The classical subspaces of the projective tensor products of lpand C(α) spaces, α< ω1

Abstract

We completely determine the lq and C(K) spaces which are isomorphic to a subspace of lp⊗ˆπC(α), the projective tensor product of the classical lp space, 1≤p<∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α<ω1. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from lp to l1, 1≤p<∞. The first main theorem is an extension of a result of E. Oja and states that the only lq space which is isomorphic to a subspace of lp⊗ˆπC(α) with 1≤p≤q<∞ and ω≤α<ω1 is lp. The second main theorem concerning C(K) spaces improves a result of Bessaga and Pelczynski which allows us to classify, up to isomorphism, the separable spaces (X,Y) of nuclear operators, where X and Y are direct sums of lp and C(K) spaces. More precisely, we prove the following cancellation law for separable Banach spaces. Suppose that K1 and K3 are finite or countable compact metric spaces of the same cardinality and $1 (a) (lp⊕C(K1),lq⊕C(K2)) and (lp⊕C(K3),lq⊕C(K4)) are isomorphic.