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OPERATOR SYSTEM NUCLEARITY VIA $C^{\ast }$-ENVELOPES

Part of: Selfadjoint operator algebras Normed linear spaces and Banach spaces; Banach lattices Methods of category theory in functional analysis Linear spaces and algebras of operators

Published online by Cambridge University Press:  11 May 2016

VED PRAKASH GUPTA  and
PREETI LUTHRA
VED PRAKASH GUPTA*
Affiliation:
School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India email vedgupta@mail.jnu.ac.in
PREETI LUTHRA
Affiliation:
Department of Mathematics, University of Delhi, Delhi 110007, India email maths.preeti@gmail.com
*
✉vedgupta@mail.jnu.ac.in
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Abstract

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We prove that an operator system is (min, ess)-nuclear if its $C^{\ast }$-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of a countable discrete group by Farenick et al. [‘Operator systems from discrete groups’, Comm. Math. Phys.329(1) (2014), 207–238] is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (respectively, a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to three (respectively, every component of the graph is complete).

Keywords

operator systemsamenable groupsgraphsnuclearityC∗ -envelopestensor products

MSC classification

Primary: 46L06: Tensor products of $C^*$-algebras 46L07: Operator spaces and completely bounded maps
Secondary: 47L25: Operator spaces (= matricially normed spaces) 46M05: Tensor products 46B28: Spaces of operators; tensor products; approximation properties
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 101 , Issue 3 , December 2016 , pp. 356 - 375
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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