Description of the automorphism group Aut(A/A(alpha)) for a minimal action of a compact Kac algebra and its application

摘要

It is shown that, for a minimal action a of a compact Kac algebra 116 on a factor A, the group of all automorphisms leaving the fixed-point algebra A(alpha) pointwise invariant is topologically isomorphic to the intrinsic group of the dual Kac algebra K. As an application, in the case where dim K < infinity, the left (in fact, two-sided) coideal of K determined by the normalizer (group) of A(alpha) 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 A(alpha) subset of or equal to A contains a common Cartan subalgebra. This result is an extension of a result due to Jones and Popa. (C) 2002 Elsevier Science (USA). [References: 28]