The multiplicative norm convergence in normed Riesz algebras

Year 2021, Volume: 50 Issue: 1, 24 - 32, 04.02.2021

Abdullah Aydın

https://doi.org/10.15672/hujms.638900

Cited By: 2

Abstract

A net $(x_\alpha)_{\alpha\in A}$ in an $f$-algebra $E$ is called multiplicative order convergent to $x\in E$ if $\lvert x_\alpha-x\rvert\cdot u \rightarrow 0$ for all $u\in E_+$. This convergence was introduced and studied on $f$-algebras with the order convergence. In this paper, we study a variation of this convergence for normed Riesz algebras with respect to the norm convergence. A net $(x_\alpha)_{\alpha\in A}$ in a normed Riesz algebra $E$ is said to be multiplicative norm convergent to $x\in E$ if $\big\lVert \lvert x_\alpha-x\rvert\cdot u\big\rVert\to 0$ for each $u\in E_+$. We study this concept and investigate its relationship with the other convergences, and also we introduce the $mn$-topology on normed Riesz algebras.

Keywords

$mn$-convergence, normed Riesz algebra, $mn$-topology, Riesz spaces, Riesz algebra, $mo$-convergence

References

• [1] Y. Abramovich and C.D. Aliprantis, An Invitation to Operator Theory, American Mathematical Society, New York, 2003.
• [2] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht, 2006.
• [3] A. Aydın, Unbounded $p_\tau$-convergence in vector lattice normed by locally solid vector lattices, in: Academic Studies in Mathematics and Natural Sciences-2019/2, 118-134, IVPE, Cetinje-Montenegro, 2019.
• [4] A. Aydın, Multiplicative order convergence in $f$-algebras, Hacet. J. Math. Stat. 49 (3), 998–1005, 2020.
• [5] A. Aydın, E. Emel’yanov, N.E. Özcan, and M.A.A. Marabeh, Compact-like operators in lattice-normed spaces, Indag. Math. 2, 633-656, 2018.
• [6] A. Aydın, E. Emel’yanov, N.E. Özcan, and M.A.A. Marabeh, Unbounded $p$-convergence in lattice-normed vector lattices, Sib. Adv. Math. 29, 153-181, 2019.
• [7] A. Aydın, S.G. Gorokhova, and H. Gül, Nonstandard hulls of lattice-normed ordered vector spaces, Turkish J. Math. 42, 155-163, 2018.
• [8] Y.A. Dabboorasad, E.Y. Emelyanov, and M.A.A. Marabeh, $u\tau$-Convergence in locally solid vector lattices, Positivity 22, 1065-1080, 2018.
• [9] Y. Deng, M. O’Brien, and V.G. Troitsky, Unbounded norm convergence in Banach lattices, Positivity 21, 963-974, 2017.
• [10] N. Gao, V.G. Troitsky, and F. Xanthos, $Uo$-convergence and its applications to Cesáro means in Banach lattices, Israel J. Math. 220, 649-689, 2017.
• [11] N. Gao and F. Xanthos, Unbounded order convergence and application to martingales without probability, Math. Anal. Appl. 415, 931-947, 2014.
• [12] C.B. Huijsmans and B.D. Pagter, Ideal theory in $f$-algebras, Trans. Amer. Math. Soc. 269, 225-245, 1982.
• [13] B.D. Pagter, $f$-Algebras and Orthomorphisms, Ph.D. Dissertation, Leiden, 1981.
• [14] V. Runde, A Taste of Topology, Springer, Berlin, 2005.
• [15] V.G. Troitsky, Measures of non-compactness of operators on Banach lattices, Positivity 8, 165-178, 2004.
• [16] B.Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Wolters- Noordhoff Scientific Publications, Groningen, 1967.
• [17] A.C. Zaanen, Riesz Spaces II, The Netherlands: North-Holland Publishing Co., Amsterdam, 1983.