On sub B-convex Banach spaces

Abstract

In the article is introduced a new class of Banach spaces that are called sub B-convex. Namely, a Banach space X is said to be B -convex if it may be represented as a direct sum l_1+ W, where W is B-convex. It will be shown that any separable sub B-convex Banach space X may be almost isometrically embedded in a separable Banach space G(X) of the same cotype as X, which has a series of properties. Namely, G(X) is an approximate envelope, i.e. any separable Banach space which is finitely representable in G(X) may be almost isometrically embedded into G(X); G(X)is almost isotropic; G(X) is existentially closed in a class of all spaces that are finitely equivalent to it; the conjugate space to G(X) is of cotype 2; every operator from its conjugate into the Hilbert space is absolutely summing; every projection P in G(X) of finite rank has a norm that infinitely growth with the dimension of its range. If, in addition, X is of cotype 2, then G(X) has more impressive properties: every operator from G(X) into the Hilbert space is absolutely summing; the injective and projective tensor products of G(X) by itself are identical.