Approximate diagonals and Følner conditions for amenable group and semigroup algebras

Abstract

Abstract. 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.

Cohen–Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca–Herz algebras

“…As mentioned already, L 1 (G) is amenable in the sense of Definition 2.6 if and only if G is amenable, and by [39], L 1 (G) is 1-amenable if and only if G is amenable. Hence, for L 1 (G), amenability and 1-amenability are equivalent.…”

Weak amenability and 2-weak amenability of Beurling algebras

“…and then deduce it for any module ( [19,Theorem 1.8] and [2,Theorem 2.9.65]). However, in general, this idea cannot be applied in its present form to the Beurling algebra L 1 ω (G) because the map γ may not be well-defined if we replace L 1 (G)…”

Dual Space and Hyperdimension of Compact Hypergroups

“…We can quantify amenability via the amenability constant, which was defined in [15]. Let For a locally compact group it is known that the group algebra is amenable if and only if its amenability constant is 1 (see [21,Corollary 1.11]). For a finite group G, the amenability constant of the center of the group algebra, denoted by Z 1 (G) has been studied before in [5,2,7].…”