2026/05/20 by Miho Mukohara, Yuhei Suzuki
#math.OA
We establish a crossed product decomposition theorem for stabilized Cuntz--Pimsner algebras. This result extends Cuntz's classical decomposition for the Cuntz algebras On 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 On. 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:=ℝ × \rm SU(2): in contrast to flows, the crossed product of a quasi-free action of G on On is either non-simple or purely infinite simple.