Involutions on algebras arising from locally compact groups

Patterson, Paul L.

doi:10.1090/S0002-9939-1994-1185273-X

Involutions on algebras arising from locally compact groups
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Proc. Amer. Math. Soc. 121 (1994), 739-745 Request permission

Abstract:

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

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