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A new proof on the characterization of vanishing subalgebras of group algebras


Author: F. Ghahramani
Journal: Proc. Amer. Math. Soc. 142 (2014), 617-621
MSC (2010): Primary 43A20
DOI: https://doi.org/10.1090/S0002-9939-2013-11787-9
Published electronically: November 13, 2013
MathSciNet review: 3134002
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Abstract: We give a new proof to a result due to T. Shimizu stating that for a locally compact group $ G$ and the associated group algebra $ L^{1}(G)$, if $ S$ is a measurable subset of $ G$, then a necessary and sufficient condition for the subspace of all functions in $ L^{1}(G)$ that vanish almost everywhere off $ S$ to be an algebra is that $ S$ is equal to a subsemigroup of $ G$, locally almost everywhere. Our proof bypasses a deep result of A. Ionescu Tulcea and C. Ionescu Tulcea, used in Shimizu's proof, and is for the most part functional analytic.


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Additional Information

F. Ghahramani
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
Email: fereidou@cc.umanitoba.ca

DOI: https://doi.org/10.1090/S0002-9939-2013-11787-9
Keywords: Arens product, contractive projections, group algebra, semigroup, vanishing algebra
Received by editor(s): July 20, 2011
Received by editor(s) in revised form: March 30, 2012
Published electronically: November 13, 2013
Additional Notes: The author’s research was supported by NSERC Grant 36640-07
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.