Continuous operators on Riesz algebras

A net (x_α )_(α∈A) in a Riesz space E is called order convergent to x∈E if there is another net (y_β )_(β∈B)↓0 such that for every β∈B there can be at least one α_β∈A index so that |x_α-x|≤y_β holds for all α≥α_β. If a Riesz space E is an associative algebra and also x∙y∈E_+ for any positive elements x and y in E then E is called Riesz algebra. For any net (x_α )_(α∈A) in a Riesz algebra E satisfying |x_α-x|∙u order convergent to zero for all u∈E_+ is called multiplicative order convergent to x. In this study, by using multiplicative order convergence, the concepts of multiplicative order continuous and multiplicative order bounded operators in Riesz algebras are given, and some relations between these operators and order continuous operators are investigated.