Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.11143

Mean dimension and radius of comparison

George A. Elliott·Zhuang Niu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Consider a minimal and free topological dynamical system $(X, Γ)$, where $Γ$ is a discrete amenable group. Assume $(X, Γ)$ has the uniform Rokhlin property (URP) and the Cuntz comparison of open sets (COS), then it is shown that $$\mathrm{rc}(\mathrm{C}(X) \rtimes Γ) = \frac{1}{2} \mathrm{mdim}(X, Γ).$$ The same statement also holds for unital simple AH algebras with diagonal maps. The arguments in both cases are based on the recent preprint of OpenAI dealing with $\mathbb Z$-actions and the topological ideas used in that paper.

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

George A. Elliott, Zhuang Niu. 2026-10-08. Mean dimension and radius of comparison. https://arxiv.org/abs/2610.11143

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Distances Between von Neumann Subalgebras: Spin Models, Commuting Squares, and Free Group Factors

We investigate the relative position of von Neumann subalgebras through the Mashood--Taylor ($\mathrm{d}_{\mathrm{MT}}$) and Kadison--Kastler ($\mathrm{d}_{\mathrm{KK}}$) distances and the interior angle. For each $n\in\mathbb{N}$, we show that the hyperfinite $\mathrm{II}_1$-factor $\mathscr{R}$ contains an uncountable family of pairwise distinct regular $n\times n$ spin model subfactors. In particular, this yields a continuous family $(\mathscr{R}_{\mathsf{H}_α})_{α\in[0,π)}$ of $2\times2$ spin model subfactors which, equipped with $\mathrm{d}_{\mathrm{MT}}$, is homeomorphic to the circle group $\mathbb{T}$ and admits a natural topological group structure. We further establish \[ \mathrm{d}_{\mathrm{MT}}(\mathscr{R}_{\mathsf{H}_α},\mathscr{R}_{\mathsf{H}_β}) =\vert\sin(α-β)\vert. \] We prove that two such spin model subfactors form a commuting square over their intersection if and only if they are maximally distant. More generally, commuting squares of $\mathrm{II}_1$-factors, under natural index conditions, force maximal distance, yielding $\mathrm{d}_{\mathrm{KK}}=1=\mathrm{d}_{\mathrm{MT}}$. We also show that diffuse subalgebras which are orthogonal in the sense of Popa are maximally distant. In contrast, no two members of the above spin model family are Popa-orthogonal. Nevertheless, maximal distance between two such subfactors implies that their interior angle over the intersection is $π/2$, highlighting the distinction between angle orthogonality and Popa orthogonality. Finally, in $L(\mathbb{F}_2)=L(\langle a,b\rangle)$, for $u\in L(\langle b\rangle)$, we obtain \[ \mathrm{d}_{\mathrm{MT}}(L(\langle a\rangle),uL(\langle a\rangle)u^*) =\sqrt{1-\vertτ(u)\vert^4}, \] and construct maximally distant masas not arising from subgroups of $\mathbb{F}_2$.

math.OA↗
arXiv2026

Three-Point Trace Positivity:Schatten Norm Compression and Interpolating Metrics

Let $(\mathcal N,ν)$ be a finite tracial von Neumann algebra and set $\mathcal{K}_r(P,Q)=ν((P^{1/2}QP^{1/2})^r)$. We prove that $$ [\mathcal{K}_r(P_i,P_j)]_{i,j=1}^3\ge0, \qquad 1/2\le r\le1. $$ The range $1/2<r<1$ is new even for matrices, and order three is sharp for $1/2\le r<1$. This single mechanism resolves two problems. First, if $(\mathcal R,ω)$ is finite tracial and $H=[H_{ij}]_{i,j=1}^3\in M_3(\mathcal R)_+$, then $$ [ω(|H_{ij}|^p)]_{i,j=1}^3\ge0, \qquad 1\le p\le2, $$ settling the Lin--van den Driessche conjecture on Schatten norm compression. Second, for a semifinite tracial von Neumann algebra $(\mathcal M,τ)$, define $$ d_t(A,B)^2=\frac{τ(A)+τ(B)}2- \|B^{t/4}A^{t/4}\|_{2/t}^{2/t}. $$ We prove that $d_t$ is a complete metric on $L^1(\mathcal M,τ)_+$ for $1\le t\le2$, including the previously open range $1<t<2$ between the Hellinger and Bures endpoints, and that $$ d_t\text{ is a metric on }L^1(\mathcal M,τ)_+ \quad\Longleftrightarrow\quad t\in[1,2]\ \text{or}\ \mathcal M\text{ is abelian}. $$ Its topology and Cauchy sequences agree with those of $L^1$. The same classification holds for unital tracial $C^*$-algebras and, with the known $t=0$ obstruction, resolves Problem~2 of Komálovics and Molnár.

math.OA↗
arXiv2026

Banach non-isomorphism of type III and semifinite noncommutative $L_p$-spaces

Let $M$ be a nonzero type III von Neumann algebra with separable predual, and let $N$ be a semifinite von Neumann algebra. We prove that $M_*$ is not isomorphic to a subspace of $N_*$. For $1<p<\infty$, $p\ne2$, we prove that $L_p(M)$ is not isomorphic to a complemented subspace of $L_p(N)$. In particular, the Banach isomorphism class of $L_p(M)$ distinguishes type III algebras from semifinite algebras for $1\le p<\infty$, $p\ne2$.

math.OA↗