Harmonic analysis of weighted LpLp-algebras

Let GG be a locally compact, compactly generated group of polynomial growth and let ωω be a weight on GG. Under proper assumptions on the weight ωω, the Banach space Lp(G,ω)Lp(G,ω) is a Banach ∗∗-algebra. In this paper we give examples of such weighted LpLp-algebras and we study some of their harmonic analysis properties, such as symmetry, existence of functional calculus, regularity, weak Wiener property, Wiener property, and existence of minimal ideals of a given hull.