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ON MINIMAL IDEALS IN SOME BANACH ALGEBRAS ASSOCIATED WITH A LOCALLY COMPACT GROUP

Published online by Cambridge University Press:  19 March 2001

J. W. BAKER
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH; [email protected]
M. FILALI
Affiliation:
Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland; [email protected]
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Abstract

Let G be a non-compact, locally compact group. The minimal ideals of the group algebra L1(G), the measure algebra M(G), and other Banach algebras (usually larger than L1(G) and M(G)) such as the second dual, L1(G)**, of L1(G) with an Arens product, or LUC(G)* with an Arens-type product, are studied in the paper. Using integrable representations, which exist on some semisimple Lie groups, it is seen that the minimal left ideals can be of infinite dimension, and that the compactness of G is not necessary for these ideals to exist in L1(G) and M(G). It is shown also that, although the coefficients of integrable representations are minimal idempotents in LUC(G)* and L1(G)** they do not generate minimal right ideals in these algebras, and that for a large class of groups, they do not generate minimal left ideals either.

Type
Research Article
Copyright
The London Mathematical Society 2001

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