arXiv subjects

Yuki Arano

Publications and source records attributed to Yuki Arano.

11 recordsLinked to original sources

arXiv2023

Tensor category equivariant KK-theory

In this paper, we introduce Kasparov's bivariant K-theory that is equivariant under symmetries of a C*-tensor category. It is motivated by some dualities in quantum group equivariant KK-theory, and the classification theory of inclusions of C*-algebras. The fundamental properties of the KK-theory, i.e., the existence of the Kasparov product, Cuntz's picture, universality, and triangulated category structure, hold true in this generalization as well. Moreover, we further prove a new property specific to this theory; the invariance of KK-theory under weak Morita equivalence of the tensor categories. As an example, we study the Baum-Connes type property for $3$-cocycle twists of discrete groups.

math.OA↗

arXiv2021

On the Baum--Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg Conjecture

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $Γ$ implies that $Γ$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition.

math.OA↗

arXiv2020

Ergodic theory of affine isometric actions on Hilbert spaces

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type $\mathrm{III}_1$.

math.DS↗

arXiv2017

The Fourier algebra of a rigid $C^{\ast}$-tensor category

Completely positive and completely bounded mutlipliers on rigid $C^{\ast}$-tensor categories were introduced by Popa and Vaes. Using these notions, we define and study the Fourier-Stieltjes algebra, the Fourier algebra and the algebra of completely bounded multipliers of a rigid $C^{\ast}$-tensor category. The rich structure that these algebras have in the setting of locally compact groups is still present in the setting of rigid $C^{\ast}$-tensor categories. We also prove that Leptin's characterization of amenability still holds in this setting, and we collect some natural observations on property (T).

math.OA↗

arXiv2017

Howe-Moore type theorems for quantum groups and rigid C*-tensor categories

We formulate and study Howe-Moore type properties in the setting of quantum groups and in the setting of rigid $C^{\ast}$-tensor categories. We say that a rigid $C^{\ast}$-tensor category $\mathcal{C}$ has the Howe-Moore property if every completely positive multiplier on $\mathcal{C}$ has a limit at infinity. We prove that the representation categories of $q$-deformations of connected compact simple Lie groups with trivial center satisfy the Howe-Moore property. As an immediate consequence, we deduce the Howe-Moore property for Temperley-Lieb-Jones standard invariants with principal graph $A_{\infty}$. These results form a special case of a more general result on the convergence of completely bounded multipliers on the aforementioned categories. This more general result also holds for the representation categories of the free orthogonal quantum groups and for the Kazhdan-Wenzl categories. Additionally, in the specific case of the quantum groups $\mathrm{SU}_q(N)$, we are able, using a result of the first-named author, to give an explicit characterization of the central states on the quantum coordinate algebra of $\mathrm{SU}_q(N)$, which coincide with the completely positive multipliers on the representation category of $\mathrm{SU}_q(N)$.

math.OA↗

arXiv2016

Comparison of unitary duals of Drinfeld doubles and complex semisimple Lie groups

We determine a substantial part of the unitary representation theory of the Drinfeld double of a $q$-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every $q$-deformation of a higher rank compact Lie group has central property ${\rm (T)}$. We also determine the unitary dual of $SL_q(n,\mathbb{C})$.

math.QA↗

arXiv2016

C*-tensor categories and subfactors for totally disconnected groups

We associate a rigid C*-tensor category $C$ to a totally disconnected locally compact group $G$ and a compact open subgroup $K < G$. We characterize when $C$ has the Haagerup property or property (T), and when $C$ is weakly amenable. When $G$ is compactly generated, we prove that $C$ is essentially equivalent to the planar algebra associated by Jones and Burstein to a group acting on a locally finite bipartite graph. We then concretely realize $C$ as the category of bimodules generated by a hyperfinite subfactor.

math.OA↗

arXiv2015

A categorical perspective on the Atiyah-Segal completion theorem in $\mathrm{KK}$-theory

We investigate the homological ideal $\mathfrak{J}_G^H$, the kernel of the restriction functors in compact Lie group equivariant Kasparov categories. Applying the relative homological algebra developed by Meyer and Nest, we relate the Atiyah-Segal completion theorem with the comparison of $\mathfrak{J}_G^H$ with the augmentation ideal of the representation ring. In relation to it, we study on the Atiyah-Segal completion theorem for groupoid equivariant $\mathrm{KK}$-theory, McClure's restriction map theorem, permanence property of the Baum-Connes conjecture under extensions of groups and a class of $\mathfrak{J}_G$-injective objects coming from $\mathrm{C}^*$-dynamical systems, continuous Rokhlin property.

math.KT↗

arXiv2015

Compact Lie group actions with continuous Rokhlin property

In this paper, we study continuous Rokhlin property of $\mathrm{C}^*$-dynamical systems using techniques of equivariant $\mathrm{KK}$-theory and quantum group theory. In particular, we determine the $\mathrm{KK}$-equivalence class and give a classification of Kirchberg $G$-algebras when the $G$ is a compact Lie group with Hodgkin condition.

math.OA↗

arXiv2015

Torsion-freeness for fusion rings and tensor C*-categories

Torsion-freeness for discrete quantum groups was introduced by R. Meyer in order to formulate a version of the Baum-Connes conjecture for discrete quantum groups. In this note, we introduce torsion-freeness for abstract fusion rings. We show that a discrete quantum group is torsion-free if its associated fusion ring is torsion-free. In the latter case, we say that the discrete quantum group is strongly torsion-free. As applications, we show that the discrete quantum group duals of the free unitary quantum groups are strongly torsion-free, and that torsion-freeness of discrete quantum groups is preserved under Cartesian and free products. We also discuss torsion-freeness in the more general setting of abstract rigid tensor C*-categories

math.RA↗

arXiv2015

Unitary spherical representations of Drinfeld doubles

We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of $SU_q(3)$, we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group $SU_q(2n+1)$ has property (T), which also implies central property (T) of the dual of $SU_q(2n+1)$.

math.QA↗