Maximal and reduced Roe algebras of coarsely embeddable spaces
Maximal and reduced Roe algebras of coarsely embeddable spaces
In [7], Gong, Wang and Yu introduced a maximal, or universal, version of the Roe C*-algebra associated to a metric space. We study the relationship between this maximal Roe algebra and the usual version, in both the uniform and non-uniform cases. The main result is that if a (uniformly discrete, bounded geometry) metric space X coarsely embeds into a Hilbert space, then the canonical map between the maximal and usual (uniform) Roe algebras induces an isomorphism on K-theory. We also give a simple proof that if X has property A, then the maximal and usual (uniform) Roe algebras are the same. These two results are natural coarse-geometric analogues of certain well-known implications of a-T-menability and amenability for group C*-algebras. The techniques used are E-theoretic, building on work of Higson, Kasparov and Trout [11], [12] and Yu [28].
35-68
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
Willett, Rufus
9f00bf4a-53ab-47b5-9f59-d12174765907
1 May 2013
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
Willett, Rufus
9f00bf4a-53ab-47b5-9f59-d12174765907
Spakula, Jan and Willett, Rufus
(2013)
Maximal and reduced Roe algebras of coarsely embeddable spaces.
Journal für die reine und angewandte Mathematik, 2013 (678), .
(doi:10.1515/crelle.2012.019).
Abstract
In [7], Gong, Wang and Yu introduced a maximal, or universal, version of the Roe C*-algebra associated to a metric space. We study the relationship between this maximal Roe algebra and the usual version, in both the uniform and non-uniform cases. The main result is that if a (uniformly discrete, bounded geometry) metric space X coarsely embeds into a Hilbert space, then the canonical map between the maximal and usual (uniform) Roe algebras induces an isomorphism on K-theory. We also give a simple proof that if X has property A, then the maximal and usual (uniform) Roe algebras are the same. These two results are natural coarse-geometric analogues of certain well-known implications of a-T-menability and amenability for group C*-algebras. The techniques used are E-theoretic, building on work of Higson, Kasparov and Trout [11], [12] and Yu [28].
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e-pub ahead of print date: 23 March 2012
Published date: 1 May 2013
Organisations:
Faculty of Social, Human and Mathematical Sciences
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Local EPrints ID: 351875
URI: http://eprints.soton.ac.uk/id/eprint/351875
ISSN: 0075-4102
PURE UUID: ede5cbb0-bd46-40c6-ae25-0575f7d4d16b
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Date deposited: 25 Apr 2013 14:17
Last modified: 15 Mar 2024 03:48
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Author:
Rufus Willett
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