Nonweakly compact operators from order-Cauchy complete $C(S)$\ lattices, with application to Baire classes

This paper is concerned with the connection between weak compactness properties in the duals of certain Banach spaces of type C(S) and order properties in the vector lattice C(S). The weak compactness property of principal interest here is the condition that every nonweakly compact operator from C(S) into a Banach space must restrict to an isomorphism on some copy of /°° in C(S). (This implies Grothendieck's property that every »'-convergent sequence in C(5)* is weakly convergent.) The related vector lattice property studied here is orderCauchy completeness, a weak type of completeness property weaker than a-completeness and weaker than the interposition property of Seever. An apphcation of our results is a proof that all Baire classes (of fixed order) of bounded functions generated by a vector lattice of functions are Banach spaces satisfying Grothendieck's property. Another application extend.«previous results on weak convergence of sequences of finitely additive measures defined on certain fields of sets. 0. Introduction. We study here /°° embedding properties of a family of Banach spaces of type C(S) which properly includes all C(S) for a-Stonian S and all Baire classes of bounded functions. (A compact space S is a-Stonian if and only if the lattice C(S) is boundedly a-complete.) The Banach lattices studied here are called "up-down semicomplete", and satisfy the vector-lattice property that whenever a sequence /, < f2 < ... sits below a sequence gx > g2 > . . . and A~iÍ8n ~ f„) = 0 in the lattice C(S), then \//n = A «?„ exists in C(S) (see Definition 1.1). This property is equivalent to order-Cauchy completeness, which has been studied by Everett [21] and Papangelou [22]. § 1 gives several characterizations of these Banach lattices by topological properties of the maximal ideal space S or by equivalent lattice conditions. C(S) is up-down semicomplete (i.e., order-Cauchy complete) if and only if every dense open Fa set V satisfies ßV = S (Theorem 1.5). The Boolean algebra A is up-down semicomplete (with the obvious meaning) if and only if C(SA) is up-down semicomplete, where SA is the Stone space (Theorem 1.8). (Much more has recently been learned about the lattice and algebra structure of these spaces, and about the topological structure of their maximal ideal spaces. See the paper [4].) Received by the editors October 10, 1977 and, in revised form, April 3, 1980. 1980 Mathematics Subject Classification. Primary 46E15, 46E05, 46A40; Secondary 26A21, 28A60. © 1981 American Mathematical Society 0002-9947/81/0000-035 3/S05.25 397 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 398 F. K. DASHIELL, JR. §2 contains the main result of this paper (Theorem 2.5). It is an extension, to certain order-Cauchy complete C(S) spaces (including all Baire classes), of the Grothendieck-Rosenthal theorem that a-Stonian spaces satisfy the following property: every nonweakly compact operator from C(S) into a Banach space restricts to an isomorphism on some isometric copy of /°° in C(S) [15, Theorem 3.7, p. 32]. This immediately yields all Baire classes (as well as a larger class of functionlattices) as new examples of Banach spaces which satisfy Grothendieck's propertynamely that w* convergence for sequences in the dual implies weak convergence (see Corollary to Theorem 3.5). In this sense, Baire classes have some isomorphic resemblance, as Banach spaces, to C(S) for a-Stonian S. This is contrasted with the fact that the Baire classes (of countable order) on an uncountable Polish space are not even linearly isomorphic to a complemented subspace of any C(S) for a-Stonian S, and their maximal ideal spaces fail even to be F-spaces in a rather strong way (these facts are proved in [2]). The methods of §2 yield results for weak convergence and uniform boundedness of sequences of finitely additive measures on certain order-Cauchy complete Boolean algebras, again generalizing known results for the a-complete case, i.e., a-fields of sets (see Theorem 2.6). §3 is devoted to application of the above results to a large class of examples. We describe a family of Baire-type vector sublattices L of R* (for an abstract set X), which can be represented as C(S) spaces satisfying the conditions of §2 (Theorem 3.4). The important property of these L is that they satisfy a pointwise version of up-down semicompleteness (Definition 3.1), and furthermore the lattice and pointwise countable suprema are the same. These lattices include all Baire classes (of fixed order) of bounded functions generated by sequential pointwise limits from a vector sublattice containing 1 (Theorem 3.5). These results yield information about weak convergence and uniform boundedness of sequences of finitely additive measures on certain fields of sets, generalizing Phillips' lemma and other known results for a-fields (see Theorem 3.6). Terminology and notation. For a set S and a mapping/: S —>R, [/ > r] denotes {x G S: fix) >/•},/• G R; similarly for [/ = r], etc; coz(f) = [/ ^ 0] is the "cozero set of /". If S is a topological space, then C(S) is the space of all continuous R-valued functions on S, a cozero set in S is coz(/) for some / E C(S) (equivalent^, for some bounded/ G C(S)); a zero set in S is the complement of a cozero set. The zero sets are just the sets /"'(/£) for closed icR and / G C(S); the cozero sets are the/-'(G) for open G c R and/ G C(S). In a normal space S, these are just the closed Gs sets and the open Fa sets, respectively. We use these facts often without further mention. The Baire sets of a space are the sets in the a-field generated by the zero sets. A subspace G of a space S is C*-embedded if every bounded/ G C(G) extends continuously to S. The standard reference for all this is [7]. 1. Order-Cauchy complete lattices. The material which follows concerns both vector lattices and Boolean algebras. The property studied in this section is a weak type of sequential completeness which can be defined in two equivalent ways. These definitions appear in (c) and (d) below. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONWEAKLY COMPACT OPERATORS 399 Definition 1.1. Suppose L is a vector lattice or a Boolean algebra. (a) A sequence {/„} in L is called order-Cauchy if there exists a decreasing sequence ux > u2 > . . . in L with /\un = 0 in L and \fn — fn+m\ < w„ for all n, m = 1, 2, 3, ... . (For a Boolean algebra, substitute/, A/„+m < «„.) (b) A sequence {/,} in L order-converges to g in L if there exists a decreasing sequence w, > a¡ > ... in L with AM„ = 0 in L and |/„ — g| < un for all « = 1, 2, 3, . . . . (For a Boolean algebra, substitute/, A g < m„.) (c) L is called order-Cauchy complete if every order-Cauchy sequence in L order-converges in L to some element of L. (d) L is called up-down semicomplete if, whenever an increasing sequence/, < f2 < . . . sits below a decreasing sequence gx > g2 > . . . such that /\TM-x(g„ ~~ /„) = 0 in L, then V/n = A 8n exists in L. Remark. With respect to (d), in the presence of the requirement A( 8n ~ /.) = 0, the existence of either \Jf„ or A Sn ls equivalent to the existence of both (and then they are equal), and this is equivalent to the existence of some h E L such that /„ < h < g„, n = 1, 2, 3, . . . . The equivalence of the definitions in (c) and (d) is not immediately obvious. In fact, the property of up-down semicompleteness is much easier to test in examples than order-Cauchy completeness, and it resembles some related interposition properties of vector lattices which have been previously studied (e.g., by Seever [18]). This paper is based entirely on definition (d), and was motivated largely on the observation that the functions of a fixed Baire class form a vector lattice which is up-down semicomplete (see Theorem 3.5). The equivalence of this property to order-Cauchy completeness is not essential to this paper. Order-Cauchy completeness was introduced by Everett [21], and his work was extended by Papangelou [22]. Both used different names for the concept; the present name occurs in Quinn [23]. Its equivalence with up-down semicompleteness follows immediately from the following lemma of Papangelou [22, 2.10]: Proposition 1.2. A sequence {/,} in L is order-Cauchy if and only if there exist an increasing sequence gx < g2 < ... and a decreasing sequence hx > h2 > . . . such that gn < /„ < hn for all n and A(K ~ gn) = 0. As a corollary, we obtain the following fact. Proposition. L is up-down semicomplete if and only if L is order-Cauchy complete. In this paper we shall usually use the terminology order-Cauchy complete. However, the reader should bear in mind that operationally the property is being used in the form of up-down semicompleteness. We now characterize the order-Cauchy complete vector lattices C(S) (for compact S), and the up-down semicomplete Boolean algebras, by simple topological conditions on the maximal ideal space and by some equivalent lattice conditions. One characterization is that every dense cozero set of S should be C* embedded (see (4) of Theorem 1.5 below). It turns out that these compact spaces S coincide with License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 400 F. K. DASHIELL, JR. certain maximal ideal spaces which arise in another context: namely, in the representation of certain abstract algebras as spaces of continuous, almost-everywhere finite, extended real-valued functions on S. See Henriksen and Johnson [12, Proposition 2.2]. Completely regular spaces S such that every dense cozero set is C*-embedded have been dubbed quasi-F spaces in the paper [4] (which in some ways is a sequel to this section), and their properties are rather extensively studied there. The characterization given here is in the same spirit as the characterization, due to Seever [18], of vector lattices and Boolean algebras which sa