Related papers: On derivations and Jordan derivations through zero…

Let $\mathcal A$ be a Banach algebra. Using the concept of module biflatness, we show that the module amenability of the second dual $\mathcal A^{**}$ (with the first Arens product) necessitates the module amenability of $\mathcal A$. We…

On a compact subset of the plane with connected complement, is it possible to uniformly approximate a continuous function, holomorphic and non-vanishing on the interior, with polynomials non-vanishing on the entire compact set? In this…

Let $T:A\to X$ be a bounded linear operator, where $A$ is a C$^*$-algebra, and $X$ denotes an essential Banach $A$-bimodule. We prove that the following statements are equivalent: $(a)$ $T$ is anti-derivable at zero (i.e. $ab =0$ in $A$…

We prove that every surjective unital linear mapping which preserves invertible elements from a Banach algebra onto a C*-algebra carrying a faithful tracial state is a Jordan homomorphism thus generalising Aupetit's 1998 result for finite…

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…

While Jordan algebras are commutative, their non-associativity makes it so that the Jordan product operators do not necessarily commute. When the product operators of two elements commute, the elements are said to operator commute. In some…

Let $\mathcal{I,J}$ be symmetric quasi-Banach ideals of compact operators on an infinite-dimensional complex Hilbert space $H$, let $\mathcal{J:I}$ be a space of multipliers from $\mathcal{I}$ to $\mathcal{J}$. Obviously, ideals…

Let $\mathcal{U}=\left[ \begin{array}{cc} \mathcal{A} & \mathcal{M} \mathcal{N}& \mathcal{B} \end{array} \right]$ be a generalized matrix ring, where $\mathcal{A}$ and $\mathcal{B}$ are 2-torsion free. We prove that if $\phi…

Let $\mathfrak{A}$ be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Jordan $n$-derivation $\Delta:\mathfrak{A}\rightarrow\mathfrak{A}$ is additive. More…

We prove that if $\mathcal M$ is a properly infinite von Neumann algebra and $LS(\mathcal M)$ is the local measurable operator algebra affiliated with $\mathcal M$, then every Jordan derivation from $LS(\mathcal M)$ into itself is…

We analyze a definition of product of Banach spaces that is naturally associated by duality with an abstract notion of space of multiplication operators. This dual relation allows to understand several constructions coming from different…

We study Lie brackets and Jordan products derived from associative operations $\circ, \bullet$ satisfying the interchange identity $(w \bullet x ) \circ ( y \bullet z ) \equiv (w \circ y ) \bullet ( x \circ z )$. We use computational linear…

Let A be a Banach algebra and let X be a Banach A -bimodule. In studying the bounded Hochschild cohomology groups H^1(A,X) it is often useful to extend a given derivation D: A-> X to a Banach algebra B containing A as an ideal, thereby…

This paper is an elaborated version of the material presented by the author in a three hour minicourse at "V International Course of Mathematical Analysis in Andalusia," Almeria, Spain, September 12-16, 2011. Part I is devoted to an…

Let $W$ be a quasiprojective variety over an algebraically closed field of characteristic zero. Assume that $W$ is birational to a product of a smooth projective variety $A$ and the projective line. We prove that if $A$ contains no rational…

We describe the ternary and the generalized superderivations of finite-dimensional semisimple Jordan superalgebras over an algebraically closed field of characteristic zero and of finite-dimensional simple Jordan superalgebras with…

We prove new results on generalized derivations on C$^*$-algebras. By considering the triple product $\{a,b,c\} =2^{-1} (a b^* c + c b^* a)$, we introduce the study of linear maps which are triple derivations or triple homomorphisms at a…

We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized derivations from $\mathcal{J}$…

Let $\mathcal{A}$ be a factor with dim$\mathcal{A}\geq2$. For $A, B\in\mathcal{A}$, define by $[A, B]_{*}=AB-BA^{\ast}$ and $A\bullet B=AB+BA^{\ast}$ the new products of $A$ and $B$. In this paper, it is proved that a map $\Phi: \mathcal…

A Jordan H\"older theorem is established for derived module categories of piecewise hereditary algebras. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module…