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Some Weighted Group Algebras are Operator Algebras

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

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, (hunheelee@snu.ac.kr)
Ebrahim Samei
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada, (samei@math.usask.ca)
Nico Spronk
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, (nspronk@math.uwaterloo.ca)
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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 

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