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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm178-3-3,
author = {Matthew Daws},
title = {Dual Banach algebras: representations and injectivity},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {231-275},
zbl = {1115.46038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm178-3-3}
}
Matthew Daws. Dual Banach algebras: representations and injectivity. Studia Mathematica, Tome 178 (2007) pp. 231-275. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm178-3-3/