Involutions on algebras arising from locally compact groups

Abstract

Two Banach algebras are naturally associated with a locally compact group G: the group algebra, L 1 ( G ) {L^1}(G) , and the measure algebra, M ( G ) M(G) . For these two Banach algebras we determine all isometric involutions. Each of these Banach algebras has a natural involution. We will show that an isometric involution, ( # ) {(^\# }) , is the natural involution on L 1 ( G ) {L^1}(G) if and only if the closure in the strict topology of the convex hull of the norm one unitaries in M ( G ) M(G) is equal to the unit ball of M ( G ) M(G) . There is a well-known relationship between the involutive representation theory of L 1 ( G ) {L^1}(G) , with the natural involution, and the representation theory of G. We develop a similar theory for the other isometric involutions on L 1 ( G ) {L^1}(G) . The main result is: if ( # ) {(^\# }) is an isometric involution on L 1 ( G ) {L^1}(G) and T is an involutive representation of ( L 1 ( G ) , # ) ({L^1}(G){,^\# }) , then T is also an involutive representation of L 1 ( G ) {L^1}(G) with the natural involution.