Some Linear Topological Properties of L« of a Finite Measure Space

We are interested here in isomorphic invariants of the various Banach spaces associated with the spaces L°°Gu) for finite measures ju. (Throughout, "/*" and v denote arbitrary finite measures on possibly different unspecified measureable spaces.) We classify the spaces L°°(M) themselves up to isomorphism (linear homeomorphism) in §3, where we also obtain information on the spaces A and A* for subspaces A of L(jx). In §2, we give a short proof of a result (Corollary 2.2) which simultaneously generalizes the result of Pelczyiiski that L(jx) is not isomorphic to a conjugate space if fi is nonpurely atomic [7], and the result of Gel'fand that L[0, l ] is not isomorphic to a subspace of a separable conjugate space (c.f. [8]). We also obtain there that an injective double conjugate space is either isomorphic to /°° or contains an isomorph of /°°(r) for some uncountable set T, if it is infinite dimensional. (Henceforth, all Banach spaces considered are taken to be infinite dimensional. Also, we recall that a Banach space is called injective if every isomorphic imbedding of it in an arbitrary Banach space Y is complemented in F.) We include brief proofs of some of the results. Full details of the work announced here and other related work will appear in [ l l ] .