Description of the automorphism group \(\text{Aut}({\mathcal A}/{\mathcal A}^\alpha)\) for a minimal action of a compact Kac algebra and its application
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 \(A\).
scientific article; zbMATH DE number 1836685
scientific article