Related papers: Nonlinear $\ast$-Jordan-Type Derivations on von Ne…

Let $\mathscr{M}$ be a $II_1$ factor acting on the Hilbert space $\mathscr{H}$, and $\mathscr{M}_{\textrm{aff}}$ be the Murray-von Neumann algebra of closed densely-defined operators affiliated with $\mathscr{M}$. Let $\tau$ denote the…

Let $\mathcal{L}$ be a subspace lattice on a Banach space $X$ and let $\delta:\mathrm{Alg}\mathcal{L}\rightarrow B(X)$ be a linear mapping. If $\vee\{L\in \mathcal{L}: L_-\nsupseteq L\}=X$ or $\wedge\{L_-:L\in \mathcal{L}, L_-\nsupseteq…

Let A be an associative algebra (or any other kind of algebra for that matter). A derivation on A is an endomorphism \del of the underlying Abelian group of A such that \del(ab)=a(\del b)+(\del a)b for all a,b\in A (1.1) A Hasse-Schmidt…

We provide a differential structure on arbitrary cleft extensions $B:=A^{\mathrm{co}H}\subseteq A$ for an $H$-comodule algebra $A$. This is achieved by constructing a covariant calculus on the corresponding crossed product algebra…

If $\Delta$ and $\Gamma$ are two derivations of a commutative algebra $A$ such that $\Delta\Gamma-\Gamma\Delta=\Delta$ is locally nilpotent, one can endow $A$ with a new product $\ast$ whose filtered semiclassical limit is the Poisson…

Let $\mathcal R$ be a ring, $\mathcal{M}$ be a $\mathcal R$-bimodule and $m,n$ be two fixed nonnegative integers with $m+n\neq0$. An additive mapping $\delta$ from $\mathcal R$ into $\mathcal{M}$ is called an \emph{$(m,n)$-Jordan…

The Jordan type of an Artinian algebra is the Jordan block partition associated to multiplication by a generic element of the maximal ideal. We study the Jordan type for Artinian Gorenstein (AG) local algebras A, and the interaction of…

We study symmetric continuous bilinear maps $V$ on a C$^*$-algebra $A$ that have the Jordan product property at a fixed element $z\in A$. We show that, whenever $A$ is a finite direct sum or a $c_0$-sum of infinite simple von Neumann…

Let \(\mathcal{A}\) be a unital Banach algebra such that any Jordan derivation from \(\mathcal{A}\) into any \(\mathcal{A}\)-bimodule \(\mathcal{M}\) is a derivation. We prove that any 2-local derivation from the algebra $M_n(\mathcal{A})$…

For von Neumann algebras M, N not isomorphic to C^2 and without type I_2 summands, we show that for an order-isomorphism f:AbSub(M)->AbSub(N) between the posets of abelian von Neumann subalgebras of M and N, there is a unique Jordan…

The solution $X_n$ to a nonlinear stochastic differential equation of the form $dX_n(t)+A_n(t)X_n(t)\,dt-\tfrac12\sum_{j=1}^N(B_j^n(t))^2X_n(t)\,dt=\sum_{j=1}^N B_j^n(t)X_n(t)d\beta_j^n(t)+f_n(t)\,dt$, $X_n(0)=x$, where $\beta_j^n$ is a…

Let $\phi: A\to A$ be a (not necessarily linear, additive or continuous) map of a standard operator algebra. Suppose for any $a,b\in A$ there is an algebra automorphism $\theta_{a,b}$ of $ A$ such that \begin{align*} \phi(a)\phi(b) =…

We consider the unital associative algebra $\mathcal{A}$ with two generators $\mathcal{X}$, $\mathcal{Z}$ obeying the defining relation $[\mathcal{Z},\mathcal{X}]=\mathcal{Z}^2+\Delta$. We construct irreducible tridiagonal representations…

A Hilbert bimodule is a right Hilbert module X over a C*-algebra A together with a left action of A as adjointable operators on X. We consider families X = {X_s :s\in P} of Hilbert bimodules, indexed by a semigroup P, which are endowed with…

Let ${\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$ and $\xi\in{\mathbb C}$ a scalar. It is shown that an additive map $L$ on $\mathcal M$ satisfies $L(AB-\xi BA)=L(A)B-\xi BL(A)+L(B)A-\xi AL(B)$ whenever…

Let $\mathscr{R}$ be a type $II_1$ von Neumann algebra. We show that every unitary in $\mathscr{R}$ may be decomposed as the product of six symmetries (that is, self-adjoint unitaries) in $\mathscr{R}$, and every unitary in $\mathscr{R}$…

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free…

Given any square matrix or a bounded operator $A$ in a Hilbert space such that $p(A)$ is normal (or similar to normal), we construct a Banach algebra, depending on the polynomial $p$, for which a simple functional calculus holds. When the…

We find a class of algebras A satisfying the following property: for every nontrivial noncommutative polynomial, the linear span of all its values in A equals A. This class includes the algebras of all bounded and all compact operators on…

Given a complex Hilbert space H, we study the differential geometry of the manifold A of normal algebraic elements in Z=L(H), the algebra of bounded linear operators on H. We represent A as a disjoint union of subsets M of Z and, using the…