Banach spaces with the Daugavet property, and the centralizer

Abstract

We introduce representable Banach spaces, and prove that the class R of such spaces satisfies the following properties: (1) Every member of R has the Daugavet property. (2) It Y is a member of R , then, for every Banach space X, both the space L ( X , Y ) (of all bounded linear operators from X to Y) and the complete injective tensor product X ⊗ ˆ ϵ Y lie in R . (3) If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, and for most vector space topologies τ on Y, the space C ( K , ( Y , τ ) ) (of all Y-valued τ-continuous functions on K) is a member of R . (4) If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, most C ( K , Y ) -superspaces (in the sense of [V. Kadets, N. Kalton, D. Werner, Remarks on rich subspaces of Banach spaces, Studia Math. 159 (2003) 195–206]) are members of R . (5) All dual Banach spaces without minimal M-summands are members of R .