On orthomorphisms, quasi-orthomorphisms and quasi-multipliers

In this paper we introduce a new concept, namely that of a quasi-orthomorphism, on a vector lattice, which generalizes the notion of an orthomorphism. We consider quasi-multipliers on ℓ-algebras and establish the relationship between quasi-orthomorphisms and quasi-multipliers. We prove that, for certain Banach algebras, the quasi-orthomorphisms QOrth(A) form a Dedekind complete Banach f-algebra with a multiplicative identity. We conclude with a Kakutani type representation theorem for QOrth(A).