On cyclic derivations and (approximate) cyclic amenability of triangular Banach algebras

Abstract

‎The inherited properties between the triangular Banach algebra in the form of $\mathcal{T}=\Mat {A}{M}{B}$ and the corner Banach algebras $A$ and $B$ are very important and studied‎. ‎In this paper‎, ‎first a method for investigating being cyclic of derivations on Banach algebras is presented and then the cyclic derivations on triangular Banach algebras is studied‎. ‎Using the above method‎, ‎we show that the unital triangular Banach algebra $\mathcal{T}$‎ ‎is cyclic amenable if and only if the corner algebras $A$ and $B$ are cyclic amenable‎. ‎Finally‎, ‎an example shows that the condition of unitary of $\mathcal{T}$ is necessary for the correctness of the results.\