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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. Introduction. Amenable Banach algebras were introduced by B. E. Johnson in [13] where he proved that a locally compact group G is amenable if and only if its associated group algebra L1(G) is amenable (Johnson’s Theorem). Shortly thereafter, Johnson proved his fundamental, intrinsic characterization of amenable Banach algebras in terms of the existence of virtual and approximate diagonals [14]. Virtual diagonals are often said to play the role in the theory of amenable Banach algebras that invariant means play in the theory of amenable groups. In this paper we will study the exact relationship between the classical invariance properties possessed by amenable locally compact groups and the virtual/approximate diagonal “invariance properties” of their amenable group algebras. We will also address these matters in the context of a discrete semigroup algebra. Amenable locally compact groups are characterized by a hierarchy of invariance properties, the deepest of which are the combinatorial Følner conditions. Følner conditions have also proven to be interesting and useful in the study of amenable semigroups, and are considered especially nice because 2000 Mathematics Subject Classification: 22D05, 22D15, 43A07, 43A20.

Approximate diagonals and Følner conditions for amenable group and semigroup algebras