Norms of tensor product identities

Abstract

For two symmetric Banach sequence spaces E and F, each either 2-convex or 2-concave, we derive asymptotically optimal estimates for the norms of identity maps $Eⁿ ⨂ _ε F^m ↪ {l₂}^nm$, where Eⁿ and F^m denote the n-th and m-th sections of E and F, respectively and $Eⁿ ⨂ __ε F^m$ their injective tensor product. This generalizes classical results of Hardy and Littlewood as well as of Schutt for l_p-spaces. Based upon this, we give applications to Banach-Mazur distances, volume ratios and projection constants of tensor products, and approximation numbers of certain tensor product identities. As examples we consider powers of sequence spaces as well as Lorentz sequence spaces. Finally, we study the more general context of tensor products of spaces with enough symmetries. In particular, we consider tensor products involving finite-dimensional Schatten classes.