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Lattice bimorphisms on $$f$$-algebras. (English) Zbl 1059.06013
Let $$A$$, $$B$$, $$C$$ be lattice-ordered algebras and let $$\psi :A\times B \to C$$ be a bilinear map. $$\psi$$ is called a lattice bimorphism if for each $$0\leq f \in A$$ and for each $$0\leq g\in B$$, the partial maps $$g\to \psi (f,g)$$ and $$f\to \psi (f,g)$$ are lattice homomorphisms of $$B$$ and $$A$$ into $$C$$, respectively. $$\psi$$ is called multiplicative if $$\psi (uf, vg) = \psi (u,v) \psi (f,g)$$ holds whenever $$u,f \in A$$ and $$v,g \in B$$.
In the paper the connection between lattice bimorphisms and multiplicative bilinear maps on $$f$$-algebras are investigated. An $$f$$-algebra is a real vector lattice $$A$$ which is simultaneously an associative algebra such that the positive cone $$A^+$$ is closed under multiplication and $$f\wedge g=0$$ yields $$fh\wedge g = hf\wedge g =0$$ for all $$h\in A^+$$. The main result is the following:
If $$A,B,C$$ are Archimedean $$f$$-algebras with unit elements $$e_A, e_B, e_C$$, respectively, and $$\psi : A\times B \to C$$ is a Markov bilinear map, then $$\psi$$ is a lattice bimorphism iff $$\psi$$ is multiplicative. The main conclusion of this result is the Cauchy-Schwarz inequality in Archimedean (not necessarily commutative) $$d$$-algebras, which is an improvement of the result of G. Buskes and A. van Rooij, who established this inequality in the commutative case.

##### MSC:
 06F25 Ordered rings, algebras, modules 47B65 Positive linear operators and order-bounded operators
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