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.


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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