Dual Banach algebras: representations and injectivity

Abstract

We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C${}^{\ast }$- and W${}^{\ast }$-algebras; we show that interpolation space techniques can be used in the place of GNS type arguments. We define a notion of injectivity for dual Banach algebras, and show that this is equivalent to Connes-amenability. We conclude by looking at the problem of defining a well-behaved tensor product for dual Banach algebras.

40 pages; Update corrects some mathematics, and merges two sections to make for easier reading

Cited by in corpus (2)

• A Representation Theorem for Completely Contractive Dual Banach Algebras
• Multiresolution Analysis for Compactly Supported Interpolating Tensor Product Wavelets (long version)