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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extreme points of the unit ball
of the Fourier-Stieltjes algebra


Authors: Peter F. Mah and Tianxuan Miao
Journal: Proc. Amer. Math. Soc. 128 (2000), 1097-1103
MSC (1991): Primary 43A30, 43A35, 43A65, 22D99
Published electronically: August 5, 1999
MathSciNet review: 1637396
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Abstract: Let $G$ be a locally compact group. Among other things, we proved in this paper that for an IN-group $G$, the extreme points of the unit ball of the Fourier-Stieltjes algebra $B(G)$ are not in the Fourier algebra $A(G)$ if and only if $G$ is non-compact, or equivalently, there is no irreducible representation of $G$ which is quasi-equivalent to a subrepresentation of the left regular representation of $G$ if and only if $G$ is non-compact. This result is a non-commutative version of the following well known result: For any locally compact group $\widehat G$, the extreme points of the unit ball of the measure algebra $M(\widehat G)$ are not in the group algebra $L^{1}(\widehat G)$ if and only if $\widehat G$ is non-discrete. On the other hand, we also showed that if $B(G)$ has the RNP, then there are extreme points of the unit ball of $B(G)$ that are in $A(G)$. Since it is well known there are non-compact locally compact group $G$ for which $B(G)$ has the RNP, there exist non-compact locally compact groups $G$ where extreme points of the unit ball of $B(G)$ can be in $A(G)$. This shows that the condition $G$ be an IN-group cannot be entirely removed.


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Additional Information

Peter F. Mah
Affiliation: Department of Mathematics and Statistics, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1
Email: pfmah@mist.lakeheadu.ca

Tianxuan Miao
Affiliation: Department of Mathematics and Statistics, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1
Email: tmiao@thunder.lakeheadu.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05104-7
PII: S 0002-9939(99)05104-7
Keywords: Locally compact groups, extreme points, weak$^{*}$-strongly exposed points, Fourier algebra, Fourier-Stieltjes algebra
Received by editor(s): November 14, 1997
Received by editor(s) in revised form: June 1, 1998
Published electronically: August 5, 1999
Additional Notes: This research is supported by an NSERC grant.
Dedicated: Dedicated to Professor Edmond E. Granirer, with our admiration and respect,on the occasion of his retirement
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society



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