Dual Banach algebras: representations and injectivity

Matthew Daws¹

¹ St. John's College Oxford, OX1 3JP, UK

Studia Mathematica, Tome 178 (2007) no. 3, pp. 231-275

Résumé

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.

Détail

DOI : 10.4064/sm178-3-3

Keywords: study representations banach algebras reflexive banach spaces algebras which admit representations which bounded below seem generalisation arens regular banach algebras class includes dual banach algebras defined runde group algebras discrete weakly cancellative semigroup algebras algebras behave similar * * algebras interpolation space techniques place gns type arguments define notion injectivity dual banach algebras equivalent connes amenability conclude looking problem defining well behaved tensor product dual banach algebras