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
MG under some assumptions. A Galois correspondence is established between closed normal subgroups and von Neumann subalgebras of
M containing
MG and globally invariant under
G and another subgroup
H of Aut
M, which commutes with
G and acts ergodically on
M.
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