Stabilization theorem and symmetric structure of Cuntz--Pimsner algebras

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.
As another application, we provide a short, alternative, and independent solution to the reduced Hao--Ng isomorphism problem. Unlike previous approaches, which heavily rely on non-self-adjoint operator algebras and their C*-envelopes, our proof builds solely on basic facts about C*-algebras.
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 isometric shift-absorption posed by Izumi for 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.