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On the structure of positive homomorphisms on algebras of real-valued continuous functions. (English) Zbl 1069.54013

This paper is a deep and elegant contribution to the study of the structure of positive homomorphisms on real functions algebras, an interesting problem tackled by a large number of authors. By an algebra on a Tychonoff space \(X\) it is understood a subalgebra \(A\) of \(C(X)\) (the algebra of all real-valued continuous functions on \(X\)) which contains the constants and such that the subset \(A^{\star}\) of all bounded functions in \(A\) separates points from closed sets, and by a homomorphism a non-zero real-valued positive multiplicative linear functional on \(A\). The symbol \(\text{Hom}\, A\) stands for the set of all homomorphisms on \(A\). A homomorphism \(\varphi\) is said to be countable evaluating if for every sequence \(\{f_{n}\}_{n=1}^{\infty}\) in \(A\), there is a point \(x\in X\) such that \(\varphi (f_{n})=f_{n}(x)\) for all \(n\). An algebra \(A\) on \(X\) is said to be countably evaluating if every homomorphism on \(A\) is countably evaluating.
In the first part of the paper the author proves that several known sufficient conditions which guarantee that an algebra is countably evaluating fail to be necessary. Among other interesting cases, this is showed for the condition given by S. Mrówka in [Studia Math. 21, 1-14 (1961; Zbl 0099.10102)], by E. Adam, P. Biström and A. Kriegl [Archiv. Math. 35, 165-192 (1999; Zbl 1050.46025)] and by J. R. Arrazola, J. Bustamante and R. Escobedo [Rev. Mat. Complut. 11, 17-29 (1998; Zbl 0910.46020)]. The author also gives a sufficient (but, as he points out by means of an example, not necessary) condition for an algebra to be countable evaluating which can be applied to obtain most of the known results: Theorem. Let \(A\) be an algebra on a space \(X\). If for each increasing sequence \(\{h_{n}\}_{n=1}^{\infty}\) in \(A\) such that each set \(F_{1/n}h_{n}\) is non-empty and \(\cap_{n=1}^{\infty}F_{1/n}h_{n}=\emptyset\) there exists \(g\in A\), \(g>0\), satisfying either \(\lim_{_{n}}\inf \{g(x) : x\in F_{1/n}h_{n}\}=+\infty\) or \(\lim_{_{n}}\sup \{g(x) : x\in F_{1/n}h_{n}\}=0\), then \(A\) is countably evaluating. (Here, for a function \(f\in R^{X}\) and \(\varepsilon >0\), \(F_{\varepsilon}f\) stands for the set \(\{x\in X : | f(x)| \leq \varepsilon\)}.
The second part of the paper undertakes the analysis of the structure of (positive) homomorphisms in the general case. The main result is the following Theorem. The map \(\varphi \rightarrow {\mathcal F}_{\ker \varphi}\) defines a one-to-one correspondence between the homomorphisms on \(A\) and the \(\varepsilon\)-families \({\mathcal F}_{B}\) where \(B\) is maximal and every function in \(A\) is bounded on a set belonging to \({\mathcal F} _{B}\). (An \(\varepsilon \)-family is a family of sets \({\mathcal F} _{B}=\{F_{\varepsilon}f: \varepsilon >0, f\in B\}\), \(B\subset A\), with the finite intersection property.)
The previous result (which brings up the intricacy of the structure of the homomorphisms on an arbitrary algebra) applies in order to obtain several characterizations of countable evaluating algebras, and also to characterize Lindelöf spaces: Theorem. A space \(X\) is Lindelöf if and only if each countably evaluating homomorphism on any algebra on \(X\) is the evaluation at some point of \(X\). The paper is finished by showing that in the case of an inverse-closed algebra \(A\) (i.e, \(1/f\in A\) for each \(f\in A\) which does not vanish on \(X\)) it is possible to obtain a representation of a homomorphism in a more usual form: Theorem. Let \(A\) be an inverse-closed algebra. The map \(\varphi \longrightarrow Z(\text{ker}\, \varphi)\) defines a one-to-one correspondence between the homomorphisms on \(A\) and the \(Z(A)\)-ultrafilters such that every function on \(A\) is bounded on some of its members.

MSC:

54C35 Function spaces in general topology
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E25 Rings and algebras of continuous, differentiable or analytic functions
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