Dual Banach algebras: representations and injectivity

Details Dual Banach algebras: representations and injectivity Dual Banach algebras: representations and injectivity

2006-04-17

arxiv.org/abs/math/0604372v2

Studia Math. 178 (2007), no. 3, 231--275.

Mathematics

Functional Analysis

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

Scientific paper

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$^*$- and W$^*$-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.

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