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DOI: 10.23671/VNC.2016.1.5951
Characterization and Multiplicative Representation of Homogeneous Disjointness Preserving Polynomials
Kusraeva, Z. A.
Vladikavkaz Mathematical Journal 2016. Vol. 18. Issue 1.
Abstract: Let \(E\) and \(F\) be vector lattices and \(P:E\rightarrow F\) 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;
\((2)\) \(\hat{d}^{\,n}P(x)(y)=0\) 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 \(x,y\in E\);
\((4)\) there exist a vector lattice \(G\) and lattice homomorphisms \(S_1,S_2:E \rightarrow G\) 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 \(x\in E\);
\((5)\) there exists an order bounded disjointness preserving linear operator \(T:E^{s\scriptscriptstyle\odot}\rightarrow F\) such that \(Px=T(x^{s\scriptscriptstyle\odot})\) for all \(x\in E\).
Theorem 4.7. 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 \(W:\mathcal{F}\rightarrow \mathcal{F}\) is the orthomorphism multiplication by \(o-\sum_{\xi\in\Xi}\rho_{\xi}P(e_{\xi})\).
Keywords: power of a vector lattice, homogeneous polynomial, disjointness preserving polynomial, orthogonal additivity, lattice polymorphism, multiplicative representation
For citation: Kusraeva Z. A. Characterization and multiplicative representation of homogeneous
disjointness preserving polynomials // Vladikavkazskii
matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp.
51-62.
DOI 10.23671/VNC.2016.1.5951
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