English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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.

The space of extended orthomorphisms in a Riesz space