Cuntz-Pimsner algebras for subproduct systems

TL;DR

This paper extends the concept of Cuntz-Pimsner algebras from $C^*$-correspondences to subproduct systems, providing a generalized framework with theoretical justification and illustrative examples.

Contribution

It introduces a new construction of Cuntz-Pimsner algebras for subproduct systems, broadening the scope of operator algebra theory.

Findings

Establishes Morita equivalence of the new algebras under certain conditions

Provides examples illustrating the naturality of the construction

Discusses limitations and differences from classical Cuntz-Pimsner algebras

Abstract

In this paper we generalize the notion of Cuntz-Pimsner algebras of $C^*$-correspondences to the setting of subproduct systems. The construction is justified in several ways, including the Morita equivalence of the operator algebras under suitable conditions, and examples are provided to illustrate its naturality. We also demonstrate why some features of the Cuntz-Pimsner algebras of $C^*$-correspondences fail to generalize to our setting, and discuss what we have instead.