Approximate diagonals and Følner conditions for amenable group and semigroup algebras
Volume 164 / 2004
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.