On C*-Extreme Maps and *-Homomorphisms of a Commutative C*-Algebra

M. C. Gregg

On C-Extreme Maps and -Homomorphisms of a Commutative C*-Algebra

Operator Algebras 2009-02-12 v2 Functional Analysis

Authors: M. C. Gregg

Abstract

The generalized state space of a commutative C*-algebra, denoted S_H(C(X)), is the set of positive unital maps from C(X) to the algebra B(H) of bounded linear operators on a Hilbert space H. C*-convexity is one of several non-commutative analogs of convexity which have been discussed in this context. In this paper we show that a C*-extreme point of S_H(C(X)) satisfies a certain spectral condition on the operators in the range of the associated positive operator-valued measure. This result enables us to show that C*-extreme maps from C(X) into K^+, the algebra generated by the compact and scalar operators, are multiplicative. This generalizes a result of D. Farenick and P. Morenz. We then determine the structure of these maps.

Keywords

C*-extreme maps *-homomorphisms commutative C*-algebra generalized state space positive operator-valued measure C*-convexity

Cite

@article{arxiv.0902.1453,
  title  = {On C*-Extreme Maps and *-Homomorphisms of a Commutative C*-Algebra},
  author = {M. C. Gregg},
  journal= {arXiv preprint arXiv:0902.1453},
  year   = {2009}
}

Comments

minor change to proof of Theorem 5, 12 pages, To appear in Integral Equations and Operator Theory