Fixed-point permanence under actions by finite quantum groups
Alexandru Chirvasitu
TL;DR
The paper extends the Jones–Takesaki permanence results to actions by finite quantum groups on von Neumann algebras, showing that $M$ and the fixed-point subalgebra $M^{\mathbb{G}}$ share the same canonical central decompositions for types $I$, $II$, and $III$, as well as for atomic, diffuse, and hyperfinite (AFD) properties. The authors develop a general framework using Pimsner–Popa (PP) finite-index or strongly PP expectations and module-theoretic lifting techniques to transfer completely ideal properties along fixed-point inclusions, with central projections coinciding across $M$ and $M^{\mathbb{G}}$. A key technical step is establishing that the fixed-point expectation $E_{\mathrm{triv}}$ is PP in the finite-quantum-group setting via a local isotypic decomposition bound $E(x^*x)\ge \lambda_{\alpha} E_{\alpha}(x)^*E_{\alpha}(x)$ for each $\alpha$, yielding a global bound $\lambda$ and enabling the permanence results. This work generalizes the classical finite-group case and provides a robust operator-algebraic framework for understanding how quantum symmetries preserve type, finiteness, and approximation properties. The results offer tools for analyzing how quantum group actions interact with the structure theory of von Neumann algebras, with potential implications for quantum symmetry in operator algebras and related fields.
Abstract
Given an action by a finite quantum group $\mathbb{G}$ on a von Neumann algebra $M$, we prove that a number of familiar $W^*$ properties are equivalent for $M$ and the fixed-point algebra $M^{\mathbb{G}}$ (i.e. hold or not simultaneously for the two algebras); these include being hyperfinite, atomic, diffuse and of type $I$, $II$ or $III$. Moreover, in all cases the canonical central projections of $M$ and $M^{\mathbb{G}}$ cutting out the summand with the respective property coincide. The result generalizes its classical-$\mathbb{G}$ analogue due to Jones-Takesaki.