Stabilization theorem and symmetric structure of Cuntz--Pimsner algebras

Abstract

We establish a crossed product decomposition theorem for stabilized Cuntz--Pimsner algebras. This extends Cuntz's classical decomposition for Cuntz algebras and reveals an implicit symmetric structure within these algebras. Exploiting this structure, we characterize their simplicity and classify their ideals, tracial weights, and KMS weights for generalized quasi-free flows, revisiting and refining seminal results by Kitamura, Schweizer, and Laca--Neshveyev. We also give a short, elementary solution to the reduced Hao--Ng isomorphism problem for locally compact groups in full generality. Bypassing non-self-adjoint operator algebra techniques used in prior work, our proof relies solely on C*-algebra theory. In contrast, we disprove the full Hao--Ng conjecture by constructing counterexamples. Combining these results, we investigate quasi-free actions on Cuntz algebras. Notably, we affirmatively answer a recent question posed by Izumi on isometric shift-absorption for compact groups.

Major revision v3: Now we give a very short solution to the reduced Hao--Ng isomorphism problem in full generality. (NB: the proof changed from (v2). The new proof does not depend on any other results in this paper.) We also give counterexamples to the full Hao--Ng isomorphism conjecture. Other minor updates. 49 pages