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

C. J. K. Batty | Zendy

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

Author(s) -

C. J. K. Batty

Publication year - 1982

Publication title -

publications of the research institute for mathematical sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.786

H-Index - 39

eISSN - 1663-4926

pISSN - 0034-5318

DOI - 10.2977/prims/1195183295

Subject(s) - mathematics , automorphism , invariant (physics) , pure mathematics , homogeneous space , uniqueness , hilbert space , algebraic number , semigroup , algebra over a field , discrete mathematics , mathematical analysis , mathematical physics , geometry

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.

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