Lebesgue weighted Lp-algebra on locally compact groups

Abstract

Let G be a locally compact group, ω a weight function on G, and 1<p<∞. We introduce the Lebesgue weighted Lp-space Lω1,p(G)= Lp(G,ω)∩ L1(G) as a Banach space and introduce its dual. Furthermore, we consider this space as a Banach algebra with respect to the usual convolution and show that Lω1,p(G)= Lp(G) admits a bounded approximate identity if and only if G is discrete. In addition, we prove that amenability of this algebra implies that G is discrete and amenable. Moreover, we discuss the converse of this result. © 2011 Akadémiai Kiadó, Budapest, Hungary.