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Subalgebras and Riesz subspaces of an $$f$$-algebra. (English) Zbl 0534.46010
The main topic of the paper is the investigation of the connection between subalgebras and Riesz subspaces of an Archimedean $$f$$-algebra with unit element and also of the relationship between algebra homomorphisms and Riesz homomorphisms between such $$f$$-algebras, respectively. Some of the results are:
(1) Let $$A$$ be an Archimedean $$f$$-algebra with unit element $$e$$.
(a) The uniformly closed Riesz subspace $$L$$ of $$A$$ is a subalgebra if and only if $$v\wedge e\in L^+$$ for all $$v\in L^+$$. (The latter condition is called the “Stone condition”.)
(b) The uniformly closed subalgebra $$L$$ of $$A$$ is a Riesz subspace if and only if $$L$$ satisfies the Stone condition.
(2) Let $$A$$ and $$B$$ be Archimedean $$f$$-algebras with unit elements $$e_ A$$ and $$e_ B$$, respectively. Denote by $${\mathcal H}$$ the convex set of all Markov operators from $$A$$ into $$B$$, so $${\mathcal H}=\{S:A\to B; S \text{ linear}, S\geq 0, Se_ A=e_ B\}$$. If $$T\in {\mathcal H}$$, then $$T$$ is an algebra homomorphism if and only if $$T$$ is a Riesz homomorphism. This result generalizes the result for the $$C(X)-C(Y)$$ case, proved by A. J. Ellis. Furthermore, $$T$$ is an extremal point of $${\mathcal H}$$ if and only if $$T$$ is a Riesz (algebra) homomorphism for which $$Te_ A=e_ B$$. The latter equivalence is for the $$C(X)-C(Y)$$ case due to A. and C. Ionescu-Tulcea, R. R. Phelps and J. Tate.

##### MSC:
 46A40 Ordered topological linear spaces, vector lattices 06F25 Ordered rings, algebras, modules 47L07 Convex sets and cones of operators
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