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

de Pagter, B.

Pac. J. Math. 112, 193-210 (1984).

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

Keywords:

complete ring of quotients; extended orthomorphisms on an Archimedean Riesz space; Archimedean f-algebra; lateral completion; Neumann regular