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.