Multiplicative norm convergence in Banach lattice $f$-algebras

Abstract

A net $(x_{\alpha })$ in an $f$-algebra $E$ is called multiplicative order convergent to $x\in E$ if $\lvert x_α-x\rvert u\oc 0$ for all $u\in E_{+}$. This convergence has been investigated and applied in a recent paper by Aydın \cite{AAydn}. In this paper, we study a variation of this convergence for Banach lattice $f$-algebras. A net $(x_{\alpha })$ in a Banach lattice $f$-algebra $E$ is said to be multiplicative norm convergent to $x\in E$ if $||x_{\alpha }-x|u\Vert \to 0$ for each $u\in E_{+}$. We study on this concept and investigate its relationship with the other convergences, and also we introduce the $mn$-topology Banach lattice $f$-algebras.

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• Multiplicative weak convergence in Banach $f$-algebras