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Finite-dimensional approximations for Nica–Pimsner algebras

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  09 August 2019

EVGENIOS T. A. KAKARIADIS
EVGENIOS T. A. KAKARIADIS*
Affiliation:
School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK email [email protected]
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Abstract

We give necessary and sufficient conditions for nuclearity of Cuntz–Nica–Pimsner algebras for a variety of quasi-lattice ordered groups. First we deal with the free abelian lattice case. We use this as a stepping-stone to tackle product systems over quasi-lattices that are controlled by the free abelian lattice and satisfy a minimality property. Our setting accommodates examples like the Baumslag–Solitar lattice for $n=m>0$ and the right-angled Artin groups. More generally, the class of quasi-lattices for which our results apply is closed under taking semi-direct and graph products. In the process we accomplish more. Our arguments tackle Nica–Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the coefficient algebra. This is the case for CNP-relative quotients in-between the Toeplitz–Nica–Pimsner algebra and the Cuntz–Nica–Pimsner algebra. We complete this study with the relevant results on exactness.

Keywords

C*-correspondencesproduct systemsNica–Pimsner algebrasexactnessnuclearity

MSC classification

Primary: 46L08: $C^*$-modules 46L05: General theory of $C^*$-algebras
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3375 - 3402
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Copyright
© Cambridge University Press, 2019

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