let x be a banach space of dimx > 2 and b(x) be the space of bounded linear operators on x. if l : b(x) → b(x) be a lie higher derivation on b(x), then there exists an additive higher derivation d and a linear map τ : b(x) → fi vanishing at commutators [a, b] for all a, b ∈ b(x) such that l = d + τ

This article presents a unified approach to the abstract notions of partial convolution and involution in $L^p$-function spaces over semi-direct product of locally compact groups. Let $H$ and $K$ be locally compact groups and $tau:Hto Aut(K)$ be a continuous homomorphism.  Let $G_tau=Hltimes_tau K$ be the semi-direct product of $H$ and $K$ with respect to $tau$. We define left and right $tau$-c...

In this paper, first, we introduce the new concept of 2-inner product on Banach modules over a $C^*$-algebra. Next,  we present the concept of 2-linear operators over a $C^*$-algebra. Our result improve  the main result of the paper  Z. Lewandowska.  In the final of this paper, we define the notions 2-adjointable mappings between 2-pre Hilbert C*-modules and prove supperstability of them ...

let $mathfrak{l}$ be the virasoro-like algebra and $mathfrak{g}$ itsderived algebra, respectively. we investigate the structure of the lie triplederivation algebra of $mathfrak{l}$ and $mathfrak{g}$. we provethat they are both isomorphic to $mathfrak{l}$, which provides twoexamples of invariance under triple derivation.

1. The norm || • || in a Banach algebra A is said to be minimal [l ] if, given any other norm || •||1 in A (with respect to which A need not be complete), the condition ||a||iá||a|| for each oG^4 implies that ||a[|i = ||a||. We shall say that || •|| is absolutely minimal if, given any other norm ||-||i whatever in A, then ||a||iè||a|| for each aEA. An absolutely minimal norm is of course minima...

In [2] Singer and Wermer showed that a bounded derivation in a commutative Banach algebra 21 necessarily maps 21 into the radical 91. They conjectured at this time that the assumption of boundedness could be dropped. It is a corollary of results proved below that if 21 is in addition regular and semi-simple, this is indeed the case. What is actually proved here is that under the above hypothese...

Let A be a unital R-algebra and M be a unital A-bimodule. It is shown that every Jordan derivation of the trivial extension of A by M, under some conditions, is the sum of a derivation and an antiderivation.

Let $mathcal M$ be a factor von Neumann algebra. It is shown that every nonlinear $*$-Lie higher derivation$D={phi_{n}}_{ninmathbb{N}}$ on $mathcal M$ is additive. In particular, if $mathcal M$ is infinite type $I$factor, a concrete characterization of $D$ is given.