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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Completely isometric representations of $M_{cb}A(G)$ and $UCB(\hat G)^*$
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by Matthias Neufang, Zhong-Jin Ruan and Nico Spronk PDF
Trans. Amer. Math. Soc. 360 (2008), 1133-1161 Request permission

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
  • MR Author ID: 718390
  • Email: mneufang@math.carleton.ca
  • Zhong-Jin Ruan
  • Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
  • MR Author ID: 249360
  • Email: ruan@math.uiuc.edu
  • Nico Spronk
  • Affiliation: Department of Mathematics, University of Walterloo, Waterloo, Ontario, Canada N2L 3G1
  • MR Author ID: 671665
  • Email: nspronk@math.uwaterloo.ca
  • 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
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 1133-1161
  • MSC (2000): Primary 22D15, 22D20, 43A10, 43A22, 46L07, 46L10, 47L10
  • DOI: https://doi.org/10.1090/S0002-9947-07-03940-2
  • MathSciNet review: 2357691