Description of the automorphism group \(\text{Aut}({\mathcal A}/{\mathcal A}^\alpha)\) for a minimal action of a compact Kac algebra and its application
type: Item
instance of: scholarly article
title: Description of the automorphism group \(\text{Aut}({\mathcal A}/{\mathcal A}^\alpha)\) for a minimal action of a compact Kac algebra and its application
zbMATH Open document ID: 1024.46020
author: Takehiko Yamanouchi
published in: Journal of Functional Analysis
publication date: 2002-11-28T00:00:00Z
review text: In the paper under review, the author continues his investigations concerning minimal actions of compact Kac algebras from [Math. Scand. 84, 297-319 (1999; Zbl 0968.46047) and Hokkaido Math. J. 28, 409-434 (1999; Zbl 0937.46061)]. Let \(\mathbb{K}\) be a compact Kac algebra and let \(\alpha \) be a minimal action of \(\mathbb{K}\) on a factor \(A\). The aim of this paper is to describe the group of all automorphisms of \(A\) leaving the fixed-point algebra \(A^\alpha \) pointwise invariant. Using a result of \textit{J. De Cannière} [Proc. Lond. Math. Soc. (3) 40, 1-20 (1980; Zbl 0436.46048)], the author shows that this group endowed with the topology of simple convergence of the predual is topologically isomorphic to the intrinsic group \(G(\widehat{\mathbb{K}}^{\prime })\) endowed with the strong-operator topology, where \(\widehat{\mathbb{K}}\) is the dual Kac algebra. As an application, it is shown that, when \(\mathbb{K}\) is finite-dimensional, the left coideal of \(\mathbb{K}\) determined by the normalizer (group) of \(A^\alpha \) in \(A\) through the Galois correspondence is the Kac subalgebra \(G(\widehat{\mathbb{K}})^{\prime \prime }\). Moreover it is shown that if \(\mathbb{K}\) is a finite-dimensional compact Kac algebra and \(\alpha \) is a minimal action of \(\mathbb{K}\) on the AFD \(II_1\) factor \(A\), then \(\mathbb{K}\) is cocommutative if and only if \(A^\alpha \subseteq A\) contains a common Cartan subalgebra. This extends to the context of Kac algebras a result due to \textit{V. Jones} and \textit{S. Popa} [Operator Theory, Adv. Appl. 6, 89-102 (1982; Zbl 0508.46041)] about actions of a finite-group \(G\) on the AFD \(II_1\) factor \(A\).
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 47L50
zbMATH DE Number: 1836685
zbMATH Keywords: compact Kac algebras
zbMATH Keywords: minimal actions
zbMATH Keywords: factors
zbMATH Keywords: Galois correspondence
zbMATH Keywords: Cartan subalgebras
reviewed by: Maria Joiţa
MaRDI profile type: Publication
full work available at URL: jfan.2002.3950
OpenAlex ID: W2007810492
cites work: Q4112980
cites work: On the Intrinsic Group of a Kac Algebra
cites work: Produit croise d'une algebre de von Neumann par une algebre de Kac
cites work: Q4529376
cites work: Produit croise d'une algebre de von Neumann par une algebre de Kac. II
cites work: Q4023638
cites work: Ergodic Equivalence Relations, Cohomology, and Von Neumann Algebras. II
cites work: Q4529383
cites work: Amenable tensor categories and their realizations as AFD bimodules
cites work: A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras
cites work: Q4746311
cites work: Extension of Jones' theory on index of arbitrary factors
cites work: A duality for Hopf algebras and for subfactors. I
cites work: Duality for crossed products of von Neumann algebras
cites work: On a problem of R.V. Kadison on maximal Abelian *-subalgebras in factors
cites work: Classification of subfactors: The reduction to commuting squares
cites work: Sur la structure des algèbres de Kac. I. (On the structure of Kac algebras. I.)
cites work: Cartan Subalgebras in Fixed Point Algebras of Finite Group Actions.
cites work: AN ANALOGUE OF PASCHKE'S THEOREM FOR ACTIONS OF COMPACT KAC ALGEBRAS
cites work: Q4310049
cites work: duality and subgroups
cites work: A Characterization of Normal Extensions for Subfactors
cites work: Normal intermediate subfactors
cites work: A minimal action of the compact quantum group \(SU_q(n)\) on a full factor
cites work: Q3702812
cites work: CONSTRUCTION OF AN OUTER ACTION OF A FINITE-DIMENSIONAL KAC ALGEBRA ON THE AFD FACTOR OF TYPE II1
cites work: On the dominancy of minimal actions of compact Kac algebras and certain automorphisms in $\mathrm{Aut}(A/A^\alpha)$
cites work: The Connes spectrum for actions of compact Kac algebras and factoriality of their crossed products
DOI: 10.1006/JFAN.2002.3950
description: scientific article; zbMATH DE number 1836685