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 Vladikavkaz. Mat. Zh., 2016, Volume 18, Number 1, Pages 51–62 (Mi vmj572)

Characterization and multiplicative representation of homogeneous disjointness preserving polynomials

Z. A. Kusraeva

Abstract: Let $E$ and $F$ be vector lattices and $P\colon E\to F$ an order bounded orthogonally additive (i.e. $|x|\wedge|y|=0$ implies $P(x+y)=P(x)+P(y)$ for all $x,y\in E$) $s$-homogeneous polynomial. $P$ is said to be disjointness preserving if its corresponding symmetric $s$-linear operator from $E^s$ to $F$ is disjointness preserving in each variable. The main results of the paper read as follows:
Theorem 3.9. The following are equivalent: (1) $P$ is disjointness preserving; (2) $\hat d^nP(x)(y)=0$ and $Px\perp Py$ for all $x,y\in E$, $x\perp y$, and $1\leq n<s$; (3) $P$ is orthogonally additive and $x\perp y$ implies $Px\perp Py$ for all $x,y\in E$; (4) {\it there exist a vector lattice $G$ and lattice homomorphisms $S_1,S_2\colon E \to G$ such that $G^{s\scriptscriptstyle\odot}\subset F$, $S_1(E)\perp S_2(E)$, and $Px=(S_1x)^{s\scriptscriptstyle\odot}-(S_2x)^{s\scriptscriptstyle\odot}$ for all $x\in E$}; (5) {\it there exists an order bounded disjointness preserving linear operator $T:E^{s\scriptscriptstyle\odot}\to F$ such that $Px=T(x^{s\scriptscriptstyle\odot})$ for all $x\in E$}.
Theorem 4.7. {\it Let $E$ and $F$ be Dedekind complete vector lattices. There exists a partition of unity $(\rho_\xi)_{\xi\in\Xi}$ in the Boolean algebra of band projections $\mathfrak P(F)$ and a family $(e_\xi)_{\xi\in\Xi}$ in $E_+$ such that $P(x)=o$-$\sum_{\xi\in\Xi}W\circ\rho_\xi S(x/e_\xi)^{s\scriptscriptstyle\odot}$ $(x\in E)$, where $S$ is the shift of $P$ and $W\colon\mathscr F\to\mathscr F$ is the orthomorphism multiplication by $o$-$\sum_{\xi\in\Xi}\rho_\xi P(e_\xi)$.

Key words: power of a vector lattice, homogeneous polynomial, disjointness preserving polynomial, orthogonal additivity, lattice polymorphism, multiplicative representation.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-51-53119 ÃÔÅÍ-à

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UDC: 517.98
MSC: 46A40, 47H60, 47H07

Citation: Z. A. Kusraeva, “Characterization and multiplicative representation of homogeneous disjointness preserving polynomials”, Vladikavkaz. Mat. Zh., 18:1 (2016), 51–62

Citation in format AMSBIB
\Bibitem{Kus16} \by Z.~A.~Kusraeva \paper Characterization and multiplicative representation of homogeneous disjointness preserving polynomials \jour Vladikavkaz. Mat. Zh. \yr 2016 \vol 18 \issue 1 \pages 51--62 \mathnet{http://mi.mathnet.ru/vmj572} 

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This publication is cited in the following articles:
1. Z. A. Kusraeva, “Powers of quasi-Banach lattices and orthogonally additive polynomials”, J. Math. Anal. Appl., 458:1 (2018), 767–780
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