The tensor product of \(f\)-algebras. (English) Zbl 1400.46003

In [Am. J. Math. 94, 777–798 (1972; Zbl 0252.46094)], D. H. Fremlin constructed the Riesz tensor product of two Riesz spaces. In the present paper, the authors construct the tensor product in another class of Archimedean ordered structures, that of \(f\)-algebras. Their approach is purely algebraic. A representational approach has been given in [Mediterr. J. Math. 14, No. 2, Paper No. 63, 10 p. (2017; Zbl 1378.06017)] by G. Buskes and A. W. Wickstead. A key result is that the Riesz subspace generated by a subalgebra of an \(f\)-algebra is an \(f\)-subalgebra. The latter was formulated for \(f\)-rings by M. Henriksen and J. R. Isbell [Pac. J. Math. 12, 533–565 (1962; Zbl 0111.04302)]. Nevertheless, the authors give here an alternative proof of this result, providing also an operator version of it. An example of an \(f\)-subalgebra is also given. Then, applying among other things the key result, they prove that the canonical multiplication on the algebraic tensor product of two unital \(f\)-algebras is extended to a multiplication on the Riesz tensor product which turn to be a unital \(f\)-algebra that satisfies a multiplicative universal property. Using this result, they provide an analogous one for semiprime \(f\)-algebras. The proof is also based on the fact that any semiprime \(f\)-algebra can be considered as \(f\)-subalgebra of the unital \(f\)-algebra of all orthomorphisms on the initial algebra. Finally, they pass to the main result of the paper that concerns the general case of two \(f\)-algebras. Some open problems, in the context of \(f\)-algebras, are also stated.


46A40 Ordered topological linear spaces, vector lattices
46H05 General theory of topological algebras
06F25 Ordered rings, algebras, modules
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