From Locally Compact Groups

Two Banach algebras are naturally associated with a locally compact group G: the group algebra, L,(G), and the measure algebra, 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 £'(<?) if and only if the closure in the strict topology of the convex hull of the norm 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 Ll{G), with the natural involution, and the representation theory of G. We develop a similar theory for the other isometric involutions on Ll(G). The main result is: if (*) is an isometric involution on Ll(G) and T is an involutive representation of (Ll(G), *), then T is also an involutive representation of LX(G) with the natural involution. Introduction and notation In the study of locally compact groups two group algebras are of special interest. In 1952 Wendel showed that the algebra LX(G), of all absolutely integrable functions on G considered as a Banach algebra (with convolution multiplication), characterizes the locally compact group G up to homeomorphic isomorphism [10]. In 1964 Johnson using Wendel's result proved that M(G), the algebra of complex regular Borel measures on G (with convolution multiplication and the total variation norm), is also a complete set of invariants for G [6]. Both of these group algebras have a natural involution, but the involution was not used by either Wendel or Johnson. Although the natural involution on L1 (G) was not used by Wendel, it is of importance in the representation theory of locally compact groups. In [8] we used the involutive structure to give characterizations of LX(G) and M(G) as Banach ""-algebras. The question arises, how unique are the natural involutions on Ll(G) and M(G) ? Using some of the results of [8] we will show that for most groups the natural involution on either Ll(G) or M(G) is far from being unique. Received by the editors July 21, 1992 and, in revised form, October 6, 1992. 1991 Mathematics Subject Classification. Primary 43A20; Secondary 43A10, 46K99.