The multiplicative norm convergence in normed Riesz algebras

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

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