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DOI: 10.23671/VNC.2018.2.14725

Characterizations of Finite Dimensional Archimedean Vector Lattices

Polat F. , Toumi M. A.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 2.
Abstract:
In this paper, we give some necessary and sufficient conditions for an Archimedean vector lattice \(A\) to be of finite dimension. In this context, we give three characterizations. The first one contains the relation between the vector lattice \(A\) to be of finite dimension and its universal completion \(A^u\). The second one shows that the vector lattice \(A\) is of finite dimension if and only if one of the following two equivalent conditions holds : (a) every maximal modular algebra ideal in \(A^u\) is relatively uniformly complete or (b) \(Orth(A,A^u)=Z(A,A^u)\) where \(Orth(A,A^u)\) and \(Z(A,A^u)\) denote the vector lattice of all orthomorphisms from \(A\) to \(A^u\) and the sublattice consisting of orthomorphisms \(\pi\) with \(|\pi(x)|\leq\lambda|x|\) \((x\in A)\) for some \(0\leq\lambda\in\mathbb{R}\), respectively. It is well-known that any universally complete vector lattice \(A\) is of the form \(C^\infty (X)\) for some Hausdorff extremally disconnected compact topological space \(X\). The point \(x\in X\) is called \(\sigma\)- isolated if the intersection of every sequence of neighborhoods of \(x\) is a neighborhood of \(x\). The last characterization of finite dimensional Archimedean vector lattices is the following. Let \(A\) be a vector lattice and let \(A^{u}(=C^{\infty}\left(X\right))\) be its universal completion. Then \(A\) is of finite dimension if and only if each element of \(X\) is \(\sigma\)-isolated. Bresar in \cite{4} raised a question to find new examples of zero product determined algebras. Finally, as an application, we give a positive answer to this question.
Keywords: hyper-Archimedean vector lattice, \(f\)-algebra, universally complete vector lattice
Language: English Download the full text  
For citation: Polat F., Toumi M. A. Characterizations of Finite Dimensional Archimedean Vector Lattices. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 2, pp. 86-94. DOI 10.23671/VNC.2018.2.14725
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