A New Proof of the Lattice Structure of Orthomorphisms

Abstract

A linear mapping T from the Archimedean Riesz space E into itself is called an orthomorphism of E if it is order bounded and band preserving (that is T(B) £ B for every band B in E). The band preserving property is easily seen to be equivalent to the statement that x J_ y in E implies that Tx ± y (we recall that x _L y means |x| A \y\ = 0). Orthomorphisms have been studied by many authors and it is not possible to quote all of them in our references.The starting point of the theory consists of proving that the collection Orth (E) of all orthomorphisms of £ is a Riesz space under pointwise operations. Several proofs of this fact are known, but some make use of non-trivial representation theorems (for example [8,9]), others involve the use of the Dedekind completion (for example [1,4]), and another one is elementary but rather technical [2]. In this short paper we give a new proof, probably the most simple and the most elegant (but not completely elementary because of its use of Zorn's Lemma, via a result on "Riesz homomorphisms").However, instead of proving the result in question directly, we first establish a more general result which is interesting on its own and from which it will immediately follow that Orth (£), and even Orth 00 (E) (the extended orthomorphisms in the sense of [4,5]), is a Riesz space. This more general result has already been obtained by Meyer [7,8] by means of a representation theorem, but the proof given here avoids this representation theorem.

Extension and inversion of extended orthomorphisms on Riesz spaces

“…In [13] it is shown (with a rather technical proof) that if £ is a uniformly complete Archimedean Riesz space, then the algebra Orth°°(.E) is von Neumann regular, that is, for every T E. Orth°°(£) there exists S E Orth°°(£) such that T = ST 2 . As a corollary it is also obtained that, when E is uniformly complete, every weak order unit in Orth°°(£') is invertible.…”

Orthomorphisms of a commutative W*-algebra

“…Let E e ReM be a projection and let 0 < S e ReM be in the principal band generated by E in ReM. Let {P n } c /?eM be a sequence of projections satisfying Lemma 3.2 for S. From 0 < SP n < S, it follows that SP n A (I -E) = 0 and from Si 5 ,, e AeM, it follows that 5P B (/ -£ ) = 0 for « = 1,2,... From part (iv) of Lemma 3.2, it follows that S(I -E) = 0 and by this the proof of the theorem is complete. LEMMA …”

“…The general theory of orthomorphisms has received much recent attention; see for example [5,6,8,16,20,21,22] and the references contained therein. For many Riesz spaces, it is possible to determine the orthomorphisms explicitly as multipliers, in a certain sense.…”

Sums of Lattice Homomorphisms

“…These can be found in [2,4]. Note, however, that the term disjointness preserving in [2] has a different definition and that [2] considers self-maps of E rather than maps from E to F .…”

“…The relevant proofs in [2] all take place in the range of the operator and apply without change. The results we need are given by the following: -Theorem [2,4,7]. Let T be an order bounded linear operator from a Riesz space E into an archimedean Riesz space F such that \Tu\/\\Tv\ = 0 for all u, v g E with \u\ A \v\ = 0.…”