SYMMETRY OF WEIGHTED $L{\uppercase{\footnotesize{L^1}}}$-ALGEBRAS AND THE GRS-CONDITION

Abstract

Let G be a compactly generated, locally compact group of polynomial growth. Removing a restrictive technical condition from a previous work, we show that the weighted group algebra L ω (G) is a symmetric Banach ∗-algebra, if and only if the weight function ω satisfies the GRS-condition. This condition expresses in a precise technical sense that ω grows subexponentially. As a fact of independent interest, we show that groups of (at most) polynomial growth have strict polynomial growth.