Integral representation of multiplicative, involution preserving operators in ℒ(𝒞(𝒮),ℰ)

Abstract

1. Introduction. Throughout, S will be a compact Hausdorff space and E will be a Banach space which is the dual of another Banach space F. C(S) will denote the space of complex-valued continuous functions on S topologized with the topology of uniform convergence. ?(C(S), E) will denote the space of continuous linear operators from C(S) to E. A theorem of Gil de Lamadrid [5, p. 103] identifies ?(C(S), E) with a space of E-valued measures, the correspondence between operator and measure being given by integration. A closely related result was given earlier by Bartle, Dunford, and Schwartz [2 ]. Now if E is a Banach algebra with involution [7, p. 178], it makes sense to consider operators which are not only continuous and linear but which also preserve multiplication and involution. A natural question arises: How are these additional properties reflected in the representing measure? We answer this question under additional restrictions on E. We also give several examples of spaces E satisfying the hypotheses of our theorems. One can use the results of this paper to prove the Spectral Theorem for bounded operators; but the proof follows standard lines and will not be included (see [6, p. 99]). We conclude this introduction with a precise description of Gil de Lamadrid's Theorem. Our description differs from Gil de Lamadrid's, but it is not difficult to verify that they are equivalent. We consider the class N(S, E) of set functions m from 6((S), the Borel class of S, to E which are countably additive and regular with respect to the weak topology a(E, F) on E induced by F. To say that m: (3(S) -E is regular with respect to the topology o(E, F) means that, for every BE@(S) and every a(E, F)-neighborhood N of 0, there exists a compact set K and an open set U such that KCBC U and, if A C U-K, then m(A) EN. Defining addition and scalar-multiplication in N(S, E) in the usual set-wise fashion; i.e., (m1+m2)(B)=m1(B) +m2(B) and (am,)(B) ==am(B), N(S, E) is a vector space. The following formula defines a norm on N(S, E) making it a Banach space: