Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-20T13:39:22.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Weighted Group Algebras are Operator Algebras

Part of: Linear spaces and algebras of operators Locally compact groups and their algebras

Published online by Cambridge University Press:  05 January 2015

Hun Hee Lee ,
Ebrahim Samei  and
Nico Spronk
Hun Hee Lee
Affiliation:
Department of Mathematical Sciences, Seoul National University, Gwanak-ro 1 Gwanak-gu, Seoul 151–747, Republic of Korea, ([email protected])
Ebrahim Samei
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada, ([email protected])
Nico Spronk
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let G be a finitely generated group with polynomial growth, and let ω be a weight, i.e. a sub-multiplicative function on G with positive values. We study when the weighted group algebra ℓ1 (G, ω) is isomorphic to an operator algebra. We show that ℓ1 (G, ω) is isomorphic to an operator algebra if ω is a polynomial weight with large enough degree or an exponential weight of order 0 < α < 1. We demonstrate that the order of growth of G plays an important role in this problem. Moreover, the algebraic centre of ℓ1 (G, ω) is isomorphic to a Q-algebra, and hence satisfies a multi-variable von Neumann inequality. We also present a more detailed study of our results when G consists of the d-dimensional integers ℤd or the three-dimensional discrete Heisenberg group ℍ3(ℤ). The case of the free group with two generators is considered as a counter-example of groups with exponential growth.

Keywords

weighted group algebrasoperator algebrasgroups with polynomial growthLittlewood multipliers

MSC classification

Primary: 22D15: Group algebras of locally compact groups
Secondary: 47L30: Abstract operator algebras on Hilbert spaces 47L25: Operator spaces (= matricially normed spaces)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 2 , June 2015 , pp. 499 - 519
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bass, H., The degree of polynomial growth of finitely generated nilpotent groups, Proc. Lond. Math. Soc. 25(3) (1972), 603–614.CrossRefGoogle Scholar
2.Blecher, D. and Le Merdy, C., Operator algebras and their modules: an operator space approach, London Mathematical Society Monographs, Volume 30 (Clarendon, Oxford, 2004).Google Scholar
3.Borwein, P., Computational excursions in analysis and number theory, CMS Books in Mathematics, Volume 10 (Springer, 2002).Google Scholar
4.Briäta, J., Buhrmana, H., Leeb, T. and Vidickc, T., All Schatten spaces endowed with the Schur product are Q-algebras, J. Funct. Analysis 262(1) (2012), 1–9.CrossRefGoogle Scholar
5.Dales, H. G. and Lau, A. T.-M., The second duals of Beurling algebras, Memoirs of the American Mathematical Society, Volume 177 (American Mathematical Society, Providence, RI, 2005).Google Scholar
6.Davie, A. M., Quotient algebras of uniform algebras, J. Lond. Math. Soc. 7(2) (1973), 31–40.CrossRefGoogle Scholar
7.Diestel, J., Jarchow, H. and Tonge, A., Absolutely summing operators, Cambridge Studies in Advanced Mathematics, Volume 43 (Cambridge University Press, 1995).CrossRefGoogle Scholar
8.Ghandehari, M., Lee, H. H., Samei, E. and Spronk, N., Some Beurling–Fourier algebras are operator algebras, preprint 2012 (available at http://arxiv.org/abs/1208.4835).Google Scholar
9.Gromov, M., Groups of polynomial growth and expanding maps, Publ. Math. IHES 53 (1981), 53–73.CrossRefGoogle Scholar
10.Guivarch, Y., Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Sér. I 272 (1971), 1695–1696.Google Scholar
11.Le Merdy, C., The Schatten space S 4 is a Q-algebra, Proc. Am. Math. Soc. 126(3) (1998), 715–719.CrossRefGoogle Scholar
12.Palmer, T. W., Banach algebras and general theorem of *-algebras II (Cambrige University Press, 2001).CrossRefGoogle Scholar
13.Pérez-García, D., The trace class is a Q-algebra, Annales Acad. Sci. Fenn. Math. 31(2) (2006), 287–295.Google Scholar
14.Pisier, G., Similarity problems and completely bounded maps: includes the solution to ‘the Halmos problem’, Lecture Notes in Mathematics, Volume 1618, 2nd edn (Springer, 2001).CrossRefGoogle Scholar
15.Pisier, G., Introduction to operator space theory (Cambridge University Press, 2003).CrossRefGoogle Scholar
16.Pisier, G., Grothendieck’s theorem, past and present, Bull. Am. Math. Soc. 49(2) (2012), 237–323.CrossRefGoogle Scholar
17.Ryan, R. A., Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics (Springer, 2002).CrossRefGoogle Scholar
18.Spronk, N., Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras, Proc. Lond. Math. Soc. 89(1) (2004), 161–192.CrossRefGoogle Scholar
19.Tomczak-Jaegermann, N., Banach–Mazur distances and finite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, Volume 38 (Longman, New York, 1989).Google Scholar
20.Varopoulos, N. T., Some remarks on Q-algebras, Annales Inst. Fourier 22(4) (1972), 1–11.CrossRefGoogle Scholar
21.Varopoulos, N. T., Sur les quotients des algèebres uniformes, C. R. Acad. Sci. Paris Sér. I 274 (1972), 1344–1346.Google Scholar