Stabilization theorem and symmetric structure of Cuntz--Pimsner algebras

TL;DR

This paper proves a decomposition theorem for stabilized Cuntz--Pimsner algebras, revealing their symmetric structure, and applies it to classify ideals, weights, and actions, extending classical results.

Contribution

It extends Cuntz's classical decomposition to Cuntz--Pimsner algebras, unveiling their symmetric structure and applying it to classification and dynamics.

Findings

Established a crossed product decomposition theorem for stabilized Cuntz--Pimsner algebras.

Characterized simplicity and classified ideals, weights, and KMS states.

Confirmed a recent question on isometrically shift-absorption for compact groups.

Abstract

We establish a crossed product decomposition theorem for stabilized Cuntz--Pimsner algebras. This result extends Cuntz's classical decomposition for the Cuntz algebras $\mathcal{O}_n$ and reveals an implicit symmetric structure within Cuntz--Pimsner algebras. By exploiting this structure, we characterize the simplicity of these algebras and classify ideals, tracial weights, and KMS weights for generalized quasi-free flows. Our findings recover and refine seminal results in the literature, including those by Kitamura, Schweizer, and Laca--Neshveyev. By combining our main results with the Hao--Ng isomorphism, we study quasi-free actions on $\mathcal{O}_n$. We confirm a recent question on isometrically shift-absorption posed by Izumi on compact groups. We also identify a new dichotomy for the group $G:=\mathbb{R} \times {\rm SU}(2)$: in contrast to flows, the crossed product of a quasi-free action of $G$ on $\mathcal{O}_n$ is either non-simple or purely infinite simple.