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 $G:=\mathbb{R}\times \mathrm{S}\mathrm{U}(2)$: 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.