Dual Banach algebras: representations and injectivity

Volume 178 / 2007

Matthew Daws Studia Mathematica 178 (2007), 231-275 MSC: Primary 47L10; Secondary 46B70, 46H05, 46H15, 46H99, 46M05, 43A10, 43A20, 46A25, 46A32, 46A35, 46L10, 46L06, 46M10. DOI: 10.4064/sm178-3-3


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 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.


  • Matthew DawsSt. John's College
    Oxford, OX1 3JP, UK

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