THES'i- This is to certify that the thesis entitled Bochner Property in Banach Spaces presented by Uttara Naik-Nimbalkar has been accepted towards fulfillment of the requirements for Ph.D- degree in Mathematics W rr’ ' Major professor Date___Ju_]_y_26,_]_9.ZSl_ 0-7 639 LIBRAR Y " , Michigan Starr . n. “' "“ 5 rain" ‘ .3} Um] T L) “M- i'WW-w "In-v I' ...- -‘ OVERDUE PINES ARE 25¢ PER DAY PER ITEM Return to book drop to ranove this checkout from your record. BOCHNER PROPERTY IN BANACH SPACES BY Uttara Naik—Nimbalkar A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1979 ABSTRACT BOCHNER PROPERTY IN BANACH SPACES By Uttara Naik-Nimbalkar In this thesis we study a class of Banach spaces in which some analogues of the following theorems hold. (1) (Bochner's Theorem) A function m defined on Rn is the characteristic functional of a probability measure on Rn iff m is positive definite, m(0) = l and it is continuous in the usual topology. (2) A necessary and sufficient condition for the tightness of a family {uo.:a'€ I} of probability measures on Rn is the equicontinuity of the corresponding character- istic functionals {30] at zero. . Let E be a real separable Banach space with metric approximation proPerty, and let E’ be its topological dual. D.H. Mnstari has shown that embeddability of E into a space of random variables is a sufficient condition for the validity of Bochner's Theorem in E. A simpler proof of this result is given by first proving an inequality analogous to the finite dimensional one which connects the Uttara Naik-Nimbalkar behaviour at zero of a characteristic functional with the probability of compact sets. A result of Lindenstrauss and Pelczynski is used to show that the embeddability condition is also necessary if E is further assumed to be of stable type_ p. For such spaces we also give an explicit form of the tOpology on E’ that is associated with the Bochner Theorem. For this we use some results of deAcosta and V. Mandrekar. It is known that the validity of Bochner's Theorem implies that the space is of cotype 2.- Results of Maurey and Rosenthal are exploited to show that the converse is not true. This also giVes cotype 2 spaces which are not embeddable. Finally, in Section IV, we consider (2). We denote by $2 the topology on E’ generated by characteristic functionals of Gaussian probability measures. Sazanov's result shows that for real separable Hilbert spaces, the equicontinuity in ‘?2 of characteristic functionals of a family of probability measures implies the tightness of the family. It is shown that the validity of this result, in fact, characterizes the cotype 2 spaces. TO MY PARENTS ii ACKNOWLEDGMENTS I wish to express my deep gratitude to Professor V. Mandrekar for his patient guidance and encouragement throughout the studies leading to and during the preparation of this dissertation. Also, I thank Professor J.C. Kurtz, C. Tsai, D. Luecking and R. Hill for reading this thesis. I am also indebted to Mary Reynolds for her excellent and swift typing of the manuscript. Finally, I am grateful to the Department of Mathematics, Michigan State University for the generous support. financial and otherwise, during my studies for the Ph.D. degree. iii TABLE OF CONTENTS Section II III IV INTRODUCTION PRELIMINARIES AND NOTATION EXAMPLES OF SPACES OF COTYPE 2 WHICH ARE NOT EMBEDDABLE IN L°(Q.u) BOCHNER PROPERTY I BOCHNER PROPERTY II FINAL REMARKS BIBLIOGRAPHY iv Page 15 17 31 4O 41 SECTION 0 INTRODUCTION Let E be a real separable Banach space with (topological) dual E’ and duality denoted by < , >. Let m be a function from E’ into the set of complex numbers C, with m(0) = l and which is positive definite in the sense that for each integer n, Z) Ciajm(yi"yj)‘2 O i,j for ci 6 C and yi E E’ (i = l,2,...,n). The solutions to the following problems have played a useful role in the case where E is one-dimensional. Problem I. Give necessary and sufficient conditions on $ so that m is a characteristic functional (or Fourier transform) of a probability measure u on the Borel sub- sets 5(E) of E; i.e.. o(y) = f expli ]u(dx) A E (= u(y). say). A . . Problem II. Let {ua.:a E I} be the characteristic functionals (ch.f. for short) of a family {ha} of probability measures on. 8(E). Give necessary and sufficient conditions on {Ca :a e I} in order that the family is tight. (For the definitions of "tight" and other concepts used, the reader is referred to Section I.) Bochner's work ([17], p.207 and p.193) provides com- plete answers to the above problems in the case where E is finite-dimensional. Namely, Problem I has a solution iff m is continuous at zero and Problem II has a solution iff {Ga :a 6 I} is an equicontinuous family. It is known that the exact analogue of the above_ solutions fails in case E is an infinite-dimensional Hilbert space ([18],[27]). In this case, however, Sazanov [29] and Gross [10] gave a complete solution to Problem I; namely, Problem I has a solution iff m is continuous in the topology S, generated by the non-negative trace class Operators. In ([25],[29j) it was shown that sufficiency part of Problem II is valid if [Go :a 6 I} is a family of functions equicontinuous at zero in the S-t0pology, but the necessary part is not, in the case where E is an infinite dimensional Hilbert space. In this work, we study a class of Banach spaces in which some analogues of the above solutions hold. To facilitate this we introduce Bochner Property I: E has Bochner Property I if there exists a topology on E’ such that continuity in it of m solves Problem I for every m. Bochger Property II: E has Bochner Property II if there exists a topology T on E’ such that {CO :a G I} is equicontinuous at zero in T implies {ua.:a‘€ I] is tight. From the previous comments we get that the class of Banach spaces with Bochner ProPerty I and Bochner Property II includes all separable Hilbert spaces. It is known that a separable Hilbert space is embeddable in L°(Q,P) [30], and has approximation property (in fact, has a basis). In ([22], Theorem 1) Mustari has shown that (0.1) All embeddable (in L°(Q,P)) Banach spaces with approximation property have Bochner PrOperty I and (0.2) All spaces with Bochner Pr0perty I are of cotype 2. \ He raises the question of whether all cotype 2 spaces have Bochner PrOperty I. We show that this is not true by means of examples. These are of general interest as they provide examples of spaces of cotype 2 which are not embeddable. We present them in Section 2. We observe that the main tool of Mustari's proof of (0.1) is an inequality first proved by Levy for which the above two hypotheses of embeddability and metric approximation property seem to be tailor-made. In order to bring this point across we give, in Section 3, a simple proof of Mustari's result, based on What we now term as a Levy Inequality. This proof shows that one can characterize subclasses of Banach spaces having Bochner Property I with respect to the smallest topologies making ch.f. of certain probability measures continuous. We give such a characterization of Banach spaces embeddable in Lp using t0pologies associated with ch.f. of symmetric stable probability measures. In the last section we study the class of Banach spaces having Bochner Property II with respect to topologies gener- ated by ch.f. of stable measures. For the case of stable measures of index p, where l g.p < 2 we only get a partial characterization. However for p = 2, i.e., in the case of Gaussian measures we get that a Banach space has Bochner ProPerty II iff it is of cotype 2. This also shows that Bochner Pr0perty I is different from Bochner ProPerty II, since by [22] we know that a Banach space has Bochner Property I with respect to the Gaussian topology iff it is isomorphic to a Hilbert space. We conclude by giving some final remarks and Open problems. For a review of other methods used in solution of Bochner Problems I and II see [18]. SECTION I PRELIMINARIES AND NOTATION This section contains a description of the tools used in later sections. We also fix the notation used throughout. A 'Banach space' E will mean a real, separable, complete normed linear space with norm H HE' We will denote its topological dual by E’. A measure space is a pair (0,“), where u is a countably additive, non-negative measure on some o-algebra of subsets of Q. It is a probability space if u(0) = l. L°(Q,u) is the equivalence classes of real valued measur- able functions where functions, equal u a.e., are identified. (Throughout this work we consider L°(Q,u) with u a probability measure and the t0pology to be that of convergence in probability). For l‘g p < a, Lp(Q,u) is the Banach space with norm H-Hp consisting of all f 6 L°(Q,u) such that nw§=flflpmll = (331 wimp”. 1 g p < ., Definition 1:1. A space E is said to be embeddable in a linear metric space P, if there exists a linear topological isomorphism of E into F. Lemma 1.2. A separable (closed) subspace G of an infinite dimensional LP(Q,u), for an arbitrary measure u: can be embedded in LP, 1 ghp < a. Proof: By ([6], Lemma 5, p.168 and [11], Theorem C, p.173) we obtain that G is embeddable in LF 6 LP. How- ever zp embeds in Lp ([16], p.133). Thus Lp Q 2p embeds in LP, giving the result. Definition 1.3 [20]. Let (Q,u) be a probability Space. Then E is said to be strongly embeddable in Lp(0,u), o < p < e, if E is embeddable in LP(Q,u) and the Lp-convergence and the L°-convergence coincide on the image of E. Definition 1.4. We say that a separable Banach space E has the metric approximation property if there exists a sequence [Wm] of Operators of finite rank such that Hwnx-xHE-+ 0 for each x E E. Definition 1.5. A Banach space E is said to be of Rademacher type p if for every sequence {xi}:=l c E Q with Z) Hxin < a we have Z>Xi€i converges a.e. Where i=1 5i are independent identically distributed (i.i.d° for short) symmetric Bernoulli random variables, i.e., _ _ ._ -1 P(ei - l) - P(ei - l) - 2. Definition 1:6, A real valued random variable with ch.f. exp(-|t]P) for t real, is called a symmetric stable random variable of index p (l g,p < 2). For p = 2 it is called a standard Gaussian random variable. Definition 1.7. A Banach space E is said to be of stable type p if for every sequence [Xj}:=l c E with Z)ijHp < a we have Z}x.nj converges a.e., where ”j J are i.i.d. symmetric stable random variables of index p, l g,p g 2. Remark 1.8. Spaces of Rademacher type 2 may be equivalently defined over the Gaussian system, i.e., E is Of Rademacher type 2 if for every sequence [xi] c E with Z)HxiH2 < m we have Z)xiyi converges a.e., where Yi are i.i.d. standard Gaussian random variables. Thus Rademacher type 2 is equivalent to stable type 2 and we refer this case as of type 2. For p # 2 we have the following Lemma ([21], p.79). Lemma 1.9. (i) E is of stable type p implies E is of Rademacher type p. (ii) E is of Rademacher type p implies E is of stable type q for all q < p. Definition 1&19. E is said to be of cotype 2 (Rademacher) if for every sequence [xi] c E satisfying ZZXiei converges a.e. we have Z)HxiH2 is finite. 0r equivalently ([21], Thms. 1.1.1.2 and Cor. 1.3) the space is of cotype 2 if a.e. convergence of ZZXiyi implies ZZHXiiz is finite, where Yi are i.i.d. standard Gaussian random variables. The following lemma shows that an analogous definition of stable cotype _p, 1 g_p < 2 does not restrict the class of Banach spaces. Lemma 1.11. If nj are i.i.d. symmetric stable random variables of index p, 1 g_p < 2, then the a.e. convergence of Z)xjnj implies Z}ijup is finite, for {xj};=1 is any Banach space E. 2529;: First note that ZEHXij converges iff Z)P(|njl >-W%;U) converges, ([7], p.544). But 2 Pqnjx > W) = § P 1) which is finite by Borel-Cautelli Lemma, since ZDnjxj converges a.e. implies HnjxjH 4 O a.e. We also need some results on the weak convergence of measures on linear spaces ([4]). we start by defining cylinder measures ([3]). Let G and F be two Hausdorff topological vector spaces in separating duality, with the duality denoted by < , >. Let M be a finite dimensional subspace of F, .L M the annahilator [xlx e G, = 0 for all x’ e M] and let GM.= G/ML. Then GM is finite dimensional. If M.l g.M2 are finite dimensional subspaces of F, we denote by p the projection of G onto G . 6(GM) will denote the o-field of Borel subsets of GM and the canonical map of G into GM. We note that 1 and call U p;l(B(GM)) = a the algebra of M pM G = p; (G M) cylinder sets. Definition 1.12. A cylinder measure u on G is a family of probability measures {uM::M a finite-dimensional (o(G,F) closed) subspace of F} such that “M is a probability measure on 6(GM) and if Ml‘gM2 then -1 P o Ml'Mz O l 2 Given a cylinder measure we can define a finitely additive measure u on (J p&106(GM)) by u(p&l(A)) = uMlA) for A 6 5(GM). anversely given any finitely additive measure u on (J p;}LB(GM)), with u(G) = 1 M and u restricted to p&}(B(GM)) countably additive, then the family {pm = u ° pgl} gives a cylinder measure. In particular, any countably additive measure u on U P;1(B(GM)) gives a cylinder measure. M 10 Definition 1.13. Given a cylinder measure u on -1 G, then for each x’ E F, u 0 x’ is a probability measure. we define the ch.f. of u on F as A (X’) = e u IR 15 u o XI-l(d5). Proposition 1.14 ([3], p.19). The ch.f. defines a one-one onto map between cylinder measures on G and comple- valued positive definite functions m on F ‘with restriction to each finite dimensional subspace continuous and m(0) = 1. DefinitiOn 1.15. Given a topological vector space F, a linear continuous map Y of F into L°(Q,u) (with (Q.u) a probability space) is called a random linear functional. Remark. Without loss of generality Y(x) is assumed to be symmetric and of the form U§(x) where I is an embedding and U is uniform on [-1,1] and is independent of [?(x), x E E]. If G and F are in separating duality and Y is a . . A _ i‘Hx) random linear functional on F, then PY(X) - I e u(dx) 0 defines a continuous complex-valued positive definite function on F and hence a cylinder measure on G ([3], Expose no. 2). If further Y is an embedding and F is a Banach space then we get, given a > 0 there exists an h(e) > 0 such that A HXHF > 8 implies I1-PY(X)] > h(€). 11 Remark 1.16. Let E be a separable Banach space with E’ its topological dual and < , > denoting the duality. Then 6(E) is the o-algebra generated by G (the algebra of cylinder sets), and hence by Pr0position 1.14 we get that the ch.f. determines u uniquely. A separable Banach space E isa complete separable metric space. In order to study convergence of measures on E, we need some general concepts and results on the weak convergence of measures on the Borel subsets 5(8) of a metric space S [4]. Definition 1.17. A sequence Of probability measures [uh] on. 5(8) is said to converge weakly to the probability measure u on .B(S) if I fdun + I fdu for every bounded S S continuous real valued function defined on S. If un converges weakly to u, we write un = u. Definition 1.l§. A family (“a.:a 6 I] of probability measures on (S,B(S)) is said to be relatively compact if every sequence Of elements {ua 12:1 n contains a weakly convergent subsequence. Definition 1.19. A probability measure u on (S,B(S)) is said to be tight if for each e > 0, there exists a compact set K such that u(K) > l-e. Note that on a separable complete metric space each probability measure is tight ([4], p.10). 12 Definition 1.20. .A family {pa :O.€ I] of probability measures is said to be tight, if for each s > 0, there exists a compact set K, such that ua(K) > 1-e for all a 6 A. The next proposition is due to Proherov, for a proof see ([4], p.37). PrOposition 1.21. -Let S be a separable and complete metric space. Then a family [pa :a.6 I] of probability measures on (S,B(S)) is tight iff the family is relatively compact. We conclude this section by giving the following known facts from the theory of probability measures on Banach spaces. Theorem 1.22 ([12], Theorem 3.1 and 4.1 and [32]). Let {xk}:=l be independent E—valued random variables with n Sn.= §>Xk and “n the distribution of Sn“ Then the following are equivalent. (a) 8n converges a.e. (b) Sn converges in probability. (c) u converges weakly. n If further [Xk} are symmetrically distributed then (a), (b),(c) are equivalent to (d) [on] is tight. (e) There exists an E—valued random variable S such that 4 for every y E E’. (f) 13 There exists a probability measure u on E such that, mu (y) * wu(y) for all y E E’. *n Definition 1:23. A probability measure u on a separable Banach space E is said to be symmetric stable of order p (1 g_p gDZ),‘-if for each y 6 E’, u o y-1 is a symmetric stable measure of order p on the real line R. For p = 2 it is called (symmetric) Gaussian measure on E. Theorem 1.24 ([2],[19], for a proof see [19]). Let E be a Banach space and p E (0.2], then the following are equivalent. (1) (2) (3) E is of stable type p. Every function of the form @(Y) = exP[-fr l|p l(du)] for a finite measure 1 on P, the boundary of the unit ball of E, is the ch.f. Of a (necessarily stable) probability measure on E. Every function of the form ®(y) = eXp[-I I[p v(du)], where [E Hpr wax)E is finite, is the ch.f. of a (necessarily stable) probability measure on E. Definition 1.25. A probability measure u on 3(E) is said to be infinitely divisible if for each integer n, there exists a probability measure ”n on E, so that * n U = Un (here * denotes convolution of measures). 14 Theorem 1.26 ([19]). For a real separable Banach space the following are equivalent: (1) E is of Rademacher type p. (2) The positive definite function given by My) = expif-1-i /1+Hxl|P)F(dx)1 where (a) F is a o-finite measure on E with F({0}) = 0 and F finite outside neighbor- hood of zero. (b) j HxHP F(dx) < co. (0 < p g 2) llelgl is a ch.f. of a non-Gaussian infinitely divisible measure. The above theorems use the concept of "flat concentration" due to deAcosta ([1], p.279). Definition 1.27. A family [“o :g.e I] of probability measures on (E,B(E)) is flatly concentrated if for every ('3 > 0 and 5 > 0, there exists a finite dimensional sub- space M Of B, such that ua[x E Elinf£Hx-z| :zEMlgelzl-s for all g 6 I. SECTION II EXAMPLES OF SPACES OF COTYPE 2 WHICH ARE NOT EMBEDDABLE IN L°(Q,u) We need the following results. Theorem 2.1 ([20], Theorem 98). Let (Q,u) be a probability space. If E embeds in L°(Q,u) then the following are equivalent for 0 < p g_2. (a) E is of stable type p. (b) E can be strongly embedded in LP(Q,u). Theorem 2.2 ([28], p.775). If LP(E) embeds in L1, then E embeds in Lq for all q < p. Lemma 2.3 ([21], p.55). If E is of cotype 2, then so is rpm) for lgpg 2. Lemma 2.4 ([21], Proposition 2.2, Corollary 2.2). If E is of stable type q it is of stable type r for all r g.q and so is £p(E) for all p, q < p. Example 2.5. Choose a real separable Banach space E of cotype 2, stable type q for q > 1, and not 15 16 q embeddable in L O for some qO (l quo < 2). Then for p, with max(q,qo) < p‘g 2, £p(E) is of cotype 2, but not embeddable in L°(Q.u). lggggf: Suppose £p(E) is embeddable in L°(Q,u), then by the assumptions on E, £p(E) is of stable type 1, thus by Theorem 2.1 it embeds in Ll(Q,u). But E is separable, giving LP(E) is separable, thus its image is a separable subspace of L1(Q,u). By Lemma 1.2 we can assume that £p(E) is embeddable in L1. USing Theorem 2.2, we get that E embeds in Lr for all r < p. This contradicts the choice of E. Thus £p(E) cannot embed in L°(Q.u). but by Lemma 2.3, is of cotype 2. The spaces Eq, (1 < q < 2), satisfy the assumptions on E. Therefore for any 1 < q < p g_2, 2p(£q) give examples of the required spaces. SECTION III BOCHNER PROPERTY I We first prove an analogue of Levy Inequality ([5], PrOposition 8.29, p.171) for a real separable Banach space E, embeddable in L°(Q,P) and having metric approximation prOperty. We denote the embedding by y, and by PY the cylinder measure induced on E’ with ch.f. Py. Without loss of generality we can assume that y maps E into the set of symmetric real valued random variables, i.e., PW is real valued. Let the sequence of finite dimensional Operators associated with the metric approximation property of E be (fin, n 6 N]. Denote by FA the transpose of ”n' 'With the above notation we get, Lemma 3.1 (Levy Inequality). Any probability measure H on E satisfies the following inequality, given 6 > 0, there exists an h(e) > 0 such that u[x[H(1rm-1rk)x(| > c) g 31-32-7-IE’U-C((1rm-1rk)'y)]PY(dy). l7 18 'ggggf: From embeddability we get that given 6 > 0, there exists h(e) > 0, such that whenever .HxH > e, 1-PY(x) > h(€) for all x E E. Using this and Chebychev's inequality ulXI [\(TTm-Trk)xll > e} g pix) <1-§Y((wm-vrk)x)) > 11(8)] 1 A g m IE[1 -Py((Fm-Wk)x)]“(dx) l i ,x (Wm-Wk), = He, J‘ [J‘ '<1-e )P‘l’ (dy)]u(dx) E E (Wm-Wk)’ where PY denotes the probability measure on E’ with the finite dimensional support (Fm-Fk)'E’. Since the measures involved above are probability measures and the function is jointly measurable, by Fubini Theorem we get 1 A , u[X[H(wm-wk)xH > a} g_5727 IE [l-u((Wm-Fk) y)]PY(dy). A NOte that PY being real valued all integrals above are real. This completes the proof. Let n denote the set of certain positive definite functions a on E’ such that, (3.2) (i) e is real valued. (ii) 6(0) = 1. (iii) 9 is continuous in the norm topology on E’. (iv) If and 92 belong to u then the 91 product 61 .92 is in n. 19 Let [I = {y E E’]1-e(y )<< 6]- etc Then the system {Ue'€:e E n. e > 0} forms a sub-basis of neighborhoods at zero and generates a tOpology with respect to which exactly all functions in n are continuous. We will denote this topology by T”. The following gives an example of such a family: For pl and “2, two probability measures on B, we denote their convolution by the probability measure A and let “0 = {NIH belongs to a subclass of symmetric probability measures on E, which is closed under convolution]. Then “0 satisfies conditions (3.2). Lemma 3.3. If E has metric approximation prOperty and is embeddable in L°(Q,P) with embedding y, then the continuity in TX of a positive definite function m, with m(0) = 1, implies - wk) ’y)]PY(dy) = 0. lim sup 1im I Re[1-m(wa(wm E! k n m Proof: Since m is continuous in Ta , given a > 0, 0 there exists a 5 > 0 and a symmetric probability measure A v with v 6 x0 such that A Re(l-o(y)) < 3 whenever 1-v(y) < 6. 20 Using the fact that Re(1-m(y)) g,2 we get, for all y E E’ Re(l-m(y)) g_§ (l-C(y))4-€ for all y E E’. Thus I Re[1-o(w’(w , n E 2 3'5 =2 6 =2 6 But Hr x-xH 4 O I m thus lim sup 1im k n m giving lim sup 1im k n m m-Wk)’y)]PY(dy) E A f ’[l-v(wg(Wm-vk)’y)]PY(dy)+-e i ( (l-e IE’ IE )v(dx))PY(dY)-+e A fE(l-PY((wm-wk)wnX))v(dx)-+e. A for all x 6 E and PY is continuous, A [Ell-PY((vm-Wk)an)]v(dx) = 0. IE’ Re[1-—m(wé(wm -Wk)’y)]PY(dy)‘g s. but 8 is arbitrary and thus we get the result. Lemma 3.4. If a function on E’ is continuous in T , then it is sequentially weak-star continuous. Ko Proof: It is enough to show that the ch.f. of any probability measure is sequentially weak-star continuous. This can be shown by using Lebesgue Dominated Convergence Theorem. USing the above lemmas we get the following result which includes a result of Mustari ([22], Theorem 1(C)). 21 Theorem 3.5. If E is embeddable in L°(Q,P) for a probability space (Q,P), and has the metric approximation prOperty, then every positive definite function m, with @(0) = l, and which is continuous in TKO , is a ch.f. of a probability measure on E. ‘ggggf: ‘We use the notations as defined above. By Lemma (3.4) m is sequentially weak-star continuous. Hence its restriction to every finite dimensional subspace of E’ is continuous. Thus there exists a cylinder measure a associated with m (1.14). Let “n = u o #31, then for each n, ”n is a Borel probability measure on E having finite dimensional support vn(E) and Gn(y) = ®(V5(Y))~ Using (3.1) we get the Levy inequality; given a > 0, there exists h(€) > 0 such that un{x[ ”(Wm-Emu > e) g 37]?)‘IEJ1‘CP‘7’A‘T’m By Lemma 3.3 we have lim sup lim I [1"@(Wé(vm"7k)'Y]Pw(dY) = 0. k n m E’ * Therefore lim sup un[x[H(I-Wk)XH > e) k n = lim sup 1im un[x|H(nm-nk)xH > g} = 0. n Thus for e > 0, g’ > 0 there exists a finite-dimensional subspace w (E) of E such that sup u {x[H(I-w )xH:>e}<:g’, nO n n nO giving [“n}neN is flatly concentrated in the sense of (1.27). 22 Now as m is sequentially weak-star continuous and » for each x 6 E, we get that (Cn(y) =)m(nfiy) converges pointwise to m(y). It also follows that restriction of m to each one-dimensional subspace of E’ is continuous. Thus by ([1], Theorem 2.4, p.280) we get the existence A of a probability measure u: such that u(y) = m(y). Corollaryi3.6. If E has metric approximation pro- perty and embeds in 'L°(Q,P), and the topology Tn is 0 such that the ch.f. of every probability measure is con- tinuous in it, then E has Bochner Property I with respect to it. Theorem 3.7. If E has metric approximation property and embeds in L°(Q,P), then E has Bochner Property I. Proof: In Theorem 3.5 we take the set to consist “o of ch.f. of all symmetric probability measures on E. Then the continuity of ch.f. in TKO of every probability measure u follows from.the inequality [l-G(y)[2lg 2(l-Re fi(y)) and the fact that Re G(y) is the ch.f. of the symmetric probability measure v given by MA) = %(H(A)+u(-A)). We next show that for embeddable spaces with metric approximation prOperty and of Rademacher or of stable type p, we can get topologies of the form TK where n can be described explicitly. 23 Corollary 3.8. Let E be of Rademacher type p, embeddable in L°(Q,P) and have the metric approximation property. Let F be a symmetric measure on E satisfying (i) F is o-finite on E with F[0} = 0 and finite outside every neighborhood of zero. (ii) I Hpr F(dx) is finite. Hxllgl Let up = [w :E’ 4 C[¢(y) = exp I (cos -—1)F(dx), for all F satisfying (i) and (ii)]. Then if a positive definite function 0, with 0(0) = 1 is continuous in 'rK , it is a ch.f. of a probability measure P on E. nggf: We note that up is the set of ch.f.‘s of symmetric non-Gaussian infinitely divisible measures on E by Theorem 1.26, and satisfies conditions (3.2). The result now follows from Theorem 3.5. Next we consider tOpologies associated with symmetric stable measures. We will denote by rp (for p 6 [1,2]) the topology TK obtained by taking n = {¢I¢(y) = exp[-f l[p v(dx), as v varies B through the set of measures for which IE llxllp v < e}- Let 7p be the topology Tn obtained by taking n to be the set of ch.f.‘s of all symmetric stable measures of index ~ p on E. Then in general, Tp is weaker than Tp [33], 24 but in the case where E is of stable type p, Tp coin- cides with 7r'p by Theorem 1.24 and r’ is a topological vector space under Tp ([30], p.125). For the sake of abbreviation we say that E has B(I,p) if it has Bochner Property I with respect to the topology T . We note that (Theorem 1.24) if E has P B(I,p) then E is of stable type p. Corollary 3.9. If E is of stable type p, embeddable in L°(Q,P) and has metric approximation prOperty, then E has B(I,p), 1 g p g 2. Proof: Since E is Of stable type p, Tp topology coincides with Tp. Therefore by Corollary 3.6 it is enough to show that every ch.f. is continuous in Tp. Suppose m is the ch.f. of a probability measure u on E, i.e., w(y) = [ em“x> u(dX). E Then ll-Cp(y)lz g 2 Re<1-cp)u(dX). E Since u is a probability measure on E, given a > 0, there exists a compact set K.C.E such that u(K) > 1-€/2, (1.19). Therefore 2) (l-COS )u(dx) g 2‘]. (1-cos )u(dx) + e E K :4)" llp .u(dX)+e. K lips?- 25 Let V be the measure on E such that for all A E @(E), v(A) = u(K[]A). Thus I )1pr New = J” 1)an mm s sup qup - MK) < ., E K xek and Il-cp12 g 4J‘E1lp wax) +6. Hence' w is continuous in Tp. It is well known that m is positive definite and m(O) = 1. Remark 3.10. The fact that Tp gives a necessary topology is true without any assumptions on E. Corollary 3.11. If E is of Rademacher type p, l < p < 2, has metric approximation prOperty and is embeddable in L°(Q,P), then E has Bochner PrOperty I with respect Egggg: By Corollary 3.8 it is enough to show that ch.f. of any probability measure u is continuous in TK Since E is Rademacher p, it is stable q for allp q < p, hence E has B(I,q) by (3.9). Thus given 6 > 0, there exists a symmetric stable measure V of index q and a 5 > 0, such that Re(1-—G(y)) < 6 whenever l-C(y) < 5. We show that 0 belongs to x . It is known [33] P A that v(y) = exP[-f [[q x(dx)] where x is a finite T measure on the boundary r of the unit ball of E. Since 26 m _ ._Jl_ = _ q [0 (cos ts 1) sl+q ds It] for a constant cq (0 < cq < m) ([5], p.205), we get C A - Q v(Y) - eXplfrjo (cos s-l) gig; ds i(dX)]. Identify E = r x [0,m) and define measure on B(E) as in ([19]: p.323) by ds F1(A) = [TIC 1A(x.s)cq sl+q 1(dX) and 32(2).) = f [0 1A(x.-s) —-d—§-;—i-1)F(dx). E .. /\ . . . giVing V E ”p' Thus a is continuous in T . . KP To prove the converse of Corollaries 3.9 and 3.11 (without metric approximation property) we need a result of [15]. Definition 3.12 ([15], p.282). Let X,Y be Banach spaces, B(X,Y) the set of bounded linear operators from X into Y. Define 27 d(X,Y) = inf{HTHHT-1H]T e B(X,Y)]. (If X and Y are not isomorphic then d(X,Y) = e). Theorem 3.13(l) ([15]. p.313). Let X be a Banach space, for p 2_1, A g,m there exists a measure V and a subspace Y of Lp(V), such that d(X,Y) g_x iff whenever for all y 6 X’ n P P n m i§1ll 2 :31 [Y v J>1 [ui}i=l. {Vj}j=l e X then AP 2 lluillpz E ijH". i=1 j=1 Theorem 3.14. If E has B(I,p), 1 g,p g_2, then B is embeddable in Lp(Q.u) for some measure u. Proof: Let 1 g_p < 2. Suppose E has B(I,p) but is not embeddable in a LP(Q,u). Then for every subspace Y of Lp(Q,u) and any integer k‘z 1, d(E,Y) > 2k. Hence by Theorem 3.13, for each K, there exists finite sequences [ u?) 2:1 and [vg]::1 in E, such that for all y e E’ r “k p p Z Il 2 21 [[ but 2 i=1 j=1 (3.15) ”k “‘k 2kP .23 nuli‘np < z Hvljfllp. i=1 j=1 (1)1 thank Professor A. Weron for bringing this result to my attention in this context. 28 Without loss of generality we can assume that. Z) HV§HP = l. mk n=1 as (Since if .2) HV§HP = Mk’ we can replace v? by W? = Miip 3:1 k u. k k and ui by xi = lip . Then (3.15) holds for the finite k “k k mk sequences [xi}i=1 and [wj}j=1 mk with z: Hw‘.‘||P= 1). i=1 3 We note that Z} Z [[utup < Z -T]é—< co, and ' k=1 2 P =1 1: Z) Hkap = m. Thus we have two sequences [u.}?_ k=1 j=1 3 1 1‘1 and {Vi]:=l such that for all y E E’, Z l[p 2 Z [lp i=1 j=1 3 Z ““1” < °° and 2 ijHP = °°° 'i=1 . j=1 We have already noted that since E has B(I,p) it is of stable type p. Thus by definition of stable p, ZZHuin is finite implies that Z3uini converges a.e., for {Bi} i.i.d. symmetric stable random variables Of index p. Thus there exists a probability measure u on .E such that .21.:1! l P 3(y) = e for each y E E’. Moreover u is symmetric p-stable measure on E. USing this and the first inequality in (3.16) we can show that the positive definite '25.:1' l P function @(y) = e is continuous in Tp. Also @(0) = 1. Thus by the hypothesis, there exists a probability measure V on E, such that ”20?... ’l P C(y) = o(y) = e 1‘1 l . 29 By Theorem 1.22 we get, Z}Vini converges a.e. for [nil i.i.d. symmetric p-stable random variables. Therefore 2: HVin < e by Lemma 1.11. which contradicts 3.16. This gives the result. For p = 2: T2 is generated by symmetric bilinear forms, hence E is isomorphic to a Hilbert space ([23]), thus embeds in L2. Remark 3.17. USing Lemma 1.2 we obtain that E is embeddable in L? if it has B(I,p). However ([30]) we know that LP, 1 g,p g_2 is embeddable in L°(Q,P) for some probability space (Q,P). Thus we obtain the following result. Theorem 3.18. Let E have metric approximation property. Then E is of stable type p and embeddable in L°(Q,P) for some probability space (Q,P) iff it has B(I,p). Lemma 3.19. If E is of stable type p and has Bochner PrOperty I, then it has B(I,p). Proof: Suppose E has Bochner PrOperty I with respect to some tOpology 7. As already noted for a space of stable type p, the Tp topology coincides with the topology Tp generated by ch.f.‘s of symmetric stable measures, but these are continuous in T. Thus a positive definite function which is continuous in T is continuous in T, and there- P fore is the ch.f. of a probability measure. 30 In general a ch.f. of a probability measure is necessarily continuous in TP (ch. 3.10). Thus E has B(I,p). Remark 3.20. Theorem 3.18 and the examples in Section II with the above Lemma, show that cotype 2 is not a sufficient condition for a space to have Bochner Property I. Though, as we have already noted that it is a necessary condition ([19],[22]). Remark 3.21. Combining a result of Maurey ([20], Theorem 98) with Theorem 3.14 and Remark 3.17 we get for real separable Banach spaces with metric approximation prOperty the following are equivalent: (1) E has B(I,p). (2) E is strongly embeddable in Lp(Q,u) for some probability space (Q,u). SECTION IV BOCHNER PROPERTY II We recall our definition of Bochner PrOperty II. A Banach space E is said to have Bochner Property II, if there exists a tOpology T on E’, such that given a family [ua::a e I} of probability measures on E, the equicontinuity in T of their ch.f. is a sufficient condition for tightness. Let u denote the set consisting of ch.f.‘s of all probability measures on E and TM the corresponding tOpology as defined in Section III. Theorem 4.1. If E has Bochner PrOperty II with respect to Tu then E is of cotype 2. Proof is as in the necessity part of our next theorem. Spaces having Bochner Property I are also of cotype 2, however the following theorem shows that Property I is not equivalent to PrOperty II in view of [23]. We first make a few definitions. 31 32 Definitions 4.2 ([24], p.154). An S-operator on a Hilbert space H is a linear, symmetric, non-negative, compact Operator having finite trace. Given an E-valued symmetric Gaussian random variable X defined on some probability space (Q,P). Let u = P<>X-l. Then by Fernique's Theorem [8], we know that IE ”x”2 u(dx) is finite. .Define an Operator A from E' into E by Ay =j‘ dex) E where the integral is in the sense of Bochner. Then A is called the covariance operator of X and the ch.f. of u is exp(- % ). Thus ?2 coincides with the topology for which a basis of neighborhoods of zero is given by the system of sets [y e E’[ < 1], where A runs through the set of Gaussian covariance Operators. We note that if E is a separable Hilbert space then :2 is the same as the S-topology and A is an S-0perator [24]. Theorem 42;. E has Bochner PrOperty II with respect to the ?2 tOpology iff E is of cotype 2. Proof: Suppose E is of cotype 2, and (“aloel a family of probability measures such that their ch.f. Ga are equicontinuous at 0 in $2. Then given 6 > 0, there exists a Gaussian covariance Operator A , such that e , , A A (4.4) : 1 implies l-Re ua(Y) g ll-oaw)! < c) v a e I. 33 Let [Xk]k 2+ be independent, E-valued symmetric 6 Gaussian random variables with Xk having covariance (Al/ky,y>. Observe that auka < m [8], where 3 denotes expectation. Define _ -n _ -1 X - :32 oan where on - auxn” , (here oan = 0 if JHXnH = 0). Then X is a symmetric Gaussian E—valued random variable with covariance = 2:2-2n o: (Al/hy,y>. Further for every 5 > 0, there exists a 5 > 0, such that A (4.5) [1"ua(Y)( < 3: whenever < 5. Let x be the Gaussian measure corresponding to X (i.e., the distribution of X). New, since E is of cotype 2, there exists a Hilbert space H, a continuous linear Operator U from H into E and a symmetric Gaussian 1 measure on H such that x = x1 0 UP ([9], Theorem 4). X1 Without loss of generality we can (and we do!) assume that U is one-one. Let T be the covariance Operatoryof x1. A _ then T is an S-Operator and Al(h) = e l/Q(Th'h). There- _. * 'k e l/2, where U denotes the adjoint. fore {(y) = * By Remark 1.16, A = UTU . * A * A Define on U (E') by va(U y) = ua(y). If A Va * * * _ * ' U yl — U’y2 then UTU'yl — UTU‘y2 i.e., Ayl - Ay2. ' A Hence by (4.5) and positive-definiteness of “a we get A A A . . ua(yl) = ua(y2). Hence Va is well defined on the range * of U and A * * * [l-va(U y)! < c if (TU y,U y)H< 6. 34 * But the range of U is dense in H and Co is uniformly * continuous on the range of U giving A (4.6) I1-—va(h)[ < 6 whenever (Th,h)H < 6. A In other words, [v :a E I] is equicontinuous in the O S-topology in the sense of ([24], p.155). By ([29]) {va_:a 6 I] is tight. Since U is continuous, [va<>U-l: o E I] is tight on E ([4], p.30). But va<’U’l = “a by Remark 1.16, completing the sufficiency part. To prove the necessity part suppose that E has Bochner Property II with respect to T2 but is not of cotype 2. Then there exists a sequence [xk}:=l such that for Yk independent standard Gaussian random variables, r". - ' 2 L’Xkyk converges a.e., but leka = a. Let 2 k n H H a = (z nx.||2)1/2, then a2+°° and 2: x“ .. ., k ._ i k 2 i-l k=l ak Define E-valued independent symmetric random variables {Yk}k=l by ’ . an H II” P(Y = kxk) = p(Y = k k) = 1 Xk 4 k kaH k kaH 4 32 k 2 - _ .1“ka a q k 2 . l Hka . Since Z}P[HYRH > e} = 5 Z) 2 = m, by Borel-Cantelli k k ak Lemma we get that Yk,fl 0 a.e., and hence ZlYk diverges a.e., Let wk denote the ch.f. of the Gaussian E-valued k random variable X) xnyn and m the ch.f. of the Gaussian n=1 @ E-valued random variable 2; xnyn. Then ok(y) and o(y) n=1 are continuous in T . Moreover 2 35 (4.7) 1 -cpk(y) g 1-cp(y) for all k. _1-W(y) < 6 implies % Z) 2 < -log(l-e) k=1 which tends to '0' as e 4 0. Thus we can take 5 small such that l-w(y) < 8 implies Il < l for all k, then ‘1/22 1 2 (4.8) e g,l - 4 . Note that 2 2 3k ak (erk> l " COS S 2 2 1|ka kau therefore 2 2 2 kaH kaH akxk 2 - 2 cos g 2 . ak ak HXRH and 2 llx II a l-- g1-- k (1-cos )' 4 2 2 ak kaH for all k. Thus if l-w(y) < c for small a then using (4.8) we get k 1 2 (4.9) wk(y) g, H (1"Z ) n=1 2 n “X H a x 1 n n n S H [l-- (l-COS)]o n=1 2 a: xn k Let vk denote the distribution of Z: Yn’ then from (4.9), n=1 A @k(y) g vk(y) for all k, therefore 1-Ck(y) g 1-wk(y) g_1-w(y) for all k. 36 ~ A Thus the family {vk);=l is equicontinuous in T Thus 2. by hypothesis thefamily [Vk};=1 is tight. [YR] are symmetric, hence by Theorem 1.22 we get that ZLYk converges a.e., contradicting’ ZIYk diverges. Thus E is of cotype 2. This completes the proof. Let TP be the topology as before. We say that a Banach space has B(II,p) if it has Bochner Property II with respect to Tp. We get that if a Banach space has B(II,2) then it is of type 2 (cf. Theorem 4.10), hence T2 = T2 (1.24), and in view of Theorem 4.3 it is isomorphic to a Hilbert space [14]. NOw we consider B(II,p), for l g_p < 2. However, the following theorem is valid for p=2- Theorem 4.10. If E has B(II,p) then E is of stable type p and embeddable in LP(u). Proof: We will first show that E is stable type p. } i=1 is a sequence in E with .Z) Hxin < m, Suppose [xi i=1 let ni be i.i.d. symmetric stable random variables of n order p. Let “n be the distribution of Z} Xini' Then I p i=1 A - i=1 | un(y) = e , Let v =35 Z (6 +6 ). Then 0 X. -X0 i=1 i i I qup v = z;uxiuP is finite, E and -[Ellpvlp 2_ Z) l[p for all y in E . i=1 i=1 NOw since E is of stable type p, we have ZJuini con- verges a.e.. Thus there exists a symmetric stable measure A -Z:=1)lp u on E, such that d(y) = e for each y E E'. . n Let Vn be the distribution of Z) vini. Then i=1 A -22=lll P vn(y) = e . Therefore for all n, 4‘3” l< -IP A -= th> A l-vnmgl-e ‘1 1 gl-uiy). A giving vn(y) equicontinuous in Tp. Thus {vn] is tight. Again using Theorem 1.22 and symmetry of ni we get that Z’Vini converges a.e. Hence Zinvin < w by Lemma 1.11, which contradicts (4.11) giving the result. Theorem 4.1;. If E has metric approximation property, is of stable type p and embeddable in L°(Q,P) then E has B(II,p). 38 Proof: Let [palael be a family of probability A measures such that [ha] are equicontinuous in Tp. Then from Levy Inequality (Lemma 3.1) we get for e > 0, there exists h(e) > 0 such that 1 A (4.13) nab!) ll(1Tm-7rk)XH > a} g m‘)‘ IE,(l-ua( (Trm- kl’y))P\P(dy) where Y denotes the embedding into the space of symmetric real random variables and {Wm} the sequence corresponding to the metric approximation prOperty. Since E is of stable type p, topology TP 5 T? (Theorem 1.24). Then using an argument as in the proof of Lemma 3.3 we get that equicontinuity of {Ga} in T P implies A lim sup 1im I [l-u ((n -v )'y)]P (dy) = 0. k a m E’ a m k Y Combining with (4.13) we get lim sup u [XIH(I'- )xH > e} = 0 k a a k Thus {ua3OEI is flatly concentrated. We can show that {“a"y-I]OEI is tight for each y E E’ using equicontinuity and ([17], p.193). Hence by ([1], Theorem 2.3, p.279) [pa :o E I] is a tight family, giving E has B(II,p). Remark 4.14. For a separable Banach space E having the metric approximation property the following are equiva- lent: (i) E has B(I,p) (ii) E has B(II.P) 39 (iii) E is strongly embeddable in Lp(Q,u) for some probability space (Q,u). Remarks 4.15. (1) Examples Of Section II show that cotype 2 does not imply B(II,p) for any p. (2) We observe that Theorem 4.3 is a generalization of a theorem of Kuelbs ([13], Theorem 6.3) and an extension of the result for Hilbert spaces [29]. (3) Theorem 4.3 does not have assumption of the approximation prOperty. In this context, however, the support of measures involved does have this property. This fact allows us to circumvent the problem. SECTION V FINAL REMARKS (I) Is the support of a stable measure on a stable , (i + :L-: 1)? 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