Related papers: Jordan and Lie derivations of $\phi $-Johnson amen…

We call a linear operator on a vector space over a field Jordanable if it has a Jordan canonical form. An almost Abelian Lie algebra has only one non-vanishing Lie bracket, which is given by a linear operator. If the latter is Jordanable…

We look for an effective description of the algebra D_{Lie}(X,B) of operators on a bimodule X over an algebra B, generated by inner derivations. It is shown that in some important examples D_{Lie}(X,B) consists of all elementary operators…

Let $\mathcal A$ be a von Neumann algebra and $\mathcal M$ be a Banach $\mathcal A-$module. It is shown that for every homomorphisms $\sigma, \tau$ on $\mathcal A$, every bounded linear map $f:\mathcal A\to \mathcal M$ with property that…

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. We consider Jordan $*$-homomorphisms on $C(X, \mathcal{A})$ and Jordan $*$-homomorphisms on $\operatorname{Lip}(X,\mathcal{A})$. More precisely, for any unital $C^{*}$-algebra $\mathcal{A}$, we…

Let $A$ and $B$ be Banach algebras and let $B$ be an algebraic Banach $A-$bimodule. Then the $\ell^1-$direct sum $A\times B$ equipped with the multiplication $$(a_1,b_1)(a_2,b_2)=(a_1a_2,a_1\cdot b_2+b_1\cdot a_2+b_1b_2),~~ (a_1, a_2\in A,…

Let $X$ be a compact Hausdorff space and $A$ a Banach algebra. We investigate amenability properties of the algebra $C(X,A)$ of all $A$-valued continuous functions. We show that $C(X,A)$ has a bounded approximate diagonal if and only if $A$…

We introduce two notions of amenability for a Banach algebra $\cal A$. Let $n\in \Bbb N$ and let $I$ be a closed two-sided ideal in $\cal A$, $\cal A$ is $n-I-$weakly amenable if the first cohomology group of $\cal A$ with coefficients in…

The notion of derivation with invertible values as a derivation of ring with unity that only takes multiplicatively invertible or zero values appeared in a paper of Bergen, Herstein and Lanski, in which they determined the structure of…

For a given ring $\mathfrak{R}$ and a locally finite pre-ordered set $(X, \leq)$, consider $I(X, \mathfrak{R})$ to be the incidence algebra of $X$ over $\mathfrak{R}$. Motivated by a Xiao's result which states that every Jordan derivation…

Given $L$ a complex solvable finite dimensional Lie Algebra of operators acting on a Banach space $E$ and $\{ x_i\}_{1\le i\le n}$ a Jordan-H\"older basis of $ L$, we study the relation between $Sp(L,E)$ and $\Pi Sp(x_i)$.

Let $A$ be a unital Banach algebra. We give a characterization of the left Banach $A$-modules $X$ for which there exists a commutative unital $C^*$-algebra $C(K)$, a linear isometry $i\colon X\to C(K)$, and a contractive unital homomorphism…

Let $A$ and $B$ be associative algebras over a field $F$ with {\rm char}$(F)\ne 2$. Our first main result states that if $A$ is unital and equal to its commutator ideal, then every Jordan epimorphism $\varphi:A\to B$ is the sum of a…

Let $A$ be a Banach algebra and $M$ be a Banach right $A$-module. A linear map $\delta : M\to M$ is called a generalized derivation if there exists a derivation $d : A \to A$ such that $$\delta(xa)=\delta(x)a + x d(a) \quad (a \in A, x \in…

In this article we prove that the elliptic, hyperbolic and nilpotent (or unipotent) additive (or multiplicative) Jordan components of an endomorphism $X$ (or an isomorphism $g$) of a finite dimensional vector space are given by polynomials…

We show that if an open arc J of the boundary of a Jordan domain $\Omega$ is rectifiable, then the derivative $\Phi$' of the Riemann map $\Phi: D\rightarrow \Omega$ from the open unit disk D onto $\Omega$ behaves as an $H^1$ function when…

We know that any element $X$ of the exceptional Jordan algebra $\gJ$ is transformed to a diagonal form by the compact exceptional Lie group $F_4$. However, its proof is used the method which is reduced a contradiction. In this paper, we…

In the present paper 2-local derivations on various algebras of infinite dimensional matrix-valued functions on a compact are considered. It is proved that every 2-local derivation on such algebra is a derivation. Also we explain that the…

Let $X$ be a Banach algebra and $B(X)$ be the set of all bounded linear operators on $X$. Suppose that $\alpha: B(X) \rightarrow B(X)$ is an automorphism. We say that a mapping $\delta$ from $B(X)$ into itself is derivable at $G \in B(X)$…

In this note, we prove that any Jordan derivation on the generalized matrix ring $T_n(R,M)$ is a derivation. This extends some well-known results of this branch due to Bre\v{s}ar et al. in the cited literature.

Let $U$ be a Banach Lie group and $G\le U$ a compact subgroup. We show that closed Lie subgroups of $U$ contained in sufficiently small neighborhoods $V\supseteq G$ are compact, and conjugate to subgroups of $G$ by elements close to $1\in…