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Ryo Toyota

Publications and source records attributed to Ryo Toyota.

9 recordsLinked to original sources
arXiv2026

Warped cones associated to isometric free actions do not have geometric property (T)

Suppose that $G$ is a finitely generated group, $M$ is a compact Riemannian manifold and $G\curvearrowright M$ is a free and isometric action. We prove that the associated discretized warped cone does not have geometric property (T). In contrast, we show that the discretized warped cone associated to the natural action $SL_m(\mathbb Z)\curvearrowright \mathbb T^m$ has geometric property (T) for $m\geq 3$. We also prove that, for free probability-measure-preserving Lipschitz actions on compact Riemannian manifolds, geometric property (T) of the discretized warped cone implies Kazhdan's property (T) of the acting group.

math.OA↗
arXiv2025

The twisted coarse Baum--Connes conjecture and relative hyperbolic groups

In this paper, we introduce a notion of stable coarse algebras for metric spaces with bounded geometry, and formulate the twisted coarse Baum--Connes conjecture with respect to stable coarse algebras. We prove permanence properties of this conjecture under coarse equivalences, unions and subspaces. As an application, we study higher index theory for a group $G$ that is hyperbolic relative to a finite family of subgroups $\{H_1, H_2, \dots, H_N\}$. We prove that $G$ satisfies the twisted coarse Baum--Connes conjecture with respect to any stable coarse algebra if and only if each subgroup $H_i$ does.

math.OA↗
arXiv2025

Non-geometric property (T) of warped cones

In this paper, we study the geometric property (T) for discretized warped cones of an action on a compact Lie group $M$ by its finitely generated subgroup. We show that if a subgroup $G$ is dense in $M$, then the associated discretized warped cone $\bigsqcup_n M\times \{t(n)\}$ does not have geometric property (T) for any sequence of positive numbers $\{t(n)\}_{n\in \mathbb{N}}$ converging to $\infty$. This result applies to certain ergodic actions of groups with property (T), for example, the action of $SO(d,\mathbb{Z}[\frac{1}{5}])$ on $SO(d)$ with $d\geq 5$. As an application, we obtain new examples of expanders without geometric property (T), including certain superexpanders.

math.GR↗
arXiv2023

An Operator-Valued Haagerup Inequality for Hyperbolic Groups

We study an operator-valued generalization of the Haagerup inequality for Gromov hyperbolic groups. In 1978, U. Haagerup showed that if $f$ is a function on the free group $\mathbb{F}_r$ which is supported on the $k$-sphere $S_k=\{x\in \mathbb{F}_r:\ell(x)=k\}$, then the operator norm of its left regular representation is bounded by $(k+1)\|f\|_2$. An operator-valued generalization of it was started by U. Haagerup and G. Pisier. One of the most complete form was given by A. Buchholz, where the $\ell^2$-norm in the original inequality was replaced by $k+1$ different matrix norms associated to word decompositions (this type of inequality is also called Khintchine-type inequality). We provide a generalization of Buchholz's result for hyperbolic groups.

math.OA↗
arXiv2023

Quantum Gromov-Hausdorff convergence of spectral truncations for groups with polynomial growth

For a unital spectral triple $(\mathcal{A}, H,D)$, we study when its truncation converges to itself. The spectral truncation is obtained by using the spectral projection $P_{\Lambda}$ of $D$ onto $[-\Lambda,\Lambda]$ to deal with the case where only a finite range of energy levels of a physical system is available. By restricting operators in $\mathcal{A}$ and $D$ to $P_{\Lambda}H$, we obtain a sequence of operator system spectral triples $\{(P_{\Lambda}\mathcal{A}P_{\Lambda},P_{\Lambda}H,P_{\Lambda}DP_{\Lambda})\}_{\Lambda}$. We prove that if the spectral triple is the one constructed using a discrete group with polynomial growth, then the sequence of operator systems $\{P_{\Lambda}\mathcal{A}P_{\Lambda}\}_{\Lambda}$ converges to $\mathcal{A}$ in the sense of quantum Gromov-Hausdorff convergence with respect to the Lip-norm coming from high order derivatives.

math.OA↗
arXiv2023

Jordan Decomposition of Non-Hermitian Fermionic Quadratic Forms

We give a rigorous proof of Conjecture 3.1 by Prosen [Prosen T 2010 J. Stat. Mech. $\textbf{2010}$ P07020] on the nilpotent part of the Jordan decomposition of a quadratic fermionic Liouvillian. We also show that the number of the Jordan blocks of each size can be expressed in terms of the coefficients of a polynomial called the $q$-binomial coefficient and describe the procedure to obtain the Jordan canonical form of the nilpotent part.

quant-ph↗
arXiv2023

Geometric Property (T) and Positive Cones of Real Algebraic Roe Algebras

We give a characterization of geometric property (T) for a coarse disjoint union of finite graphs with bounded degree using the idea of noncommutative real algebraic geometry. In the proof, we define a $*$-subalgebra $I_u[X]$ of real algebraic Roe algebra $\mathbb{R}_u[X]$ over a graph $X$ with bounded degree. Then we show that $I_u[X]$ contains the Laplacian $\Delta$ as an order unit with respect to the positive cone ${\sum \limits}^2I_u[X]$ which consists of sums of hermitian squares.

math.OA↗
arXiv2022

Controlled $K$-theory and $K$-Homology

Motivated by the idea that our access to the spacetime is limited by the resolution of our measuring device, we give a new description of $K$-homology with a finite resolution. G. Yu introduced a $C^*$-algebra called the localization algebra $C^*_L(X)$ which consists of functions from $[1,\infty)$ to the Roe algebra $C^*(X)$ whose propagations converge to $0$ and he showed that for any finite dimensional simplicial complex $X$ endowed with the spherical metric, the $K$-theory of the localization algebra is isomorphic to the $K$-homology of $X$. We give a coarse graining version of this theorem using controlled $K$-theory (also known as quantitative $K$-theory). Namely, instead of considering families of operators whose propagations converge to $0$, we prove that for each dimension $n$, there exists a threshold $r_n>0$ such that the $K$-homology of $n$-dimensional finite simplicial complex $X$ is isomorphic to a certain group of equivalence classes of operators whose propagation is less than $r_n$. This picture also enables us to represent any element in the $K$-homology group $K_*(X)$ by a finite matrix for a finite simplicial complex $X$.

math.KT↗