Invariant States for Strongly Positive Operators on $C^*$-Algebras

Abstract

A linear operator a on a C*-algebra A induces a contraction 6$ on the Hilbert space provided a satisfies the Schwarz inequality: 0(a*a) >ff(a)*0(a). If (f> is invariant under a class sr of such operators, the following four properties are closely connected: (i) abelianness of the reduction of ~6(A) to the ^-invariant part of %$, (ii) asymptotic abelianness of 0, (iii) abelianness of ^^(A)' H .$%, (iv) uniqueness of decompositions of 9 into extremal ^-invariant states. If sr 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.