$$\varvec{\varepsilon }$$-Shading Operator on Riesz Spaces and Order Continuity of Orthogonally Additive Operators

Abstract

Given a Riesz space E and $$0 < e \in E$$ 0 < e ∈ E , we introduce and study an order continuous orthogonally additive operator which is an $$\varepsilon $$ ε -approximation of the principal lateral band projection $$Q_e$$ Q e (the order discontinuous lattice homomorphism $$Q_e :E \rightarrow E$$ Q e : E → E which assigns to any element $$x \in E$$ x ∈ E the maximal common fragment $$Q_e(x)$$ Q e ( x ) of e and x). This gives a tool for constructing an order continuous orthogonally additive operator with given properties. Using it, we provide the first example of an order discontinuous orthogonally additive operator which is both uniformly-to-order continuous and horizontally-to-order continuous. Another result gives sufficient conditions on Riesz spaces E and F under which such an example does not exist. Our next main result asserts that, if E has the principal projection property and F is a Dedekind complete Riesz space then every order continuous regular orthogonally additive operator $$T :E \rightarrow F$$ T : E → F has order continuous modulus |T|. Finally, we provide an example showing that the latter theorem is not true for $$E = C[0,1]$$ E = C [ 0 , 1 ] and some Dedekind complete F. The above results answer two problems posed in a recent paper by O. Fotiy, I. Krasikova, M. Pliev and the second named author.

Diffuse orthogonally additive operators

“…The solution of the von Neumann problem depends substantially on the presence of 'rich' order structure on the set of regular linear integral operators on the L 2 [0, 1]. Various order properties of the space OA r (E, F ) were considered in [10]- [12], [26] and [32].…”

Lateral order on complex vector lattices and narrow operators

“…The lateral order on real vector lattices introduced in [25] is turned out to be a useful tool for investigation orthogonally additive operators [2,17,26,29,37]. Recently, some connections of the lateral order with the theory of derivations on von Neumann algebras were observed [6].…”