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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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