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

Abstract

AbstractGiven 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.