Co-universality and controlled maps on product systems over right LCM-semigroups

TL;DR

This paper investigates the structure and properties of C*-algebras associated with product systems over right LCM-semigroups, introducing controlled maps and elimination methods to generalize results from single C*-correspondences.

Contribution

It develops a framework using controlled maps and C*-envelope theory to analyze co-universal C*-algebras, extending results to product systems over right LCM-semigroups.

Findings

Co-universal C*-algebra coincides with Fock algebra quotient when the controlling group is exact.

Product system is amenable if the controlling group is amenable.

Characterization of nuclearity and exactness for Fock C*-algebra and representations.

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 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 co-universal 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 co-universal 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 co-universality, 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.