Extreme positive operators and homomorphisms

Abstract

Consider the following theorem, first proved (in the real case) by Arens and Kelley [1]: Let X be a compact Hausdorff space and C(X) the space of allreal (or complex) valued continuous functions on X, with the supremum norm. Let K be the convex set of linear functionals L on C(X) such that L(1) = 1 = |J LJJ. The set of extreme points of K coincides with the set of all nontrivial multiplicative linear functionals on C(X). Their proof of this result depends on the representation of linear functionals by means of measures (which makes it possible to identify a multiplicative functional with evaluation at a point of X). Using methods of a more algebraic nature, Tate [8] has proved the above equivalence for certain partially ordered real commutative algebras (which include the algebra C(X)). The fact that such methods are at all feasible arises, essentially, from the fact that the convex set K admits a second description, namely, it is those linear functionals L on C(X) such that L > 0 and L(1) = 1. Suppose, now, that we represent the algebra C(X) by A and the algebra of scalars by B. The above result then asserts that the set of nontrivial homomorphisms of a certain algebra A into B coincides with the set of extreme points of a certain convex set of linear transformations from A into B. The purpose of the present paper is to try to determine the extent to which this type of result remains valid if the algebra of scalars B is replaced by a more general algebra (over the same field as A). To the best of our knowledge, the first proof of a theorem of this type was given by A. Ionescu-Tulcea and C. Ionescu-Tulcea [5] (announced in [6, footnote 3]), where A and B were taken to be (real) C(X) and C(Y), respectively. Their method is an extension of the method of Arens and Kelley. Working more in the spirit of Tate's paper, and with sets of transformations analogous to the second description of K (above) we will give an extremely simple proof of a similar result (Theorem 1.1) for certain algebras of functions which include algebras of the form C(X)(2). It would be possible to work with