Banach spaces with the Daugavet property, and the centralizer

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 ⊗ 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. © 2007 Elsevier Inc. All rights reserved