It is shown that a L∞-space with separable dual constructed by Bourgain and Delbaen has small Szlenk index and thus does not have a quotient isomorphic to C(ω). It follows that this is a L∞-space which is the same size as c0 in the sense of the Szlenk index but does not contain c0. This has some consequences in the theory of uniform homeomorphism of Banach spaces.

motivated by an arens regularity problem, we introduce the concepts of matrix banach space and matrix banach algebra. the notion of matrix normed space in the sense of ruan is a special case of our matrix normed system. a matrix banach algebra is a matrix banach space with a completely contractive multiplication. we study the structure of matrix banach spaces and matrix banach algebras. then we...

Let (T; T) be an arbitrary measurable space, X a Banach space whose dual X is strongly separable and f an integrand deened on T X. If f is normal in the sense of Rockafellar 18], then the conjugate integrand f is normal. Moreover the subdiierential multifunction is EEros (or weakly) measurable. These results extend those of Rockafellar 18] and of Hess 6], and provide an alternate approach to th...

We show that a separable real Banach space embeds almost isometrically in a space Y with a shrinking 1-unconditional basis if and only if limn→∞ ‖x ∗ + x∗ n ‖ = limn→∞ ‖x ∗ − x∗ n ‖ whenever x∗ ∈ X∗, (x∗ n )∞ n=1 is a weak∗-null sequence and both limits exist. If X is reflexive then Y can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng.

We construct a family of separable Hilbertian operator spaces, such that the relation of complete isomorphism between the subspaces of each member of this family is complete Kσ . We also investigate some interesting properties of completely unconditional bases of the spaces from this family. In the Banach space setting, we construct a space for which the relation of isometry of subspaces is equ...

We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following: let X be a Banach space with an unconditional basis {ei}i∈N; then either there exists a perfect set P of infinite subsets of N such that for any two distinct A,B ∈ P, [ei]i∈A ≇ [ei]i∈B , or for a residual set of infinite subsets A of N, [ei]i∈A is isomorphic to X, and in that case, X is is...

We prove that for any separable Banach space X, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to X. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that ther...

In the present paper we obtain a Caratheodory-type selection theorem. This result is used in [9] to extend the theory of Nash equilibria, developed in [12, 4, 2, 7, 8, 10, and 13-171, to the setting of an arbitrary measure space of agents and an infinite dimensional strategy space. More specifically, we consider the following problem. Let T be a measure space, Z be a complete separable metric s...

If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A 0 , A 1) such that A 0 and A 1 are isometric to X ⊕ V , and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype...

Suppose J is a two-sided quasi-Banach ideal of compact operators on a separable innnite-dimensional Hilbert space H. We show that an operator T 2 J can be expressed as nite linear combination of commutators A;B] where A 2 J and B 2 B(H) if and only its eigenvalues (n) (arranged in decreasing order of absolute value, repeated according to algebraic multiplicity and augmented by zeros if necessar...