The multiplicative norm convergence in normed Riesz algebras

A net (x(alpha))(alpha is an element of A) in an f-algebra E is called multiplicative order convergent to x is an element of E if vertical bar x(alpha )- x vertical bar . u ->(o) 0 for all u is an element of 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 is an element of A) in a normed Riesz algebra E is said to be multiplicative norm convergent to x is an element of E if parallel to vertical bar x(alpha) - x vertical bar . u parallel to -> 0 for each u is an element of E+. We study this concept and investigate its relationship with the other convergences, and also we introduce the mn-topology on normed Riesz algebras.