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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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) $\left |\frac 1p - \frac 12\right |=\left |\frac 1q - \frac 12\right |$. 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.
- David P. Blecher and Christian Le Merdy, On quotients of function algebras and operator algebra structures on $l_p$, J. Operator Theory 34 (1995), no. 2, 315–346. MR 1373327
- F. F. Bonsall, A minimal property of the norm in some Banach algebras, J. London Math. Soc. 29 (1954), 156–164. MR 61283, DOI https://doi.org/10.1112/jlms/s1-29.2.156
- Michael Cowling, The predual of the space of convolutors on a locally compact group, Bull. Austral. Math. Soc. 57 (1998), no. 3, 409–414. MR 1623239, DOI https://doi.org/10.1017/S0004972700031828
- M. Daws and N. Spronk, The approximation property implies that convolvers are pseudo-measures, Preprint (arXiv:1308.1073 [math.FA]), 2013.
- Cédric Delmonico, Convolution operators and homomorphisms of locally compact groups, J. Aust. Math. Soc. 84 (2008), no. 3, 329–344. MR 2453684, DOI https://doi.org/10.1017/S1446788708000669
- Antoine Derighetti, Relations entre les convoluteurs d’un groupe localement compact et ceux d’un sous-groupe fermé, Bull. Sci. Math. (2) 106 (1982), no. 1, 69–84 (French, with English summary). MR 654425
- S. Dirksen, M. de Jeu, and M. Wortel, Crossed products of Banach algebras. I, to appear in Dissertationes Mathematicae. Preprint (arXiv:1104.5151 [math.FA]), 2011.
- A. H. Dooley, Sanjiv Kumar Gupta, and Fulvio Ricci, Asymmetry of convolution norms on Lie groups, J. Funct. Anal. 174 (2000), no. 2, 399–416. MR 1768980, DOI https://doi.org/10.1006/jfan.2000.3573
- E. Gardella and H. Thiel, Functoriality of group algebras acting on $L^p$-spaces. Preprint (arXiv:1408.6137), 2014.
- Eusebio Gardella and Hannes Thiel, Banach algebras generated by an invertible isometry of an $L^p$-space, J. Funct. Anal. 269 (2015), no. 6, 1796–1839. MR 3373434, DOI https://doi.org/10.1016/j.jfa.2015.05.004
- Eusebio Gardella and Hannes Thiel, Group algebras acting on $L^p$-spaces, J. Fourier Anal. Appl. 21 (2015), no. 6, 1310–1343. MR 3421918, DOI https://doi.org/10.1007/s00041-015-9406-1
- Eusebio Gardella and Hannes Thiel, Quotients of Banach algebras acting on $L^p$-spaces, Adv. Math. 296 (2016), 85–92. MR 3490763, DOI https://doi.org/10.1016/j.aim.2016.03.040
- E. Gardella and H. Thiel, Isomorphisms of algebras of convolution operators. In preparation, 2016.
- E. Gardella and H. Thiel, Extending representations of Banach algebras to their biduals. Preprint (arXiv:1703.00882), 2017.
- Carl Herz, The theory of $p$-spaces with an application to convolution operators, Trans. Amer. Math. Soc. 154 (1971), 69–82. MR 272952, DOI https://doi.org/10.1090/S0002-9947-1971-0272952-0
- Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 355482
- C. Herz, On the asymmetry of norms of convolution operators. I, J. Functional Analysis 23 (1976), no. 1, 11–22. MR 0420138, DOI https://doi.org/10.1016/0022-1236%2876%2990055-0
- B. E. Johnson, An introduction to the theory of centralizers, Proc. London Math. Soc. (3) 14 (1964), 299–320. MR 159233, DOI https://doi.org/10.1112/plms/s3-14.2.299
- Barry Edward Johnson, Cohomology in Banach algebras, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 127. MR 0374934
- John Lamperti, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), 459–466. MR 105017
- Matthias Neufang and Volker Runde, Column and row operator spaces over ${\rm QSL}_p$-spaces and their use in abstract harmonic analysis, J. Math. Anal. Appl. 349 (2009), no. 1, 21–29. MR 2455727, DOI https://doi.org/10.1016/j.jmaa.2008.08.021
- Daniel M. Oberlin, $M_{p}(G)\not =M_{q}(G)$ $(p^{-1}+$ $q^{-1}=1)$, Israel J. Math. 22 (1975), no. 2, 175–179. MR 387956, DOI https://doi.org/10.1007/BF02760165
- N. C. Phillips, Crossed products of $L^p$ operator algebras and the K-theory of Cuntz algebras on $L^p$ spaces. Preprint (arXiv:1309.6406), 2013.
- N. C. Phillips, Multiplicative domain for $L^p$ operator algebras. In preparation (draft from May 2014).
- Volker Runde, Representations of locally compact groups on ${\rm QSL}_p$-spaces and a $p$-analog of the Fourier-Stieltjes algebra, Pacific J. Math. 221 (2005), no. 2, 379–397. MR 2196641, DOI https://doi.org/10.2140/pjm.2005.221.379
- L. Tzafriri, Remarks on contractive projections in $L_{p}$-spaces, Israel J. Math. 7 (1969), 9–15. MR 248514, DOI https://doi.org/10.1007/BF02771741
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Additional Information
Eusebio Gardella
Affiliation:
Mathematisches Institut, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
MR Author ID:
1118291
Email:
gardella@uni-muenster.de
Hannes Thiel
Affiliation:
Mathematisches Institut, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
MR Author ID:
930802
Email:
hannes.thiel@uni-muenster.de
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