Characterizations of Finite Dimensional Archimedean Vector Lattices

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. 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 + References 1. Bresar M. Finite Dimensional Zero Product Determined Algebras are Generated by Idempotents. Expo Math., 2016, vol. 34, no. 1, pp. 130-143. 2. Rickart C. E. General Theory of Algebras, Princeton, Van Nostrand, 1960. 3. Huijsmans C. B. Lattice-Ordered Division Algebras. Proc. R. Ir. Acad., 1992, vol. 92A, no. 2, pp. 239-241. 4. Uyar A. On Banach lattice algebras. Turkish J. Math., 2005, vol. 29(3), pp. 287-290. DOI: 10.1007/11117-015-0358-0. 5. Ercan Z., Onal S. Some Observations on Riesz Algebras. Positivity, 2006, vol. 10, pp. 731-736. 6. Aliprantis C. D., Burkinshaw O. Positive Operators, London, Acad. Press, 1985. 7. Bernau S. J., Huijsmans C. B. Almost \(f\)-algberas and \(d\)-algebras. Math. Proc. Cambridge Philos. Soc., 1990, vol. 107, no. 2, pp. 287-308. 8. Buskes G., van Rooij A. Almost \(f\)-Algebras. Commutativity and the Cauchy-Schwarz Inequality. Positivity, 2000, vol. 4, pp. 227-331. DOI : 10.1016/j.exmath.2015.07.002. 9. De Pagter B. \(f\)-Algebras and Orthomorphisms. Ph.D. Thesis, Leiden, Leiden Univ., 1981. DOI: 10.1016/j.jmaa.2008.03.009. 10. Luxemburg W. A. J., Moore L. C. Archimedean Quotient Riesz Spaces. Duke. Math. J., 1967, vol. 34, pp. 725-739. DOI: 10.1016/j.indag.2014.03.001. 11. Luxemburg W. A. J., Zaanen A. C. Vector Lattices. I, Amsterdam, North-Holland, 1971. 12. Huijsmans C. B. Some Analogies Between Commutative Rings, Riesz Spaces and Distributive Lattices with Smallest Element. Nederl. Akad. Wet., Proc., Ser. A77, 1974, vol 36, pp. 132-147. DOI: 10.1007/s11117-006-0047-0. 13. Toumi M. A. When Orthomorphisms are in the Ideal Center. Positivity, 2014, vol. 18(3), pp. 579-583. DOI: 10.1007/s00012-012-0166-3. 14. Nouki N., Toumi M. A. Derivations on Universally Complete \(f\)-Algebras. Indag. Math., 2015, vol. 26(1), pp. 1-18. 15. Toumi M. A. A relationship between the space of orthomorphisms and the centre of a vector lattice revisited. Positivity, 2016, vol. 20, no. 2, pp. 337-342. DOI: 10.1007/s11117-013-0263-3. 16. Marinacci M., Montrucchio L. On concavity and supermodularity. J. Math. Anal. Appl., 2008, vol. 344, pp. 642-654. 17. Toumi M. A. Characterization of Hyper-Archimedean Vector Lattices Via Disjointness Preserving Bilinear Maps. Algebra Univers, 2012, vol. 67, no. 1, pp. 29-42. ← Contents of issue


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