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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.

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