)(,D,j) CNw) (Pro-
position 2.7.9). Hence, if (D,j) is associative, then 1m =,£(&)
(Proposition 2.7.14). Suppose c&,j) is alternative. If legs)
is the Smiley radical of .D, then 771 C RED) by the same proof
as in Proposition 2.7.14. Since mmj = (9(3) and 771 is the
unique maximal ideal of (,D,j), we have 771 = E0).

Following the next two lemmas, we will be in a position
to establish the main structure theorem.

Lemma 4.3.2. If J is a subalgebra of m+, m an associative

k-l
(k)CK3 ’ k=1,2,ooo o
(k) 3k-l
MOreover, if m is commutative, then K = K , k = 1,2,... .

algebra, and K is an ideal of J, then K

3
Proof. If x,y E K, then Ux(y) = xyx E K . Hence, K(2) = UK(K)

C K3. Since K(k+1) = (10), the first inclusion follows

U K
Kao‘
by induction. Now assume m is commutative, and let x,y,z E K.
1 1 1
Then xyz = 2 (xyz + zyx) 2 UX’Z(Y) 2 [UX+Z(Y) ' UX(Y) ' UZ(Y)]
E 1((2) . Thus, K3 C K(2) . The remaining inclusions follow by

induction as in the first part.

85

Iema 4.3.3. Let J = bwn’ja) be a Jordan matrix algebra of

(k) 3k-l
order n23. If (,D,j) is associative, then Q(J) = (771 ) OJ
m

n

for all k E N and fl Q(J)(k) = 0 if and only if H 771k = O.
k=l k=l
If (,D,j) is alternative and Q(J)(k) = A(k)r1 O J for all

k en, then (1 (2mm = o if and only if n A(k) = o.
k=1 k=1

Proof. Since f(J) (k) is an ideal of J = 8368“,;18) for all
k = 1,2,..., (Proposition 2.2.3), for each k E N there exists
an ideal A(k) of (.8,j) such that Q(J)(k) = A(k) (1 J (Pro-

position 2.7.10). In this notation A(l) = 771, and in either case

we have n EKJ)(k) = n [A(k)n n J] = [ n A(k)n] n J
1

k= k=l k=1
= [ n A(k)]n n J. Now, if B is an ideal of (b,j) and
k=1

co

0 # b e B, then b[12] # 0 e Bn 0 J. Thus Bn 0 J 0 if and

co
only if Bn = 0. Thus it follows that O 8(J)(k) 0 if and
k=l

CD
only if (1 A(k) = 0.
k=l

Assume that (,D,j) is associative. First note that 771
an ideal of (,D,j) and j an involution implies that 771k is an
ideal of (,B,j) for all k E N. By Proposition 2.7.10,

A(k) = {d e .19 : am] 6 (2mm, i,j = 1,2,...,n} for

each k e N. If d e A(k), then d[12] e A(k)n n J = 90,00
k-l k-l
c «n3 = (7711 n J)3 by Lemma 4.3.2. Thus d[12] is equal

. . k-l ,
to a finite sum of products of 3 matrices each of whose

entries are in 771, and hence each entry of d[12] is a finite sum
of products of 3k"1 elements of 771. In particular then, the
k-l
(1,2) entry of d[12], which is d, is in 7713 . Hence
3k-l
A(k) C771 .

86

We will use induction on k to establish the reverse

k-l
inclusion. If k = 1, then A(l) =‘Wb ASSume that W1 C A(k)
for k21. Ifit d ,ooo,d ,d ’00.,d ,d 9".)d em,
1 p1 p1+l p2 p2+l p3
k-l k-l k
‘where p1 = 3 , p2 = 2.3 , and p3 = 3 . Let X = d1...dp1:
k-l
y = d d , and Z = d d 0 Then x,y,z E W? 2

p1+1... p2 p2+1000

p3
and.therefore, by the induction hypothesis, x,y,z E A(k). By
.JMAl (p. 42), and Proposition 2.4.1, xyz[l3] = 2xy[12].2[23]

= 4(x[l3].y[32]).z[23] E (EKJ)(k))'3 = EKJ)(k+1), since

x[13],y[32],z[23] E A(k)n H J = RKJ)(k). Hence dldz...d

P3
= xyz E A(k+l), and SO W1 C A(k+l). By induction W1 C A(k)
for all k E N, and so A(k) =7713 for all k E N.
From the first part we now have that 0 EKJ)( = 0 if
k=l
” 3k'1 ” k
and only if FIW1 = 0. Since F1W1 ==0 if and only if
m 3k-1 k—l k-l
fl W1 = 0, the remainder of the lemma follows.
k=l

It is immediate from the setting of Lemma 4.3.3, that if

C&,j) is associative, then m EKJ)(k)

k=1

3
fl Rw)k = 0, Since 7)1 = RED) by Proposition 2.7.14.
k=l

= 0 if and only if

We can now establish the main Structure theorem. Note
that the two cases previously considered are also included for

the sake of completeness.

Theorem 4.3.1. Let g be a local Jordan algebra with completion
J, and let J have radical capacity n E_N.
(1) If n = 1, then J is a completely primary local Jordan

algebra, and EKJ) is the set of noninvertible elements

87

of J.

(2) If n = 2, then J = J <3 J2<3 S, where J and J are

l l 2

completely primary local Jordan algebras with identities
e1 and e2 reSpectively Such that l = e1 + e2, and S

is a subspace of J.

(3) If n 2 3, then J 25©(0h,ja), a Jordan matrix algebra, and

4:135!”

J/xem = 32.- sans.) = 82(CB/7R)n.j_). where marina,»
a a

1-.‘( 'sh‘!.":-_,-, ot‘ -.l.

==Wh.“ @(Bh,ja). (B,j) admits the following possibilities:

(i) (D,j) is a (not associative) alternative algebra with

 

involution and identity over an extension field of Q
such that bw,j) C N(.&). 771 = [8(5) is the Smiley
radical of .D, and C§,j) is either an Octonian algebra
with Standard involution or a simple Artinian algebra
with involution. In the first instance 3 is simple
and E19) is the unique maximal ideal of .B. In either
case ((8) is the unique maximal ideal of (&,j).

(ii) (D,j) is an associative algebra with involution such
that .D is a semi-local algebra and E0) is the
uniQue maximal ideal of CD,j).

If n 2 3 and Exg)(k)* = A(k)n fl bcbn’ja) for k E N, then

.3 is complete in the topology induced by the decreasing sequence

of ideals ,QCD) = A(1) 2 A(2) 2 ... , where we define A(O) =.D.
Proof. (1), (2) and the first part of (3) were established in the
discussion preceeding Lemma 4.3.2. In (3-i) we have not yet Shown
that 3 simple implies that ,QQB') is the unique maximal ideal of B.

Since the proof is identical to the Jordan case (Proposition 2.6.1),

88

it will not be repeated here. From (3-ii), it remains to be shown
that if n 2 3 and (B,j) is associative, then .3 is a semi-
1ocal algebra. By a semi-local algebra m we mean an associative
algebra with identity such that (a) m has only a finite number of

«no
maximal ideals, (b) EKM) is a nucleus for m; i.e., fl gKm)k = O,
k=l

and (c) m/RKM) is Artinian. For .3, (b) follows from Lemma 4.3.3
00

m

and the fact that O RKJ)
k=l ,_

proved. By III and IV (p. 83),.D = A, an associative division

0, and (c) has previously been

0
algebra with involution or .D A(D A , A an associative division

algebra and j the exchange involution. From the discussion in

§l.2, it is clear that in either case .3 ‘has at most two ideals.
Thus .3 has at most two ideals which contain RED). Since every
maximal ideal of .8 must contain 9w), 3 has at most two maximal
ideals. Note that this also yields that ,D is a local algebra
(not necessarily right Noetherian) if and only if .& is completely
primary in the sense of Jacobson [4].

We will now Show that .5 is complete in the topology

induced by the ideals A(k), k = 0,1,2,... . In what follows J
«:0

is identified with bcbh’ja)' By Lemma 4.3.3, 0 A(k) = 0, since
k=l

°° (10*
n EKg) = 0. Thus .5 is actually a metric Space. Let 1d,}
k=1

be a Cauchy sequence in .D, and let k E N be given. Then there

exists N E N Such that i ,i 2 N implies d, - d, E A(k).
1 2 11 12

Since A(k) is an ideal of (p,j), d1 , d1 6 A(k). Now for any
1 2

x E .9, x[12] E bwn’ja)’ and hence for i1,i2 2 N, d11[12] - d12[12]

= -1 j _ -1 j (k)* .
dilelz + a2 d1111821 d12612 ' a2 (112611821 6 1"?) , Since

89

- J' '1 j .
di ,di ,a2 di a1,a2 di 81 E A(k). Thus {di[12]} 18 a Cauchy
1 2 l 2
sequence in awn’ja) = J, and since J is complete relative to

(10*

the topology induced by the ideals Q(y) , there exists

qu E .5, p,q = 1,2,...,n, such that 1im di[12] = 2 que . There-
1 qu
fore given any k E N, there exists N' E N such that i 2 N'

. . _ <k)* _
implies di[12] Z‘. quepq E Q(J) , and so (11 x12 E A(k).

Thus lim di = x12 E .3 and ,8 is complete.
i

90

4.4. Complete Local Jordan Algebras

Definition 4.4.1. A Quadratic Jordan algebra J over a commutative
associative ring with identity is said to be complete, if EKJ)

is a nucleus and J is a complete metric Space with respect to

the Q(J) -topo logy .

If J iS a complete 10C31wJordan algebra, and J*

(k) (10*

I u.(ew

is the completion

71‘
of J, then J = J , and fiKJ)

= go) for all k e N.

Hence we can apply Theorem 4.3.1 to obtain

 

Theorem 4.4.1. Let J be a complete local Jordan algebra of i
radical capacity n. Then:
(1) If n = 1, J is a complete, completely primary local
Jordan algebra and EKJ) is the set of noninvertible
elements of J.
(2) If n = 2, then J = J

®J2®S, where J ,J ,S are

l l 2

as described in Theorem 4.3.1.
(3) If n 2 3, then J a: bwn’ja)’ a Jordan matrix algebra
such that momma» = 909)“ n scandal), and
(i) (B,j) is a (not associative) alternative
algebra as in Theorem 4.3.1 (n = 3 only) or
(ii) (B,j) is an associative algebra with involution
which is a complete semi-local algebra Such that
,QCD) is the unique maximal ideal of Cb,j).
Conversely, if J is a complete, completely primary Jordan algebra,
or J = ©CDh,ja), a Jordan matrix algebra of order n 2 3 with

CD,j) as in. (3), then J is a complete local Jordan algebra.

91

Proof. (1), (2) and (3-i) are immediate from Theorem 4.3.1.
(3-ii) is immediate once it is noted that in the notation of
Theorem 4.3.1, A(k) = Emb) k-1 by Lemma 4.3.3.

For the converse, first assume that J is a complete,
completely primary Jordan algebra. Then EKJ) is a nucleus for
J, J/EKJ) is a Jordan division algebra so that EKJ) is the
unique maximal ideal of J, and J/EKJ) trivially satisfies the
minimum condition. Thus J is a complete local Jordan algebra.

Next let n 2 3, J = ©(fin,ja), and ,9(J)<k)

= A(k) H J,
where A(l) = RED). We will prove the completeness of J for
g0,j) associative only, but it will be clear how the proof can
be adjusted for the alternative case. So assume .3 is an
associative semi-local algebra which is complete in the,QcD)-
topology, and 18(5) is the unique maximAI ideal of CD,j). Then
Q(J) '-" RC3)“ G J is the unique maximal ideal of J and Q(fi)

a nucleus for .8 implies EKJ)iS a nucleus f0r~1 (Lemma 4.3.3).
Since CDLQCb),j) is a simple Artinian algebra with involution,
and J/,€(J) 2 $(3n,j_) by Corollary 2.7.1, J/,?(J) satisfies the
minimum condition byathe Second Structure Theorem (Proposition

2.7.16). Thus J is a local Jordan algebra.

Let {Xr} be a Cauchy sequence in J. Then for each

(I) (r)

r EEN, X = 2 x,, e,,, x,, 6.8. If k is given, then there
r . 13 1] 13
1,1
exists N EEN such that r1,r2 2‘N implies X - X E EKJ)(k)
k 1 r1 r2
3 -
= A(k)n n J = E%9)n fl.J (Lemma 4.3.3). Thus when r1,r2 2 N,
(r1) (r2) k-l
x - x E 8C5) . Therefore, if p EN, choose k such
ij ij k 1
k-l (r1) (r2) 3 ' p
that 3 2 p. Then for r1,r2 2 N, xij - xij E [3(3) C 3(3) '

U

92

i.e., for each pair i,j = l,2,...,n,{x§§)} is a Cauchy sequence
in 3. Thus for each (i,j) there exists xij E3 such that

lim x§?) = x... and hence for any p E N there exists N.. E N
r 13 13 13

such that if r 2 Nij’ then xij - XEE) EIQL3)p. If k E N,
k-l

let = 3 . Then for r 2 max {Ni : i,j = 1,2,...,n},
:(r) 3k-l 3
ij Xij G 9(3)!) = 19C” for every i,j = 1,2,...,n. Thus,
if x= 2x,,e,,,then‘ x-x EE(J)(k),and limX =xe.&.
i,j 1] 13 r r r n

It remains to be shown that X E QCD; ,j a) = J. Now for all r E.N,

Xr E 9C9n,ja), and thus XI. = ?[ij]+ x(r £)[ii]. Since
1<jx i
x [ij]+ +'l§ xi i[ii] - Xr = 2 (x - x Hr )[i' +‘l 2 (x -x(r))[ii]
. . ij 2213' ij 3] 2 . ii ii ’
1<J L<j 1
X = lim Xr = ij[ij] +’-2 xi i[ii] E bc3n ,j 8). Therefore J
r i<jX

is a complete local Jordan algebra.
If n = 3 and (3,j) is alternative with fifib) the unique
maximal ideal of c3,j) and EL3)n n J = EKJ), then 0°,€>(J) is the

Q
unique maximal ideal of J. If D A(k) = 0, then H :EKJ)(k )= 0:

k=1
so that EKJ) is a nucleus for J. If .3 is complete in the

topology induced by the A(k), k = 0,1,2,..., then J is complete
in the EKJ) topology as in the associative case. Since J/EKJ)
2 96nd), where 3 = films), and (3,j) is either an Octonian
algebra :ith standard involution or a simple Artinian algebra
with involution, J/EKJ) satisfies the minimum condition by the
Second Structure Theorem. Thus J is a complete local Jordan

algebra.

The concepts of commutative associative algebra with identity

and associative (linear) Jordan algebra coincide over a field of

 

93

(k)

characteristic not 2. If m is such an algebra, then fiKu)
= EKM) k-l, k.E N, by lemma 4.3.2. Thus m is a not-necessarily
Noetherian local associative algebra if and only if M is a local
Jordan algebra. m is a metric space with reSpect to both the
associative and Jordan EKM) topologies. Suppose {xn} is a
Cauchy sequence in m relative to the associative EKm)-t0pology
and assume m is complete with reSpect to the Jordan EKm)-topology.
Then for any k E N, there exists N E_N such that m,n 2 N

implies xm - xn E fiKm)k. Hence given k' E;N, there exists

3"”1 (k')
N E N such that m,n 2 N implies xm - xn 6 RT”) = EKM)

and {xn} is Cauchy in the Jordan topology. Thus there exists

x E m such that lim xn = x in this topology. That is for any

n
k EN there exists N E N such that n 2 N implies x - xn
k-l
k -
EE<91><> =I?(91)3 . Thus if pEN, and 3k12p,x -Xn
e Rwy) : gmm)p for all n 2 N. Hence 1im xn = x in the

n
associative topology, and m is complete in this topology. The

converse is equally clear. Therefore 3 is a complete local
(associative) algebra if and only if m is a complete local Jordan
algebra. Note that in this case m has capacity 1.
We close this section with several questions which remain
open.
1. Is there a corresponding local algebra theory for
alternative algebras? The difficulty here lies in
finding ideals correSponding to Ak in the associative

(k)

case and A in the Jordan case.
2. Can the structure for radical capacity 2 be more

completely determined? This problem seems to be

 

94

laden with difficulties.
Under what conditions is the completion J of a
local Jordan algebra g itself complete? A trivial

condition is that f(g*)(k) = £19)(k)*.

 

B IB LIOGRAPHY

*1“; 7; ‘- 2.71:ij

10.

ll.

12.

BIBLIOGRAPHY

Batho, E.H., Non-commutative semi-local and local rings,
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Cohen, 1.8., On the structure and ideal theory of complete
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Herstein, I.N., Lie and Jordan structures in simple associative
rings, Bulletin of American Mathematical Society, 67 (1961),
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Jacobson, N., "Structure of Rings", American Mathematical
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Mathematical Society, Providence, 1956.

, "Structure and Representations of Jordan Algebras",
American Mathematical Society Colloquium Publications, Vol.
XXXIX, American Mathematical Society, Providence, 1969.

_________, "Lectures on Quadratic Jordan Algebras", Tata
Institute, Bombay, 1970.

McCrimmon, K., A general theory of Jordan rings, Proceedings
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Nagata, M., "Local Rings", Interscience Tracts in Pure and
Applied Mathematics, No. 13, John.Wiley and Sons, Inc.,
New York, 1962.

Schafer, R.D., "An Introduction to Nonassociative Algebras",
Academic Press, New York, 1966.

Smiley, M.F., The radical of an alternative ring, Annals of
Mathematics, (2) 49 (1948), 702-709.

Tsai, C. and Foster, D., Primary ideal theory for quadratic
Jordan algebras, to appear.

Zariski, 0. and Samuel, P., "Commutative Algebra", Vol. II,
VanNostrand, Princeton, 1958.

95

 

    

 

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