A remark on compact automorphism groups of \(C^*\)-algebras (Q1074071)

The following results is proved. A \(C^*\)-system over a compact group whose fixed-point algebra has trivial centre and relative commutant coinciding with the centre of the full algebra is obtained canonically by inducing up from a \(C^*\)-system over a closed subgroup which now has trivial relative commutant. This result is used in an essential way to determine the compact Lie group associated with certain endomorphisms of \(C^*\)-algebras, in the course of the development of a new duality theory for compact groups in terms of \(C^*\)-algebras [cf. by the same authors: Bull. Am. Math. Soc., New Ser. 11, 333-338 (1984; Zbl 0561.46029); ''Duals of compact groups realized in the Cuntz algebras and their actions on \(C^*\)-algebras'', J. Funct. Anal., to appear; Proceedings of the Marseille Conference on Mathematical Physics, July 1986, World Scientific P. Co., to appear].