On C ∗-extreme Maps and ∗-homomorphisms of a Commutative C ∗-algebra

The generalized state space of a commutative C-algebra, denoted SH(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 SH(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. Several non-commutative analogs of convexity have appeared in the literature including CP-convexity [4] and matrix convexity [2], as well as C-convexity [3], [6], which is the topic of this paper. In [6], Hopenwasser, Moore, and Paulsen characterized operators which are C-extreme in the unit ball of B(H) and obtained results about other C-convex sets and their extreme points. In [3], Farenick and Morenz extend the idea of C-convexity to the space of completely positive maps on a C-algebra. They show that C-extreme maps with their range in K are also extreme (in the classical sense) and obtain a characterization of C-extreme maps on a commutative C-algebra which have their range in Mn, the C -algebra of n × n complex matrices. Subsequently, Zhou [9] gave two necessary and sufficient conditions for a completely positive map to be C-extreme, and described the structure of C-extreme maps with range in Mn. The main results presented here are Theorem 5, which gives a necessary condition for a map φ : C(X) → B(H) to be C-extreme, and Theorem 10, which then shows that a positive unital map φ : C(X) → K is C-extreme if and only if it is multiplicative. We then determine the structure of such maps. The author wishes to express her gratitude to Professor David Pitts for many lively conversations which led to significant improvements in the paper. Throughout, let X be a compact Hausdorff space, C(X) the C-algebra of continuous functions on X , and H a Hilbert space. Definition 1. The generalized state space of C(X) is SH(C(X)) = {φ : C(X) → B(H) |φ is positive and φ(1) = I}. 1991 Mathematics Subject Classification. Primary 46L05; Secondary 46L30.