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Transactions of the American Mathematical Society

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

 
 

 

Representations of Banach algebras subordinate to topologically introverted spaces


Authors: M. Filali, M. Neufang and M. Sangani Monfared
Journal: Trans. Amer. Math. Soc. 367 (2015), 8033-8050
MSC (2010): Primary 46H15, 46B10, 43A20, 47L10
DOI: https://doi.org/10.1090/tran/6435
Published electronically: April 24, 2015
MathSciNet review: 3391908
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Abstract: Let $ A$ be a Banach algebra, $ X$ a closed subspace of $ A^*$, $ Y$ a dual Banach space with predual $ Y_*$, and $ \pi $ a continuous representation of $ A$ on $ Y$. We call $ \pi $ subordinate to $ X$ if each coordinate function $ \pi _{y,\lambda }\in X$, for all $ y\in Y, \lambda \in Y_*$. If $ X$ is topologically left (right) introverted and $ Y$ is reflexive, we show the existence of a natural bijection between continuous representations of $ A$ on $ Y$ subordinate to $ X$, and normal representations of $ X^*$ on $ Y$. We show that if $ A$ has a bounded approximate identity, then every weakly almost periodic functional on $ A$ is a coordinate function of a continuous representation of $ A$ subordinate to $ WAP(A)$. We show that a function $ f$ on a locally compact group $ G$ is left uniformly continuous if and only if it is the coordinate function of the conjugate representation of $ L^1(G)$, associated to some unitary representation of $ G$. We generalize the latter result to an arbitrary Banach algebra with bounded right approximate identity. We prove the functionals in $ LUC(A)$ are all coordinate functions of some norm continuous representation of $ A$ on a dual Banach space.


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Additional Information

M. Filali
Affiliation: Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland
Email: mahmoud.filali@oulu.fi

M. Neufang
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada – and – Université Lille 1, U.F.R. de Mathématiques, Laboratoire Paul Painlevé, 59655 Villeneuve d’Ascq, France
Email: mneufang@math.carleton.ca

M. Sangani Monfared
Affiliation: Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario N9B 3P4, Canada
Email: monfared@uwindsor.ca

DOI: https://doi.org/10.1090/tran/6435
Received by editor(s): August 28, 2013
Published electronically: April 24, 2015
Additional Notes: The second and third authors were partially supported by NSERC
Article copyright: © Copyright 2015 American Mathematical Society

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