Some stability properties of c 0 - saturated spaces Denny

A Banach space is c0-saturated if all of its closed infinite dimensional subspaces contain an isomorph of c0. In this article, we study the stability of this property under the formation of direct sums and tensor products. Some of the results are: (1) a slightly more general version of the fact that c0-sums of c0saturated spaces are c0-saturated; (2) C(K,E) is c0-saturated if both C(K) and E are; (3) the tensor product JH⊗̃ǫJH is c0-saturated, where JH is the James Hagler space. Let E be a Banach space. Following Rosenthal [10], we say that a Banach space F is E-saturated if every infinite dimensional closed subspace of F contains an isomorphic copy of E. In this article, we will be concerned with the stability properties of c0saturated spaces under the formation of direct sums and tensor products. In §1, we prove a result which implies that c0-sums of c0-saturated spaces are c0-saturated. In §2, it is shown that the tensor product E⊗̃ǫF is c0-saturated if E is isomorphically polyhedral (see §2 for the definition) and F is c0-saturated. As a corollary, we obtain that C(K,E) is c0-saturated if and only if both C(K) and E are. Finally, in §3, we show that JH⊗̃ǫJH is c0-saturated, where JH denotes the James Hagler space [4]. Standard Banach space terminology, as may be found in [7], is employed. The (closed) unit ball of a Banach space E is denoted by UE . The space c00 consists of all finitely non-zero real sequences. If (xn) and (yn) are sequences residing in possibly different Banach spaces, we say that (xn) dominates (yn) if there is a constant K < ∞ such that ‖ ∑ anyn‖ ≤ K‖ ∑ anxn‖ for all (an) ∈ c00. Two sequences which dominate each other are said to be equivalent. A sequence (xn) in a Banach space is seminormalized if 0 < inf ‖xn‖ ≤ sup ‖xn‖ < ∞. If A is an arbitrary set, |A| denotes the cardinality of A. For an infinite set A, P∞(A) is the set of all infinite subsets of A. 1991 Mathematics Subject Classification 46B20, 46B28.