Concluding remarks

Abstract

Summary: We have not addressed certain important problems that remain unsolved after many years concerning the classical Banach spaces themselves. (Q13) Let K be a compact metric space. Is every complemented sub-space of C(K) isomorphic to C(L) for some compact metric space L? It is known that if K is uncountable then C(K) is isomorphic to C[0,1]. If if is countable then C(K) is isomorphic to C(ω^(ωα)) for some α < ω_1. Every complemented subspace of c_0 (isomorphic to C (ω)) is either finite dimensional or isomorphic to c_0 ([Pel]). If X is complemented in C[0,1] and X* is nonseparable then X is isomorphic to C[0,1] [R6]. Every quotient of c0 embeds isomorphically into c0 but this does not hold in general for C(ω^(ωα)). A discussion of these and related results may be found in [A1, A2, A3, A4], [Gal, Ga2], [Bo2]. The isomorphism types of the complemented subspaces of L_1[0,1] remain unclassified. (Q14) Let X be a complemented (infinite dimensional) subspace of L_1[0,1]. Is X isomorphic to L_1 or l_1? Every X which is complemented in l_p (1 ≤ p < ∞) or c_0 is isomorphic to l_p or c_0. There are known to be uncountably many mutually nonisomorphic complemented subspaces of L_p[0,1] (1 < p < ∞, p ≠ 2) [BRS] and all are known to have a basis [JRZ]. These spaces have been classified as ℒ_p spaces ([LP], [LR]), provided they are not Hilbert spaces.