The space of extended orthomorphisms in a Riesz space

Abstract

We study the space Orth°°(L) of extended orthomorphisms in an Archimedean Riesz space L and its analogies with the complete ring of quotients of a commutative ring with unit element. It is shown that for any uniformly complete /-algebra A with unit element, Orth°°(Λ) is isomorphic with the complete ring of quotients of A. Furthermore, it is proved that for any uniformly complete Riesz space L the space Orth°°( L) is isomorphic to the lateral completion of L. Finally, it is shown that for any uniformly complete Riesz space L the ring Orth°°(L) is von Neumann regular. The main subject in this paper is the space Orth°°(L) of extended orthomorphisms in an Archimedean Riesz space L. By an extended orthomorphism we mean an order bounded linear mapping π from an order dense ideal D in L into L with the property that πf ± g for all / E D and g E L with / ± g. As shown in [10], Orth°°(L) is an Archimedean /-algebra with unit element which is, in addition, laterally complete. The definition of Orth°°(L) for an Archimedean Riesz space is in some sense analogous to the definition of the complete ring of quotients Q(R) of a commutative ring R with unit element (see [8], §2.3). A natural thing to do, therefore, is to compare these two objects for Archimedean /-algebras with unit element. In §2 of this paper it is proved that for any uniformly complete /-algebra A with unit element, the algebras Orth°°(,4) and Q(A) are indeed isomorphic. For any/-algebra A = C(X)9 where X is a completely regular Hausdorff space, the complete ring of quotients of A is precisely the lateral completion Aλ of A. So, by the above-mentioned result, in this case Orth°°(v4) is the lateral completion of A. In §3 we study the relation between Orth°°(L) and the lateral completion Lλ for an arbitrary Archimedean Riesz space, and it will be shown that Orth°°(L) = L λ holds for uniformly complete Riesz spaces. Another interesting property of the ring of quotients Q(R) of a semiprime commutative ring R with unit element is that Q(R) is von Neumann regular. In the last section of this paper it will be shown that Orth°°(L) is a von Neumann regular/-algebra for any uniformly complete Riesz space L.