Description of the Automorphism Group Aut([formula omitted]/[formula omitted]α) for a Minimal Action of a Compact Kac Algebra and Its Application

Abstract

It is shown that, for a minimal action α of a compact Kac algebra K on a factor A, the group of all automorphisms leaving the fixed-point algebra Aα pointwise invariant is topologically isomorphic to the intrinsic group of the dual Kac algebra K̂. As an application, in the case where dim K<,∞, the left (in fact, two-sided) coideal of K determined by the normalizer (group) of Aα in A through the Izumi–Longo–Popa (Galois) correspondence is identified. As a consequence, we prove that, when A is the AFD II1 factor, K is cocommutative if and only if Aα⊆A contains a common Cartan subalgebra. This result is an extension of a result due to Jones and Popa.