RACSAM. On copies of c 0 in some function spaces. 1 Preliminaries. Juan Carlos Ferrando

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1 RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 103 (1), 2009, pp Análisis Matemático / Mathematical Analysis On copies of c 0 in some function spaces Juan Carlos Ferrando Abstract. If (, Σ, µ) is a probability space and X a Banach space, a theorem concerning sequences of X-valued random elements which do not converge to zero is applied to show from a common point of view that the F -normed space L 0(µ, X) of all classes of X-valued random variables, as well as the p-normed space L p(µ, X) of all X-valued p-integrable random variables with 0 < p < 1 and the space P 1(µ, X) of the µ-measurable X-valued Pettis integrable functions, all contain a copy of c 0 if and only if X does. We also show that if is a noncompact hemicompact topological space, then the Banach space C 0() of all scalarly valued continuous functions defined on vanishing at infinity, equipped with the supremum-norm, contains a norm-one complemented copy of c 0. Sobre copias de c 0 en algunos espacios de funciones Resumen. Si (, Σ, µ) es un espacio de probabilidad y X un espacio de Banach, aplicamos un teorema sobre sucesiones de variables aleatorias con valores en X que no convergen a cero para demostrar, desde un punto de vista común, que el espacio F -normado L 0(µ, X) de todas las clases de variables aleatorias X-valoradas, así como el espacio p-normado L p(µ, X) de las variables aleatorias p-integrables X-valoradas, con 0 < p < 1, y el espacio P 1(µ, X) de las funciones X-valoradas µ-medibles y Pettis integrables, todos ellos contienen una copia de c 0 si y sólo si X también la contiene. Asimismo probamos que si es un espacio topológico hemicompacto no compacto, el espacio de Banach C 0() de las funciones continuas con valores escalares definidas en que se anulan en el infinito, equipado con la norma supremo, contiene una copia de c 0 norma-uno complementada. 1 Preliminaries Throughout denotes the product { 1, 1} N, Γ the σ-algebra of subsets of generated by the n-cylinders of for each n N, and ν the Borel probability ν i on Γ, where ν i : 2 { 1,1} [0, 1] is defined by ν i ( ) = 0, ν i ({ 1}) = ν i ({1}) = 1/2 and ν i ({ 1, 1}) = 1 for each i N. Coordinate mappings on are denoted by ε i, as well as its values when considered as mappings into { 1, 1}. In the sequel (, Σ, µ) will be a nontrivial complete probability space, X a Banach space over the field K of real or complex numbers and L 0 (µ, X) will stand for the F -space of all [classes of] µ-measurable functions f : X equipped with the (continuous) F -norm f 0 = f(ω) dµ (ω) 1 + f(ω) Presentado por / Submitted by Manuel Valdivia Ureña. Recibido / Received: 11 de enero de Aceptado / Accepted: 14 de enero de Palabras clave / Keywords: Banach space, copy of c 0, vector-valued random element, F -norm, p-norm, space L p(µ, X) with 0 p < 1, space P 1 (µ, X), hemicompact space, space C 0 (). Mathematics Subject Classifications: 46E40, 46B09. c 2009 Real Academia de Ciencias, España. 49

2 J. C. Ferrando of the convergence in probability. We shall represent by L p (µ, X) with 0 < p < 1 the complete p-normed topological vector space of all µ-measurable X-valued [classes of] p-bochner integrable functions equipped with the (continuous) p-norm f p = f(ω) p dµ (ω). We shall shorten by wuc the sentence weakly unconditionally Cauchy. We stand for P 1 (µ, X) the normed space of all those [classes of] µ-measurable X-valued Pettis integrable functions f defined on provided with the semivariation norm { } f = sup P1(µ,X) x f(ω) dµ (ω) : x X, x 1. If A is a subset of a Banach space X then [A] will represent the closed linear span of A in X. If is endowed with a Hausdorff topology, the Banach space over K of all continuous functions f : K vanishing at infinity (that is, for each ǫ > 0 there is a compact set K f,ǫ such that f(ω) < ǫ for ω \ K f,ǫ ) equipped with the supremum-norm will be denoted by C 0 () and the linear subspace of C 0 () consisting of all those functions f of compact support (suppf) will be represented by C C (). 2 Copies of c 0 in L p (µ, X) with 0 p < 1 As is well known, a classic conjecture of Hoffmann-Jørgensen [7] on the X-inheritance of copies of c 0 in the Banach space L p (µ, X) with 1 p < was established by Kwapień in [9]. Later on Hoffmann- Jørgensen conjecture was shown also to be true for p = 0 [10, Theorem 2.11] (see [2, Theorem 9.1]) and for 0 < p < 1 [2, Theorem 10.4]. In this section a theorem concerning certain sequences of X-valued random elements which do not converge to zero (Theorem 2 below) is used to provide an independent proof of these facts. The following result is contained in the proof of [9, Proposition]. Theorem 1 (Kwapień) Let X be a Banach space and {x n } a sequence in X. If ({ }) n ν ε : sup ε i x i = = 0, n N then the series n=1 x n has a wuc subseries x n i. Lemma 1 Let {f n } be a sequence of X-valued random variables for (, Σ, µ). If the series n=1 ζ nf n converges in L 0 (µ, X) for every ζ c 0, then for each A Σ with µ (A) > 0 there is ω A A such that the series n=1 f n (ω A ) has a wuc subseries. PROOF. If we set ϕ n (ε, ω) = ε n f n (ω) for each (ε, ω, n) N and put S n = n ϕ i for each n N, it suffices to show that the sequence {S n } n=1 is stochastically bounded, that is, that {S n : n N } is a bounded set in L 0 (ν µ, X). In fact, since {ϕ i } is a symmetric sequence of (ν µ)- measurable functions, if this property holds the sequence {S n } n=1 is (ν µ)-almost surely bounded (see for instance [7, Theorem 2.4]). This means that the set { } n Q = (ε, ω) : sup ε i f i (ω) n N =, verifies that (ν µ)(q) = 0. Hence, if Q ω denotes the ω-slice of Q, the fact that (ν µ)(q) = ν (Q ω )dµ (ω) 50

3 On copies of c 0 in some function spaces implies that ν (Q ω ) = 0 for µ-almost all ω. Since µ (A) > 0, there is an ω A A with ν (Q ωa ) = 0, that is ({ }) n ν ε : sup ε i f i (ω A ) = = 0. n N So by Theorem 1 there exists in X a wuc subseries of n=1 f n (ω A ). Let us show that the sequence {S n } n=1 of the partial sums of the sequence {ϕ i} L 0 (ν µ, X). Since ζ iε i f i converges in L 0 (µ, X) for each (ζ, ε) c 0, setting h ζ k (ε) = sup m,n k ζ i ε i f i n ζ i ε i f i 0 is bounded in for (ζ, k) c 0 N, then h ζ k (ε) 0 for every ε with fixed ζ c 0 as k. Due to the fact that h ζ k is ν-measurable and h ζ k 1 for every (ζ, k) c 0 N, Lebesgue s dominated convergence theorem implies that hζ k (ε)dν (ε) 0 for each ζ c 0. Hence given ǫ > 0 there is k 0 (ζ) N such that n ζ i ε i f i ζ i ε i f i dν (ε) < ǫ 0 for every m, n k 0 (ζ) and Fubini s theorem shows that { n ζ iϕ i } n=1 is a Cauchy sequence in L 0(ν µ, X) for all ζ c 0. Since L 0 (ν µ, X) is a complete linear space, we conclude that the series ζ iϕ i converges in L 0 (ν µ, X) for every ζ c 0. Therefore an application of Banach-Steinhaus theorem [8, (3)] shows that the linear mapping T : c 0 L 0 (ν µ, X) defined by Tζ = ζ iϕ i is continuous. So T ({ n e i : n N }) is a bounded subset of L 0 (ν µ, X) and consequently the sequence {S n } n=1 is bounded in L 0 (ν µ, X), as stated. Theorem 2 Let {f n } n=1 be a sequence of X-valued random variables for (, Σ, µ). If the two following conditions hold 1. n=1 ζ nf n converges in L 0 (µ, X) for every ζ c 0 2. {f n } n=1 does not converge almost surely to zero then X contains a copy of c 0. PROOF. Let A = { ω : f n (ω) 0 }. If µ (A) > 0, Lemma 1 yields an ω A A and a strictly increasing sequence {n i } N such that the series f n i (ω A ) is wuc in X. So there exists ǫ > 0 and a subsequence {x i } of {f n i (ω A )} with x i ǫ for all i N. Since inf i N x i > 0 and x i is wuc in X, the selection principle along with [1, Chapter V, Corollary 7] assures that X contains a copy of c 0. On the other hand, if µ (A) = 0 then f n 0 almost surely, contradicting condition 2 above. Corollary 1 L 0 (µ, X) contains a copy of c 0 if and only if X does. PROOF. If J is an isomorphism from c 0 into L 0 (µ, X) and f n = Je n for each n N then inf f n 0 > 0. n N Otherwise there is a subsequence {f nk } of {f n } such that f nk 0 in J (c 0 ), considered as a subspace of L 0 (µ, X). So e nk 0 in c 0, a contradiction. If a sequence {f n } of representatives of the original sequence of classes of functions verifies that f n 0 almost surely, then f n 0 in L 0 (µ, X), contradicting the fact that inf n N f n 0 > 0. Thus condition 2 of the preceding theorem holds for {f n }. On the other hand, if ζ c 0, since n=0 ζ ne n converges (to ζ) in c 0 then n=0 ζ nf n converges (to Jζ) in L 0 (µ, X). Consequently the sequence {f n } n=1 satisfies condition 1 of the preceding theorem as well. Hence Theorem 2 guarantees that X contains a copy of c 0. For the converse note that X is isomorphic to a linear subspace of L 0 (µ, X). 51

4 J. C. Ferrando Lemma 2 Let J be an isomorphism from c 0 into L p (µ, X) with 0 < p < 1. If f n = Je n for each n N, where {e n } n=1 stands for the unit vector basis of c 0, then the sequence { f n ( ) p } n=1 is uniformly integrable. PROOF. Assuming by contradiction that the sequence { f n ( ) p } n=1 is not uniformly integrable, there are ǫ > 0, a sequence {A n } n=1 of pairwise disjoint elements of Σ and a subsequence of {f n}, which we denote in the same way, with A n f n (ω) p dµ (ω) > ǫ for every n N. Since n k=1 ξ ke k 1 for each n N and ξ l with ξ 1, there is K > 0 such that n ξ if i p K for each n N and ξ l with ξ 1. Hence, choosing m N such that (m 1)ǫ > K, classic Rosenthal s lemma provides a strictly increasing sequence {n i } of positive integers satisfying that f ni (ω) p dµ (ω) < ǫ m S j N,j i An j for each i N. Since (a + b) p a p + b p for a, b 0 then x + y p x y p x p y p, which applies to get p K f nj f ni (ω) A p p dµ (ω) f nj (ω) dµ (ω). ni j=1 So the fact that x + y p x p + y p yields K f ni (ω) p dµ (ω) f ni (ω) p dµ (ω) j=1 j i j=1 j=1,j i f nj (ω) p dµ (ω) S i N,i j An i f nj (ω) p dµ (ω), which implies that K (m 1)ǫ > K, a contradiction. Corollary 2 The space L p (µ, X) with 0 < p < 1 contains a copy of c 0 if and only if X does. PROOF. Assume that J is an isomorphism from c 0 into L p (µ, X) and set f n = Je n for each n N. As in the proof of Corollary 1 one may show that inf n N f n p > 0. By Lemma 2 the sequence { f n ( ) p } n=1 is uniformly integrable. Hence if a sequence of representatives verifies that f n 0 almost surely, an application of Vitali s lemma [3, Theorem ] yields f n p 0, a contradiction. Thus {f n } n=1 does not converge almost surely to zero. As in addition the series n=1 ζ nf n converges in L p (µ, X), so in L 0 (µ, X), for each ζ c 0, Theorem 2 assures that X contains a copy of c 0. Remark 1 The same argument can be applied to show that L p (µ, X), 1 p <, contains a copy of c 0 if and only if X does since, if {f n } n=1 is a normalized basic sequence in L p(µ, X) equivalent to the unit vector basis of c 0, it can be easily shown that the sequence { f n ( ) p } n=1 of L 1(µ) is also uniformly integrable. So Vitali s lemma together with Theorem 2 assure that X contains a copy of c 0. 3 Copies of c 0 in P 1 (µ, X) Theorem 2 also applies to get the following well known result concerning copies of c 0 in the space of µ-measurable X-valued Pettis integrable functions. Theorem 3 (Ferrando [4], Freniche [6]) P 1 (µ, X) contains a copy of c 0 if and only if X does. 52

5 On copies of c 0 in some function spaces PROOF. Assume by contradiction that X contains no copy of c 0 and let {f n } n=1 be a normalized basic sequence in P 1 (µ, X) equivalent to the unit vector basis of c 0. Since for each x X and E Σ the mapping f E x f dµ is a bounded linear functional on P 1 (µ, X) and the series n=1 f n is wuc in P 1 (µ, X), then the series n=1 E f n dµ is wuc in X for all E Σ. If F f n dµ 0 for some F Σ then for sure that X contains a copy of c 0, a contradiction. Hence E f n dµ 0 for all E Σ. Consequently, if we assume that there is a sequence of representatives verifying that f n 0 almost surely, then [6, Lemma 3] assures that f n P1(µ,X) 0, another contradiction. So we must conclude that no sequence of representatives {f n } converges almost surely to zero. Since the canonical inclusion map T from P 1 (µ, X) into L 0 (µ, X) has closed graph [6, Lemma 4], the closed graph theorem for topological vector spaces [8, (3)] guarantees that the restriction S of T to the copy [f n ] of c 0 in P 1 (µ, X) into L 0 (µ, X) is continuous. Since n=1 ζ nf n converges in P 1 (µ, X) for every ζ c 0, it follows that n=1 ζ nf n converges in L 0 (µ, X) for every ζ c 0. Since {f n } does not converge almost surely to zero and n=1 ζ nf n converges in L 0 (µ, X) for every ζ c 0, Theorem 2 applies to show that X contains a copy of c 0. This last contradiction completes the proof. 4 Complemented copies of c 0 in C 0 () Throughout this section will stand for a non-empty Hausdorff topological space. The following result adapts some ideas of [5] to show that if is a noncompact hemicompact topological space then C 0 () contains a norm-one complemented copy of c 0. Let us recall that for compact then C 0 () = C() may contain no complemented copy of c 0, which happens for instance if is extremally disconnected. Theorem 4 If is a non compact hemicompact space, then the Banach space C 0 () contains a norm-one complemented copy of c 0 PROOF. Since is a noncompact hemicompact space there is a strictly increasing sequence {K n } of compact subsets satisfying the property that every compact subset K of is contained in a member of this sequence. For every n N, set n := K n+1 \ K n. Given that n is a relative open subset of the compact space K n+1, choose ω n n and apply Urysohn s lemma in K n+1 to get f n C C (K n+1 ) with 0 f n 1 such that f n (ω n ) = 1 and suppf n n. Assuming each f n extended to the whole space in a trivial way, then {f n } is a normalized basic sequence in C 0 () equivalent to the unit vector basis {e n } of c 0. In fact, if f = n=1 a n f n [f i ] with a n 0, given ǫ > 0 choose m N such that a n < ǫ for each n > m. Then K := m suppf i is a compact set and f(ω) < ǫ for each ω \ K, hence f C 0 (). If µ := 1 2 iδ ω i, since each f C 0 () is µ-integrable, the conditional expectation value E n (f) of the random variable f relative to the event n, given by E n (f) = 1 µ ( n ) n f dµ, is well-defined for each n N. Moreover, it must be pointed out that n f n dµ = µ ( n ) = 1 2 n (1) for every n N. Given f C 0 (), for each ǫ > 0 there exists a compact set K f,ǫ such that f(ω) < ǫ for all ω \ K f,ǫ. Since is hemicompact, there is p N such that K f,ǫ K p. Hence f(ω) < ǫ for each ω n with n p, which implies that E n (f) 1 µ ( n ) n f(ω) dµ (ω) sup ω n f(ω) ǫ 53

6 J. C. Ferrando whenever n p. This assures that E n (f) 0 and, consequently, that the linear operator P : C 0 () C 0 () given by Pf = n=1 E n (f)f n is well-defined as well. Given ω, since (i) f n (ω) = 0 if ω / n, (ii) f n = 1 and (iii) E n (f) f for each f C 0 (), if ω m then ( (Pf)(ω) = E n (f)f n )(ω) E m (f) f. n=1 This shows that P is a bounded linear operator with P 1. Finally, as a consequence of (1) and of the fact that the f n are disjointly supported it follows that E i (f j ) = δ ij, so that Pf i = E n (f i ) f n = f i n=1 for every i N. So we conclude that P is a norm-one bounded linear projection operator from C 0 () onto [f i ]. Acknowledgement. This research has been supported by the project MTM of the Spanish Ministry of Science and Innovation. References [1] DIESTEL, J., (1984). Sequences and series in Banach spaces,springer-verlag. GTM 92 New York Berlin Heidelberg Tokyo. [2] DREWNOWSKI, L. AND LABUDA, I., (2002). Topological vector spaces of Bochner measurable functions, Illinois J. Math, 46, [3] DUNFORD, N. AND SCHWARTZ, J. T., (1988). Linear Operators, I. Wiley-Interscience. [4] FERRANDO, J. C., (2002). On sums of Pettis integrable random elements, Quaestiones Math., 25, [5] FERRANDO, J. C. AND LÓPEZ PELLICER, M., (2001). Complemented copies of c 0 in C 0 (), Rev. R. Acad. Cien. Serie A Mat., 95, [6] FRENICHE, F. J., (1998). Embedding c 0 in the space of Pettis integrable functions, Quaestiones Math., 21, [7] HOFFMANN-JØRGENSEN, J., (1974). Sums of independent Banach space valued random variables, Studia Math., 52, [8] KÖTHE, G., (1983). Topological Vector Spaces I, Grundlehren der mathematischen Wissenschaften, 159. Springer-Verlag. Berlin Heidelberg New York. [9] KWAPIEŃ, S., (1974). On Banach spaces containing c 0, Studia Math., 52, [10] LABUDA, I., (1979). Spaces of measurable functions, Comment. Math., Special Issue 2, Juan Carlos Ferrando Centro de Investigación Operativa, Universidad Miguel Hernández Elche (Alicante). Spain. jc.ferrando@umh.es 54