A linear operator σ on a
C*-algebra
A induces a contraction \hat{σ}
φ on the Hilbert space \mathscr{H}
φ associated with a σ-invariant state φ provided σ satisfies the Schwarz inequality: σ(
a*a)≥σ(
a)
*σ(
a). If φ is invariant under a class \mathscr{S} of such operators, the following four properties are closely connected:
(i) abelianness of the reduction of π
φ(
A) to the \hat{\mathscr{S}}
φ-invariant part of \mathscr{H}
φ,
(ii) asymptotic abelianness of φ,
(iii) abelianness of π
φ(
A)'∩\hat{\mathscr{S}}
φ',
(iv) uniqueness of decompositions of φ into extremal \mathscr{S}-invariant states.
If \mathscr{S} consists of 2-positive operators, almost all the same relationships between these properties hold as for the case of automorphism groups which has already been thoroughly investigated.
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