Approximate diagonals and Følner conditions for amenable group and semigroup algebras

Tom 164 / 2004

Ross Stokke Studia Mathematica 164 (2004), 139-159 MSC: 22D05, 22D15, 43A07, 43A20. DOI: 10.4064/sm164-2-3


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.


  • Ross StokkeDepartment of Mathematics and Statistics
    University of Winnipeg
    515 Portage Avenue
    Winnipeg, Manitoba
    Canada, R3B 2E9

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