On orthomorphisms, quasi-orthomorphisms and quasi-multipliers (Q2577148)

For lattice ordered algebras, comparisons between lattice maps related to orthomorphism and algebra maps related to multipliers are studied. It is shown that for certain Dedekind complete Banach \(f\)-algebras the orthomorphisms are isometric and lattice isomorphic to the quasi-multipliers. In analogy to quasi-multipliers on algebras, quasi-orthomorphims are defined for a vector lattice \(A\) as bilinear maps with appropriate orthogonal properties from \(A\times A\) to \(A\). It is then established that for certain Banach \(f\)-algebras that the quasi-orthomorphisms coincide with the quasi-multipliers and for certain Banach \(f\)-algebras the quasi-orthomorphisms form a Dedekind complete Banach \(f\)-algebra with multiplicative identity. Applying the Kakutani representation theorem, it is shown that these quasi-orthomorphisms can be identified with the space of all continuous functions on a compact space.