担当区分：筆頭著者   記述言語：英語   掲載種別：研究論文（学術雑誌）

In this paper, we investigate certain submodules in C∗-algebras associated to effective étale groupoids. First, we show that a submodule generated by normalizers is a closure of the set of compactly supported continuous functions on some open set. As a corollary, we show that discrete group coactions on groupoid C∗-algebras are induced by cocycles of étale groupoids if the fixed point algebras contain C∗-subalgebras of continuous functions vanishing at infinity on the unit spaces. In the latter part, we prove the Galois correspondence result for discrete group coactions on groupoid C∗-algebras.

DOI： 10.1515/forum-2023-0182

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Scopus

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担当区分：筆頭著者   記述言語：英語   掲載種別：研究論文（学術雑誌）

For a given inverse semigroup action on a topological space, one can associate an étale groupoid. We prove that there exists a correspondence between the certain subsemigroups and the open wide subgroupoids in case that the action is strongly tight. Combining with the recent result of Brown et al., we obtain a correspondence between the certain subsemigroups of an inverse semigroup and the Cartan intermediate subalgebras of a groupoid C∗-algebra.

DOI： 10.1017/S0013091522000402

Kyutacar

Scopus

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担当区分：筆頭著者   記述言語：英語   掲載種別：研究論文（学術雑誌）

For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept for considering quotients of inverse semigroups. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization.

DOI： 10.1017/S1446788721000021

Kyutacar

Scopus

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担当区分：筆頭著者   記述言語：英語   掲載種別：研究論文（学術雑誌）

In this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by a Cuntz-Krieger uniqueness theorem for a universal groupoid C*-algebra.

DOI： 10.1017/S1446788720000154

Kyutacar

Scopus

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記述言語：英語   掲載種別：研究論文（学術雑誌）

Kernel methods have been among the most popular techniques in machine learning, where learning tasks are solved using the property of reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis framework with reproducing kernel Hilbert C∗-module (RKHM) and kernel mean embedding (KME) in RKHM. Since RKHM contains richer information than RKHS or vector-valued RKHS (vvRKHS), analysis with RKHM enables us to capture and extract structural properties in such as functional data. We show a branch of theories for RKHM to apply to data analysis, including the representer theorem, and the injectivity and universality of the proposed KME. We also show RKHM generalizes RKHS and vvRKHS. Then, we provide concrete procedures for employing RKHM and the proposed KME to data analysis.

Kyutacar

Scopus

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