For a separable Banach space X, let L denotes the algebra of bounded Operators on X endowed with the strong Operator topology. In this note we provide a sufficient condition, in terms of open set in the strong Operator topology, for a bounded linear mapping acting on L to be supercyclic with this topology. The main result has some useful consequences and make easy the proof of some related results.

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The main goal of this paper is to introduce analogously modal Operators on $L$-algebras and study their properties. To begin with, we introduce the notion of modal Operators on $L$-algebras and investigate some important properties of this Operator. In order for the kernel of modal Operator to be ideal, we investigate what conditions are required. Relations between modal Operator and endomorphism of $L$-algebras are investigated. Also, we define the concept of positive $L$-algebra and some characterizations of positive $L$-algebra are established. Finally, we introduce a map $k_{a}$ and show that $k_{a}$ is a modal Operator and we prove that the set of all $k_{a}$ on a positive $L$-algebra makes a dual BCK-algebra.

It is known that even duals of a Banach algebra A with one of Arens products are Banach algebras, these products are natu-ral multiplications extending the one on A. But the essence of A ∗,is completely di, erent. By de, ning new products, we investigate some algebraic and spectral properties of odd duals of A. We will show re-lations between these products and Arens products, weak or weak-star continuity, commutativity and unit elements of these algebras. We also determine the spectrum and multiplier algebra for A ∗, , and we calculate the quasi-inverses, spectrum and spectral radius for elements of these kinds of algebras.