Some weighted group algebras are operator algebras

2012/08/18 by Hun Hee Lee, Lee, Hun Hee, Ebrahim Samei +3
Mathematics · #47L30 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 22D15 #Secondary 47L25 #math.FA #math.OA #msc:22D15 #msc:47L25 #msc:47L30

paper · pdf · doi:10.48550/arxiv.1208.3791

Errors concerning Grothendieck's inequality are fixed. The related constants appearing in the results are corrected accordingly

arxiv created 2013/04/04 · arxiv updated 2013/04/05

Abstract

Let G be a finitely generated group with polynomial growth, and let \om be a weight, i.e. a sub-multiplicative function on G with positive values. We study when the weighted group algebra ℓ¹(G,\om) is isomorphic to an operator algebra. We show that ℓ¹(G,\om) is isomorphic to an operator algebra if \om is a polynomial weight with large enough degree or an exponential weight of order 0<α<1. We will demonstrate the order of growth of G plays an important role in this question. Moreover, the algebraic centre of ℓ¹(G,\om) 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 is the d-dimensional integers \Z^d and 3-dimensional discrete Heisenberg group ℍ₃(\Z). The case of the free group with two generators will be considered as a counter example of groups with exponential growth.

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