## The space of extended orthomorphisms in a Riesz space.(English)Zbl 0541.46006

Some properties are discussed of the space $$Orth^{\infty}(L)$$ of extended orthomorphisms on an Archimedean Riesz space L. An order bounded linear mapping $$\pi$$ from an order dense ideal $$D\subseteq L$$ into L is called an extended orthomorphism if $$\inf(| \pi f|,| g|)=0$$ whenever $$f\in D$$, $$g\in L$$ and $$\inf(| f|,| g|)=0.$$ The space $$Orth^{\infty}(L)$$ is itself an Archimedean f- algebra. First of all it is shown that for a uniformly complete f-algebra A the algebra $$Orth^{\infty}(A)$$ can be identified with the complete ring of quotients of A. Furthermore it is proved that for a uniformly complete Riesz space L, the space $$Orth^{\infty}(L)$$ is isomorphic to the lateral completion $$L^{\lambda}$$ of L (which shows in particular that in this situation $$L^{\lambda}$$ has the structure of an f- algebra). Finally some algebraic properties of $$Orth^{\infty}(L)$$ are discussed, and in particular it is shown that $$Orth^{\infty}(L)$$ is von Neumann regular for any uniformly complete space L.

### MSC:

 46A40 Ordered topological linear spaces, vector lattices 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 47B60 Linear operators on ordered spaces
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