Remarks on Compact Automorphism Groups of a Certain von Neumann Algebra

Abstract

Two remarks are given for a (faithful) continuous action of a compact group G on a von Neumann algebra M. For a G-invariant normal state ω with central support 1 and with a one-sided G-spectrum, M is shown to be isomorphic to the ω-cyclic part of the fixed point subalgebra M^G under some assumptions. A Galois correspondence is established between closed normal subgroups and von Neumann subalgebras of M containing M^G and globally invariant under G and another subgroup H of Aut M, which commutes with G and acts ergodically on M.