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Involutions on algebras arising from locally compact groups


Author: Paul L. Patterson
Journal: Proc. Amer. Math. Soc. 121 (1994), 739-745
MSC: Primary 46K05; Secondary 43A10, 43A20, 46K10
DOI: https://doi.org/10.1090/S0002-9939-1994-1185273-X
MathSciNet review: 1185273
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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, $ {(^\char93 })$, 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 $ {(^\char93 })$ is an isometric involution on $ {L^1}(G)$ and T is an involutive representation of $ ({L^1}(G){,^\char93 })$, then T is also an involutive representation of $ {L^1}(G)$ with the natural involution.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1185273-X
Keywords: Locally compact groups, Banach $ \ast $-algebras, isometric involutions
Article copyright: © Copyright 1994 American Mathematical Society

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