Multiplicative Functionals on Semigroups of Continuous Functions

Let X be a compact Hausdorff space. We denote by C(X) the multiplicative semigroup of all continuous real-valued functions on X. Milgram [2] has shown that C(X) as a semigroup determines X. In this paper we investigate the set %(X) of all continuous positive nontrivial2 multiplicative functionals on C(X), where C(X) has the topology of uniform convergence. If multiplication is defined pointwise, (F-G)(f) = F(f)-G(f) for F, GE%(X), fEC(X) then %(X) as a semigroup determines the space X for spaces satisfying the first axiom of countability, but not, in general, otherwise. We find the general form of semigroup isomorphisms of 5(Xi) onto \§(X2) which are "continuous" in a suitable sense in the case where the spaces satisfy the first axiom of countability. Bourgin [l] and Turowicz [3] have shown that if FE%(X), there is a uniquely determined countable closed set {xn} and a summable sequence {a,} of positive numbers such that F(f) =Yl\f(xi) | "'• Conversely, any functional F defined in this way is continuous. To obtain this representation of the multiplicative functional F, one represents as an integral the corresponding linear functional L defined by L(f) = log F(e'). Thus L(f) =ffdu, and Fig) = exp flog gdu, for positive functions g. One can then prove that u is a positive measure whose support is a countable closed set {x,j, and the representation for F follows. We denote by D(F) the countable closed set {xn} which occurs in the representation of F. If S(X) is the semigroup of all continuous multiplicative functionals on C(X), then i$(X) can be algebraically characterized in S(X) as the set of functionals which are squares. Therefore if S(Xi) is isomorphic to S(X2), it follows that fjC<^i) is isomorphic to %(X2) and our results for %(X) imply analogous results for S(X). By an ideal in %(X) we mean a subset I with the property that FEI, GE%(X) imply that FGEL An ideal I is called a P-ideal if for each pair of elements 7*i, F2 in I there exist GEI, and Hi, H2 in