$$\varvec{\varepsilon }$$-Shading Operator on Riesz Spaces and Order Continuity of Orthogonally Additive Operators
Volodymyr Mykhaylyuk, M. M. Popov
Abstract
Abstract Given a Riesz space E and $$0 < e \in E$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>e</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> , we introduce and study an order continuous orthogonally additive operator which is an $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> -approximation of the principal lateral band projection $$Q_e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:math> (the order discontinuous lattice homomorphism $$Q_e :E \rightarrow E$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>e</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>E</mml:mi> <mml:mo>→</mml:mo> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> which assigns to any element $$x \in E$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> the maximal common fragment $$Q_e(x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>e</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$T :E \rightarrow F$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:mi>E</mml:mi> <mml:mo>→</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> has order continuous modulus | T |. Finally, we provide an example showing that the latter theorem is not true for $$E = C[0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> 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.