Extended orthomorphisms on Archimedean Riesz spaces

The space of extended orthomorphisms in a Riesz space

“…It follows in particular from Theorem 3.6 that any uniformly complete Riesz space L can be embedded as a Riesz subspace in Orth°°(L). Recently this result has also been proved by M. Duhoux and M. Meyer [5] (Theorem 2.14 and Corollary 2.15). One of the main differences between their approach and ours is that we always consider Orth°°(L) as an/-subalgebra of L u , whereas they do not make use of the multiplicative structure of ZΛ In the same paper also the results of Corollaries 3.10 and 3.11 appear (see [5], Remarks 2.8.2 and 2.13.2).…”

Extension and inversion of extended orthomorphisms on Riesz spaces

“…If (1) (4). If K is a compact topological space, then 6(K) has an ultrarich center (/-isomorphic to Q(K)); hence if £ is a Banach lattice with a quasi-interior point u, it follows from the Yosida representation theorem that F = i(u) satisfies (2) and (3) …”

“…A a-orthomorphism T on an Archimedean Riesz space E is an extended orthomorphism that can be defined on a super order dense ideal of E or, in other words, such that D™ is super order dense in E; the set Orth 0 (£) of all a-orthomorphisms on E has been introduced and studied in [3]. When we shall consider a possible domain D T for a a-orthomorphism T, it will be always supposed that D T is super order dense.…”

“…Using 1.1, it is easily seen that Orth"(£) is an/-subalgebra of Orth 00^) ; since moreover / E Orth"(£), it follows that Orth(Orth°(£)) = 0rth o (£), the members of Orth"(£) operating by product on Orth°(£). Note that a-orthomorphisms are useful to characterize the universal a-completion (see [3]) and the lateral a-completion (see [4]) of many almost Dedekind a-complete Riesz spaces (that is, Riesz spaces which are Riesz isomorphic to a super order dense Riesz subspace of some Dedekind a-complete Riesz space).…”

“…The space Orth(£) of all orthomorphisms on E, that is, of everywhere defined extended orthomorphisms, is an /-subalgebra of Orth°°(£). Extended orthomorphisms have been introduced in a special setting by Nakano [11], used by Luxemburg and Schep [7] and were recently studied by the authors [3]. They form a natural generalization of the orthomorphisms and the richness of their structure allows to characterize many properties of E. [ 3 ] Extended orthomorphisms on Riesz spaces 225…”