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Couniversality and controlled maps on product systems over right LCM semigroups

Evgenios T.A. Kakariadis, Elias G. Katsoulis, Marcelo Laca and Xin Li

Vol. 16 (2023), No. 6, 1433–1483
Abstract

We study the structure of C-algebras associated with compactly aligned product systems over group embeddable right LCM  semigroups. Towards this end we employ controlled maps and a controlled elimination method that associates the original cores to those of the controlling pair, and we combine these with applications of the C-envelope theory for cosystems of nonselfadjoint operator algebras recently produced. We derive several applications of these methods that generalize results on single C-correspondences.

First we show that if the controlling group is exact then the couniversal C-algebra of the product system coincides with the quotient of the Fock C-algebra by the ideal of strong covariance relations. We show that if the controlling group is amenable then the product system is amenable. In particular if the controlling group is abelian then the couniversal C-algebra is the C-envelope of the tensor algebra.

Secondly we give necessary and sufficient conditions for the Fock C-algebra to be nuclear and exact. When the controlling group is amenable we completely characterize nuclearity and exactness of any equivariant injective Nica-covariant representation of the product system.

Thirdly we consider controlled maps that enjoy a saturation property. In this case we induce a compactly aligned product system over the controlling pair that shares the same Fock representation, and preserves injectivity. By using couniversality, we show that they share the same reduced covariance algebras. If in addition the controlling pair is a total order then the fixed point algebra of the controlling group induces a super product system that has the same reduced covariance algebra and is moreover reversible.

Keywords
product systems, Nica–Pimsner algebras, $C^*$-envelope
Mathematical Subject Classification
Primary: 46L05, 46L08
Milestones
Received: 20 August 2021
Accepted: 20 December 2021
Published: 23 August 2023
Authors
Evgenios T.A. Kakariadis
School of Mathematics, Statistics and Physics
Newcastle University
Newcastle upon Tyne
United Kingdom
Elias G. Katsoulis
Department of Mathematics
East Carolina University
Greenville, NC
United States
Marcelo Laca
Department of Mathematics and Statistics
University of Victoria
Victoria, BC
Canada
Xin Li
School of Mathematics and Statistics
University of Glasgow
Glasgow
United Kingdom

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