Maps not necessarily linear but monotone (order preserving) between function spaces are analyzed. Characterizations of maps T from functions on X to those on Y with the property that the image Tf(y) depends on the value of f at one point $$x\in X$$ x ∈ X are established. Then T has a functional representation, namely, Tf(y) is equal to a function $$F_y$$ F y composed with f(x). In particular, for X extremally disconnected, T satisfies the above property for non-negative functions on X if and only if is finitely disjointness preserving ( $$\wedge f_i=0 \Rightarrow \wedge T(f_i)=0$$ ∧ f i = 0 ⇒ ∧ T ( f i ) = 0 ), orthogonally additive ( $$f\wedge g=0 \Rightarrow T(f+g) = T(f) + T(g)$$ f ∧ g = 0 ⇒ T ( f + g ) = T ( f ) + T ( g ) ), and satisfies a continuity condition. In the absence of continuity conditions, the above order theoretic conditions are equivalent to a local condition, specifically, $$Tf(y)=Tg(y)$$ T f ( y ) = T g ( y ) whenever $$f = g$$ f = g on a neighborhood of x. Results for more general domains are provided as well as consequences for bijections.
Positivity – Springer Journals
Published: Mar 24, 2016
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