arXiv subjects

Kan Kitamura

Publications and source records attributed to Kan Kitamura.

9 recordsLinked to original sources

arXiv2025

Prime and semiprime Lie ideals in C*-algebras

Using the theory of Dixmier ideals developed in previous work, we show that every semiprime Lie ideal in a C*-algebra arises as the full normalizer subspace of a semiprime two-sided ideal. This leads to a concise description of all semiprime Lie ideals in terms of semiprime two-sided ideals, and an analogous description of prime Lie ideals in terms of prime two-sided ideals. For unital C*-algebras without characters, we obtain a natural bijection between (semi)prime two-sided ideals and (semi)prime Lie ideals, and it follows that a Lie ideal is fully noncentral if and only if it is not contained in any prime Lie ideal.

math.OA↗

arXiv2024

The stable uniqueness theorem for unitary tensor category equivariant KK-theory

We introduce the Cuntz-Thomsen picture of $\mathcal{C}$-equivariant Kasparov theory, denoted $\mathrm{KK}^\mathcal{C}$, for a unitary tensor category $\mathcal{C}$ with countably many isomorphism classes of simple objects. We use this description of $\mathrm{KK}^\mathcal{C}$ to prove the stable uniqueness theorem in this setting.

math.OA↗

arXiv2024

Actions of tensor categories on Kirchberg algebras

We characterize the simplicity of Pimsner algebras for non-proper C*-correspondences. With the aid of this criterion, we give a systematic strategy to produce outer actions of unitary tensor categories on Kirchberg algebras. In particular, every countable unitary tensor category admits an outer action on the Cuntz algebra $\mathcal{O}_2$. We also study the realizability of modules over fusion rings as K-groups of Kirchberg algebras acted on by unitary tensor categories, which turns out to be generically true for every unitary fusion category. Several new examples are provided, among which actions on Cuntz algebras of 3-cocycle twists of cyclic groups are constructed for all possible 3-cohomological classes, thereby answering a question asked by Izumi.

math.OA↗

arXiv2024

A note on quantum subgroups of free quantum groups

In this short note, quantum subgroups in finite free products of the Pontryagin duals of free unitary quantum groups are classified. They correspond to pairs of a subgroup $\Gamma$ and a subset $S$ of the free group $\mathbb{F}_n$ such that $S$ is $\Gamma$-invariant, containing $\Gamma$, and connected in the Cayley graph of $\mathbb{F}_n$

math.OA↗

arXiv2023

Semiprime ideals in C*-algebras

We show that a not necessarily closed ideal in a C*-algebra is semiprime if and only if it is idempotent, if and only if it is closed under square roots of positive elements. Among other things, it follows that prime and semiprime ideals in C*-algebras are automatically self-adjoint. To prove the above, we isolate and study a particular class of ideals, which we call Dixmier ideals. As it turns out, there is a rich theory of powers and roots for Dixmier ideals. We show that every ideal in a C*-algebra is squeezed by Dixmier ideals from inside and outside tightly in a suitable sense, from which we are able to deduce information about the ideal in the middle.

math.OA↗

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.

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arXiv2023

Discrete quantum subgroups of complex semisimple quantum groups

We classify discrete quantum subgroups in the quantum double of the $q$-deformation of a compact semisimple Lie group, regarded as the complexification. We also record their classifications in some variants of quantum groups. Along the way, we show that quantum doubles of non-Kac type compact quantum groups do not admit the quantum analog of lattices considered by Brannan-Chirvasitu-Viselter.

math.QA↗

arXiv2022

Partial Pontryagin duality for actions of quantum groups on C*-algebras

We compare actions on C*-algebras of two constructions of locally compact quantum groups, the bicrossed product and the double crossed product. We give a duality between them as a generalization of Baaj-Skandalis duality. In the case of quantum doubles, this duality also preserves monoidal structures given by twisted tensor products. We also discuss its consequences for equivariant Kasparov theories in relation to the quantum analogue of the Baum-Connes conjecture.

math.OA↗

arXiv2021

Induced coactions along a homomorphism of locally compact quantum groups

We consider induced coactions on C*-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coactions and restriction which is analogous to base change formula for modules over algebras. As an application, we give an example that shows several kinds of 1-categories of coactions with forgetful functors cannot recover the original quantum group.

math.OA↗