When Does ca(Σ, X) Contain a Copy of l ∞ or C 0 ?

Abstract

For a a-algebra 1 and a Banach space X, ca(X, X) is the Banach space of all vector measures from X to X . If E admits a nonzero atomless finite positive measure, then ca(X, X) D 1. (or c0) if and only if there is a noncompact bounded linear operator from 12 to X (Theorem 1). Otherwise, ca(?, X) D 1. (or c0) if and only if X D l00 (or c0) (Theorem 2). Let E be a a-algebra on a nonempty set S, and X be a Banach space. Then ca(X, X) denotes the Banach space of all countably additive vector measures ,u: E X under the sup-norm Jl,ull = supE Ilu(E)HI. (This norm is of course equivalent with the more commonly used semivariation norm.) If Z is a Banach space, we write Z D 1 (or c0) to indicate that Z contains an isomorphic copy of lo (or c0). The purpose of this paper is to give a complete answer, in terms of X and X, to the question stated in the title. In [5, Corollary 4] it has been recently shown that the subspace cca(X, X) of ca(X, X), consisting of measures with relatively norm compact ranges, contains an isomorph of l1 if and only if X D 100. Somewhat surprisingly, it is in general not true that ca(X, X) D 1. implies X D l0 (the converse is always valid trivially). Here are our main results. Theorem 1. Suppose the u-algebra X admits a nonzero atomless finite positive measure. Then the following are equivalent. (a) ca(l, X) :) 100' (b) There exists a noncompact bounded linear operator from 12 to X. (c) ca(X, X) D co . Theorem 2. Suppose every nonzero finite positive measure on E is purely atomic. Then: (A) ca(X, X) D c0 X X D 10. (B) ca(l, X) D co X* X D co . Received by the editors June 20, 1989; the results of this paper were presented at the Polish-GDR Seminar on Functional Analysis held in Georgenthal (GDR), March 31-April 5, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46G 10, 46E27, 46B20, 46B25.