We study the relationship between the classical invariance properties of amenable locally compact groups G and the approximate diagonals possessed by their associated group algebras L¹(G). From the existence of a weak form of approximate diagonal for L¹(G) we provide a direct proof that G is amenable. Conversely, we give a formula for constructing a strong form of approximate diagonal for any amenable locally compact group. In particular we have a new proof of Johnson's Theorem: A locally compact group G is amenable precisely when L¹(G) is an amenable Banach algebra. Several structural Følner-type conditions are derived, each of which is shown to correctly reflect the amenability of L¹(G). We provide Følner conditions characterizing semigroups with 1-amenable semigroup algebras.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm164-2-3,
author = {Ross Stokke},
title = {Approximate diagonals and F\o lner conditions for amenable group and semigroup algebras},
journal = {Studia Mathematica},
volume = {162},
year = {2004},
pages = {139-159},
zbl = {1056.22002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm164-2-3}
}
Ross Stokke. Approximate diagonals and Følner conditions for amenable group and semigroup algebras. Studia Mathematica, Tome 162 (2004) pp. 139-159. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm164-2-3/