A new proof on the characterization of vanishing subalgebras of group algebras
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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.References
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Additional Information
- F. Ghahramani
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
- MR Author ID: 196713
- Email: fereidou@cc.umanitoba.ca
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 617-621
- MSC (2010): Primary 43A20
- DOI: https://doi.org/10.1090/S0002-9939-2013-11787-9
- MathSciNet review: 3134002