arXiv subjects

Shuoxing Zhou

Publications and source records attributed to Shuoxing Zhou.

8 recordsLinked to original sources

arXiv2026

ICC property (T) groups without W$^*$-superrigidity

We construct two explicit countable discrete groups $\Gamma_1$ and $\Gamma_2$ that are both ICC and have Kazhdan's property (T). Although $\Gamma_1$ and $\Gamma_2$ are not isomorphic as groups, their group von Neumann algebras are isomorphic: $L(\Gamma_1)\cong L(\Gamma_2)$. This provides a counterexample to Connes' rigidity conjecture for ICC property (T) groups. This result was obtained with assistance from GPT-5.6 Sol, independently of and concurrently with work by OpenAI.

math.OA↗

arXiv2026

On invariant subalgebras of noncommutative Poisson boundaries for higher rank lattices

Let $G$ be a real connected semisimple Lie group with trivial center, no non-trivial compact factors, and all simple factors of real rank at least two. Let $\Gamma<G$ be an irreducible lattice and $P<G$ be a minimal parabolic subgroup. Amrutam--Hartman conjectured that every $\Gamma$-invariant von Neumann subalgebra $M\subset L^\infty(G/P,\nu_P)\rtimes \Gamma$ is of the form $L^\infty(G/Q,\nu_Q)\rtimes\Lambda$. We prove this classification whenever either $M\cap L^\infty(G/P,\nu_P)\neq\mathbb{C}1$ or $M\cap L(\Gamma)\neq\mathbb{C}1$, thereby reducing the Amrutam--Hartman conjecture to the extreme case $M\cap L^\infty(G/P,\nu_P)=M\cap L(\Gamma)=\mathbb{C}1$.

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arXiv2026

Non-commutative Factor theorem for tensor products of lattices in product groups

We establish a non-commutative version of the Intermediate Factor Theorem for crossed products associated with product lattices. Given an irreducible lattice $Γ< G= G_1 \times \dots \times G_d$ in higher rank semisimple algebraic groups and a trace-preserving irreducible action $G \curvearrowright (\mathcal{N}, τ)$, we show that every intermediate von Neumann algebra between $\mathcal{N}\rtimesΓ$ and $(L^\infty(G/P,ν_P)\overline{\otimes}\mathcal{N})\rtimesΓ$ is again a crossed product of the form $(L^\infty(G/Q,ν_Q)\overline{\otimes}\mathcal{N})\rtimesΓ$.

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arXiv2025

Non-commutative Intermediate Factor theorem associated with $W^*$-dynamics of product groups

Let $G = G_{1} \times G_{2}$ be a product of two locally compact, second countable groups and $μ\in \mathrm{Prob}(G)$ be of the form $μ= μ_{1} \times μ_{2}$, where $μ_{i} \in \mathrm{Prob}(G_{i})$. Let $(B,ν_B)$ be the associated Poisson boundary. We show that every intermediate $G$-von Neumann algebra $\mathcal{M}$ with \[ \mathcal{N} \subseteq \mathcal{M} \subseteq \mathcal{N} \,\bar{\otimes}\, L^{\infty}(B,ν) \] splits as a tensor product of the form $\mathcal{N}\bar{\otimes}L^{\infty}(C,ν_C)$, where $(C,ν_C)$ is a $(G,μ)$-boundary. Here, $\mathcal{N}$ is a tracial von Neumann algebra on which $G$ acts trace-preservingly. This generalizes the Intermediate Factor Theorem proved by Bader--Shalom (\cite[Theorem~1.9]{BS06}) in the measurable setup. In addition, we give various other examples of the splitting phenomenon associated with $W^{*}$-dynamics. We also show that certain assumptions are necessary for the intermediate algebras to split, and ideals in the ambient tensor product algebra obstruct the splitting phenomenon. We also use the Master theorem from \cite{glasner2023intermediate} to resolve the second part of \cite[Problem~5.2]{jiangskalski} in the affirmative.

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arXiv2025

Noncommutative topological boundaries and amenable invariant random intermediate subalgebras

As an analogue of the topological boundary of discrete groups $Γ$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M, τ)$ and apply it to generalize the main results of [AHO23], showing that for a trace-preserving action $Γ\curvearrowright (A, τ_A)$ on an amenable tracial von Neumann algebra, any $Γ$-invariant amenable intermediate subalgebra between $A$ and $Γ\ltimes A$ is necessarily a subalgebra of $\mathrm{Rad}(Γ) \ltimes A$. By taking $(A, τ_A) = L^\infty(X, ν_X)$ for a free pmp action $Γ\curvearrowright (X, ν_X)$, we obtain a similar result for the invariant subequivalence relations of $\mathcal{R}_{Γ\curvearrowright X}$.

math.OA↗

arXiv2025

Rigidity of Furstenberg entropy under ucp maps

Given a tracial von Neumann algebra $(M,τ)$, we prove that a state preserving $M$-bimodular ucp map between two stationary W$^*$-extensions of $(M,τ)$ preserves the Furstenberg entropy if and only if it induces an isomorphism between the Radon-Nikodym factors. With a similar proof, we extend this result to quasi-factor maps between stationary spaces of locally compact groups and prove an entropy separation between unique stationary and amenable spaces. As applications, we use these results to establish rigidity phenomena for unique stationary Poisson boundaries.

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arXiv2024

Hypertrace and entropy gap characterizations of property (T) for $\mathrm{II}_1$ factors

We establish a hypertrace characterization of property (T) for $\mathrm{II}_1$ factors: Given a $\mathrm{II}_1$ factors $M$, $M$ does not have property (T) if and only if there exists a von Neumann algebra $\mathcal{A}$ with $M\subset \mathcal{A}$ such that $\mathcal{A}$ admits a $M$-hypertrace but no normal hypertrace. For $M$ without property (T), such an inclusion $M\subset \mathcal{A}$ also admits almost vanishing Furstenberg entropy. With the same construction of $M\subset \mathcal{A}$, we also establish similar characterizations of Haagerup property for $\mathrm{II}_1$ factors.

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arXiv2024

Noncommutative Poisson boundaries, ultraproducts and entropy

We construct the noncommutative Poisson boundaries of tracial von Neumann algebras through the ultraproducts of von Neumann algebras. As an application of this result, we complete the proof of Kaimanovich-Vershik's fundamental theorems regarding noncommutative entropy. We also prove the Amenability-Trivial Boundary equivalence and Choquet-Deny-Type I equivalence for tracial von Neumann algebras.

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