Compact elements of the algebrasPMp(G)andPFp(G)of a locally compact group

Abstract

Compact and weakly compact elements of the group algebra L 1 (G) of a locally compact group G, have been considered by a number of authors. In these investigations it has been shown that, if G is non-compact, then the only weakly compact element of L 1 (G ) is zero. Conversely, if G is compact, then every element of L 1 (G) is compact. For 1<p<∞, let PM p (G)and PF p (G) denote the closure of L 1 (G), considered as an algebra of convolution operators on L p (G), with respect to the weak operator topology and the norm topology, respectively, in B(L p (G), b), the bounded linear operators on L 1 (G). We study the question of characterizing compact and weakly compact elements of the algebras PM p (G)and PF p (G).