# Classification of irreversible and reversible Pimsner operator algebras

## TL;DR

This paper establishes a hierarchy linking the classification of self-adjoint and non-self-adjoint operator algebras, applying it to algebras from $C^*$-correspondences and directed graphs to unify and resolve isomorphism issues.

## Contribution

It introduces a hierarchical framework connecting different classes of operator algebras and applies it to resolve classification problems for algebras from $C^*$-correspondences and directed graphs.

## Key findings

- Unified classification hierarchy for operator algebras.
- Resolved isomorphism problems for algebras from $C^*$-correspondences.
- Complete elucidation of the hierarchy for graph-related operator algebras.

## Abstract

Since their inception in the 30's by von Neumann, operator algebras have been used in shedding light in many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two was sought since their emergence in the late 60's. We connect these seemingly separate type of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^*$-algebras with additional $C^*$-algebraic structure. Our approach naturally applies to algebras arising from $C^*$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.