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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Characterizations of algebras arising from locally compact groups

Author: Paul L. Patterson
Journal: Trans. Amer. Math. Soc. 329 (1992), 489-506
MSC: Primary 43A10; Secondary 22D15, 46K05
MathSciNet review: 1043862
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Abstract: Two Banach $ ^{\ast}$-algebras are naturally associated with a locally compact group, $ G$: the group algebra, $ {L^1}(G)$, and the measure algebra, $ M(G)$. Either of these Banach algebras is a complete set of invariants for $ G$.

In any Banach $ ^{\ast}$-algebra, $ A$, the norm one unitary elements form a group, $ S$. Using $ S$ we characterize those Banach $ ^{\ast}$-algebras, $ A$, which are isometrically $ ^{\ast}$-isomorphic to $ M(G)$. Our characterization assumes that $ A$ is the dual of some Banach space and that its operations are continuous in the resulting weak $ ^{\ast}$ topology. The other most important condition is that the convex hull of $ S$ must be weak$ ^{\ast}$ dense in the unit ball of $ A$.

We characterize Banach $ ^{\ast}$-algebras which are isomerically isomorphic to $ {L^1}(G)$ for some $ G$ as those algebras, $ A$, whose double centralizer algebra, $ D(A)$, satisfies our characterization for $ M(G)$. In addition we require $ A$ to consist of those elements of $ D(A)$ on which $ S$ (defined relative to $ D(A)$) acts continuously with its weak$ ^{\ast}$ topology. Using another characterization of $ {L^1}(G)$ we explicitly calculate the above isomorphism between $ A$ and $ {L^1}(G)$.

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Article copyright: © Copyright 1992 American Mathematical Society

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