The structure of C*-algebras of product systems

Abstract

A prevalent trend in the theory of operator algebras is the study of geometric/topological structures via bounded linear operators on a Hilbert space. The goal is to establish a rigid correspondence between such a structure and a C*-algebra, and use the rich theory of the latter to study the former. This approach has been met with much success in recent years, revealing surprising links with quantum mechanics, graphs, groups, dynamics, subshifts and more. Initially these applications were studied individually; however, the introduction of C*-correspondences and product systems within the past thirty years has presented a unifying framework. Broadly speaking, C*-correspondences and their C*-algebras account for low-rank examples (e.g., directed graphs) and are by now well explored. The more general product systems and their C*-algebras account for higher-rank examples (e.g., higher-rank graphs) and less is known in this context. In turn, there is motivation to analyse the structure of C*-algebras of product systems and interpret the results with respect to the applications that these objects encompass. The current work falls within the remit of this programme, and focuses on the gauge invariant ideal structure of C*-algebras associated with the subclass of strong compactly aligned product systems. We parametrise the gauge-invariant ideals of every equivariant quotient of the Toeplitz-Nica-Pimsner algebra (most importantly the Cuntz-Nica-Pimsner algebra) via tuples of ideals of the coefficient algebra. We describe the conditions defining these families via product system operations alone. In the process, we prove a Gauge Invariant Uniqueness Theorem. We characterise the lattice operations on the parametris ing families such that the bijection is a lattice isomorphism. We then interpret the main re sult in the settings of regular product systems, C*-dynamical systems, higher-rank graphs and product systems on finite frames. We close by examining the case of proper product systems in further detail.