Weighted group algebras on groups of polynomial growth

Abstract

Let G be a compactly generated group of polynomial growth and ω a weight function on G. For a large class of weights we characterize symmetry of the weighted group algebra L 1 (G, ω). In particular, if the weight ω is sub-exponential, then the algebra L 1 (G, ω) is symmetric. For these weights we develop a functional calculus on a total part of L 1 (G, ω) and use it to prove the Wiener property. (2000): 43A20, 22D15, 22D12. Mathematics Subject Classification

Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices

“…In a sense they extend our results for locally compact groups [15] to Banach algebras of operators which have much less structure. In this regard, Barnes's results on Banach algebras of integral operators [3] need to be mentioned.…”

“…Nevertheless, the two concepts are closely related, and almost always is the symmetry of a Banach algebra A proved by showing that it is inverse-closed in a C * -algebra. Technically, this is accomplished by the following lemma of Hulanicki [23] (see also [15] for a corrected proof). We give a formulation that is most suitable for our purposes.…”

“…Solving a 30-year old conjecture, Losert [25] succeeded in showing that the group algebra of a compactly generated, locally compact group of polynomial growth is symmetric. Subsequently, we have developed techniques to verify the symmetry of weighted L 1 -algebras on locally compact groups of polynomial growth [15] and of Banach algebras of twisted convolution [20].…”

Reflexivity and hyperreflexivity of bounded $N$-cocycles from group algebras

“…The functional calculus for a compactly generated locally compact group G with polynomial growth is a powerful tool in studying its group algebra. For example, using this idea, it is shown in [6,Theorem 5.6] that L 1 (G) has the Wiener property. This was achieved by constructing a certain well-behaved bounded approximate identity for L 1 (G).…”

“…This was achieved by constructing a certain well-behaved bounded approximate identity for L 1 (G). In the following lemma, we expand the ideas in the proof of [6,Theorem 5.6] and use the existence of functional calculus for groups with polynomial growth to construct "locally unital" bounded approximate identities in their group algebras. We will use this fact crucially later on to obtain our main results.…”