Several New Characterizations of Banach Spaces Containing

Several new characterizations of Banach spaces containing a subspace isomorphic to , are obtained. These are applied to the question of when 1 embeds in the injective tensor product of two Banach spaces. Notations and terminology. All Banach spaces are taken as infinite dimensional, “subspace” means “closed linear subspace,” “operator” means “bounded linear operator.” If W is a subset of a Banach space, [W ] denotes its closed linear span. c denotes the cardinal of the continuum, i.e., c = 2א0 ; this is also identified with the least ordinal of cardinality c. For 1 ≤ p < ∞, c denotes the family f of all scalar valued functions defined on c with ‖f‖p = ( ∑ α<c |f(α)|p)1/p < ∞. Finally, we recall that a scalar-valued function defined on a compact metric space K is called universally measurable if it is measurable with respect to the completion of every Borel measure on K. Throughout this paper, the symbols X,Y,Z,B,E shall denote Banach spaces. BaX denotes the closed unit ball of X. Recall that an operator T : X → Y is called Dunford-Pettis if T maps weakly compact sets in X to norm compact sets in Y . Also, L(X,Y ) (resp. K(X,Y )) denotes the space of operators (resp. of compact operators) from X to Y , L(X) = L(X,X), K(X) = K(X,X). A bounded subset W of X∗ is said to isomorphically norm X if there exists a C < ∞ such that ‖x‖ ≤ C sup w∈W |w(x)| for all x ∈ X . In case C = 1 and W ⊂ BaX∗, we say that W isometrically norms X. X ∨ ⊗Y , X ∧ ⊗Y denote the injective, respectively projective, tensor products of X and Y . See [DU], [Gr2] for terminology and theorems in this area. Main results. Our first main result gives several equivalences for a Banach space to contain an isomorph of 1. We have included many previously known ones, to round out the list; also, we use some of them later on. As far as I know, the equivalences of 1. with the following are new: 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 19. (Of course some of the implications were previously *Research partially supported by NSF DMS-0700126.