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Transactions of the American Mathematical Society

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Representations of $ p$-convolution algebras on $ L^q$-spaces


Authors: Eusebio Gardella and Hannes Thiel
Journal: Trans. Amer. Math. Soc. 371 (2019), 2207-2236
MSC (2010): Primary 47L10, 43A15; Secondary 43A65, 46E30
DOI: https://doi.org/10.1090/tran/7489
Published electronically: September 4, 2018
MathSciNet review: 3894050
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Abstract: For a nontrivial locally compact group $ G$, consider the Banach algebras of $ p$-pseudofunctions, $ p$-pseudomeasures, $ p$-convolvers, and the full group $ L^p$-operator algebra. We show that these Banach algebras are operator algebras if and only if $ p=2$. More generally, we show that for $ q\in [1,\infty )$, if any of these Banach algebras can be represented on an $ L^q$-space, then one of the following holds: (a) $ p=2$ and $ G$ is abelian; or (b) $ \left \vert\frac 1p - \frac 12\right \vert=\left \vert\frac 1q - \frac 12\right \vert$. This result can be interpreted as follows: for $ p,q\in [1,\infty )$, the $ L^p$- and $ L^q$-representation theories of a group are incomparable, except in the trivial cases when they are equivalent.

As an application, we show that, for distinct $ p,q\in [1,\infty )$, if the $ L^p$- and $ L^q$-crossed products of a topological dynamical system are isomorphic, then $ \frac 1p + \frac 1q=1$. In order to prove this, we study the following relevant aspects of $ L^p$-crossed products: existence of approximate identities, duality with respect to $ p$, and existence of canonical isometric maps from group algebras into their multiplier algebras.


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Additional Information

Eusebio Gardella
Affiliation: Mathematisches Institut, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
Email: gardella@uni-muenster.de

Hannes Thiel
Affiliation: Mathematisches Institut, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
Email: hannes.thiel@uni-muenster.de

DOI: https://doi.org/10.1090/tran/7489
Keywords: Locally compact group, algebra of $p$-pseudofunctions, algebra of $p$-pseudomeasures, contractive approximate identity, multiplier algebra, group amenability, crossed product
Received by editor(s): October 13, 2016
Received by editor(s) in revised form: August 28, 2017, and December 18, 2017
Published electronically: September 4, 2018
Additional Notes: The first named author was partially supported by the D. K. Harrison Prize from the University of Oregon and by a Postdoctoral Research Fellowship from the Humboldt Foundation. The first and second named authors were partially supported by the Deutsche Forschungsgemeinschaft (SFB 878). Part of this work was completed while the authors were taking part in the Research Program Classification of operator algebras, complexity, rigidity and dynamics, held at the Institut Mittag-Leffler, between January and April of 2016. We would like to thank the staff and organizers, and Søren Eilers in particular, for the hospitality and financial support.
Article copyright: © Copyright 2018 American Mathematical Society