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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Completely isometric representations of $ M_{cb}A(G)$ and $ UCB(\hat G)^*$


Authors: Matthias Neufang, Zhong-Jin Ruan and Nico Spronk
Journal: Trans. Amer. Math. Soc. 360 (2008), 1133-1161
MSC (2000): Primary 22D15, 22D20, 43A10, 43A22, 46L07, 46L10, 47L10
Published electronically: October 16, 2007
MathSciNet review: 2357691
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Abstract: Let $ G$ be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier algebra $ M_{cb}A(G)$, which is dual to the representation of the measure algebra $ M(G)$, on $ \mathcal{B}(L_2(G))$. The image algebras of $ M(G)$ and $ M_{cb}A(G)$ in $ \mathcal{CB}^{\sigma} (\mathcal{B}(L_2(G)))$ are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group $ G$, there is a natural completely isometric representation of $ UCB(\hat G)^*$ on $ \mathcal{B}(L_2(G))$, which can be regarded as a duality result of Neufang's completely isometric representation theorem for $ LUC(G)^*$.


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Additional Information

Matthias Neufang
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
Email: mneufang@math.carleton.ca

Zhong-Jin Ruan
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: ruan@math.uiuc.edu

Nico Spronk
Affiliation: Department of Mathematics, University of Walterloo, Waterloo, Ontario, Canada N2L 3G1
Email: nspronk@math.uwaterloo.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-07-03940-2
PII: S 0002-9947(07)03940-2
Received by editor(s): October 26, 2004
Received by editor(s) in revised form: December 22, 2004
Published electronically: October 16, 2007
Additional Notes: The first and third authors were partially supported by NSERC
The second author was partially supported by the National Science Foundation DMS-0140067 and DMS-0500535
The third author was partially supported by an NSERC PDF
Article copyright: © Copyright 2007 American Mathematical Society