INVOLUTIONS ON ALGEBRAS ARISING FROM LOCALLY COMPACT-GROUPS

Two Banach algebras are naturally associated with a locally compact gr oup G: the group algebra, L1(G), and the measure algebra, M(G). For th ese two Banach algebras we determine all isometric involutions. Each o f these Banach algebras has a natural involution. We will show that an isometric involution, (#), is the natural involution on L1(G) if and only if the closure in the strict topology of the convex hull of the n orm one unitaries in M(G) is equal to the unit ball of M(G). There is a well-known relationship between the involutive representation theory of L1(G), with the natural involution, and the representation theory of G. We develop a similar theory for the other isometric involutions on L1(G). The main result is: if (#) is an isometric involution on L1( G) and T is an involutive representation of (L1(G), #), then T is also an involutive representation of L1(G) with the natural involution.