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

Abstract.

The generalized state space of a commutative C^*-algebra, denoted $S_{{\mathcal{H}}}(C(X))$, is the set of positive unital maps from C(X) to the algebra ${\mathcal{B}}({\mathcal{H}})$ of bounded linear operators on a Hilbert space ${\mathcal{H}}$. 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 $S_{{\mathcal{H}}}(C(X))$ 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 ${\mathcal{K}}^{+}$, 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.