On Toeplitz algebras of product systems

TL;DR

This paper investigates the nuclearity of Toeplitz C*-algebras associated with product systems over group-embeddable monoids, establishing conditions under which nuclearity of the algebra relates to that of the coefficient algebra.

Contribution

It extends known results to broader classes of product systems over monoids, including abelian, $ax+b$, and Baumslag-Solitar monoids, under certain technical assumptions.

Findings

Nuclearity of the Toeplitz algebra is equivalent to nuclearity of the coefficient algebra for specific product systems.

The fixed-point algebra of the gauge action is nuclear iff the coefficient algebra is nuclear, under certain conditions.

For amenable groups, the Toeplitz algebra's nuclearity coincides with that of the coefficient algebra.

Abstract

In the setting of product systems over group-embeddable monoids, we consider nuclearity of the associated Toeplitz C*-algebra in relation to nuclearity of the coefficient algebra. Our work goes beyond the known cases of single correspondences and compactly aligned product systems over right LCM monoids. Specifically, given a product system over a submonoid of a group, we show, under technical assumptions, that the fixed-point algebra of the gauge action is nuclear iff the coefficient algebra is nuclear; when the group is amenable, we conclude that this happens iff the Toeplitz algebra itself is nuclear. Our main results imply that nuclearity of the Toeplitz algebra is equivalent to nuclearity of the coefficient algebra for every full product system of Hilbert bimodules over abelian monoids, over $ax+b$-monoids of integral domains and over Baumslag-Solitar monoids $BS^+(m,n)$ that admit an amenable embedding, which we provide for $m$ and $n$ relatively prime.