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On one-sided primitivity of Banach algebras

Part of: Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  12 January 2010

M. J. Crabb ,
J. Duncan  and
C. M. McGregor
M. J. Crabb
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK, Email: ([email protected]; [email protected])
J. Duncan
Affiliation:
Department of Mathematical Sciences, SCEN 301, University of Arkansas, Fayetteville, AR 72701, USA, Email: ([email protected])
C. M. McGregor
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK, Email: ([email protected]; [email protected])
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Abstract

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Let S be the semigroup with identity, generated by x and y, subject to y being invertible and yx = xy2. We study two Banach algebra completions of the semigroup algebra ℂS. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that ℂS is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for ℂS is finite dimensional and hence that ℂS has a separating family of such modules.

Keywords

Banach algebraprimitiveleft-primitiveright-primitiveirreducible representation

MSC classification

Secondary: 46H20: Structure, classification of topological algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 1 , February 2010 , pp. 111 - 123
Copyright
Copyright © Edinburgh Mathematical Society 2010