Approximate diagonals and Følner conditions for amenable group and semigroup algebras

Ross Stokke¹

¹ Department of Mathematics and Statistics University of Winnipeg 515 Portage Avenue Winnipeg, Manitoba Canada, R3B 2E9

Studia Mathematica, Tome 164 (2004) no. 2, pp. 139-159

Résumé

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 $L^1(G)$. From the existence of a weak form of approximate diagonal for $L^1(G)$ 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 $L^1(G)$ is an amenable Banach algebra. Several structural Følner-type conditions are derived, each of which is shown to correctly reflect the amenability of $L^1(G)$. We provide Følner conditions characterizing semigroups with $1$-amenable semigroup algebras.

Détail

DOI : 10.4064/sm164-2-3

Mots-clés : study relationship between classical invariance properties amenable locally compact groups approximate diagonals possessed their associated group algebras existence weak form approximate diagonal provide direct proof amenable conversely formula constructing strong form approximate diagonal amenable locally compact group particular have proof johnsons theorem locally compact group amenable precisely amenable banach algebra several structural lner type conditions derived each which shown correctly reflect amenability provide lner conditions characterizing semigroups amenable semigroup algebras