Multiplicative norm convergence in Banach lattice $f$-algebras

A net $(x_\alpha)$ in an $f$-algebra $E$ is called multiplicative order convergent to $x\in E$ if $\lvert x_\alpha-x\rvert u\oc 0$ for all $u\in E_+$. This convergence has been investigated and applied in a recent paper by Ayd{\i}n \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 $\lvert \lvert x_\alpha-x\rvert u\rVert\to 0$ 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.