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

DOI:
https://doi.org/10.1090/S0002-9939-99-05104-7

Published electronically:
August 5, 1999

MathSciNet review:
1637396

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a locally compact group. Among other things, we proved in this paper that for an IN-group , the extreme points of the unit ball of the Fourier-Stieltjes algebra are not in the Fourier algebra if and only if is non-compact, or equivalently, there is no irreducible representation of which is quasi-equivalent to a subrepresentation of the left regular representation of if and only if is non-compact. This result is a non-commutative version of the following well known result: For any locally compact group , the extreme points of the unit ball of the measure algebra are not in the group algebra if and only if is non-discrete. On the other hand, we also showed that if has the RNP, then there are extreme points of the unit ball of that are in . Since it is well known there are non-compact locally compact group for which has the RNP, there exist non-compact locally compact groups where extreme points of the unit ball of can be in . This shows that the condition 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:
https://doi.org/10.1090/S0002-9939-99-05104-7

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