the aim of this note is to study the submajorization inequalities for $tau$-measurable operators in a semi-finite von neumann algebra on a hilbert space with a normal faithful semi-finite trace $tau$. the submajorization inequalities generalize some results due to zhang, furuichi and lin, etc..

The aim of this note is to study the submajorization inequalities for $tau$-measurable operators in a semi-finite von Neumann algebra on a Hilbert space with a normal faithful semi-finite trace $tau$. The submajorization inequalities generalize some results due to Zhang, Furuichi and Lin, etc..

In this paper, we define and study two operators $Phi^s$ and $Psi^s$ with grill. Characterization and basic properties of these operators are obtained. Also, we generalize a grill topological spaces via topology $tau^s$ induced from operators $Phi^s$ and $Psi^s$.

Let $X$ be a Banach space of $dim X > 2$ and $B(X)$ be the space of bounded linear operators on X. If $L : B(X)to B(X)$ be a Lie higher derivation on $B(X)$, then there exists an additive higher derivation $D$ and a linear map $tau : B(X)to FI$ vanishing at commutators $[A, B]$ for all $A, Bin B(X)$ such that $L = D + tau$.

Abstract We study, from a quite general point of view, CQ*-algebra (X,