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Let $\mathcal{A}$ and $\mathcal{B}$ be two unital complex $\ast $-algebras such that $\mathcal{A}$ has a nontrivial projection. In this paper, we study the structure of bijective nonlinear maps $\Phi :\mathcal{A}\rightarrow \mathcal{B}$…

Let $A$ and $B$ be unital complex Banach algebras having no quotients isomorphic to $\mathbb{C}$ or $M_2(\mathbb{C})$. Assume additionally that $B$ is semisimple. If a surjective additive mapping $\Phi\colon A\to B$ satisfies…

Given Hilbert spaces $H_1,H_2,H_3$, we consider bilinear maps defined on the cartesian product $S^2(H_2,H_3)\times S^2(H_1,H_2)$ of spaces of Hilbert-Schmidt operators and valued in either the space $B(H_1,H_3)$ of bounded operators, or in…

Given a sequence of bounded operators $a_j$ on a Hilbert space $H$ with $\sum a_j^*a_j=1=\sum a_ja_j^*$, we study the map $\Psi$ defined on $B(H)$ by $\Psi(x)=\sum a_j^*xa_j$ and its restriction $\Phi$ to the Hilbert-Schmidt class $C^2(H)$.…

We prove that for any two elements $A$, $B$ in a factor $M$, if $B$ commutes with all the unitary conjugates of $A$, then either $A$ or $B$ is in $\mathbb{C}I$. Then we obtain an equivalent condition for the situation that the $C$-numerical…

Given a ring $R$ with center $Z(R)$, we say a linear map $f:R\rightarrow R$ is commuting if $[f(x),x]=0$ for all $x\in R$. Such a map has a standard form if there exists $\lambda\in R$ and additive $\mu:R\rightarrow Z(R)$ such that…

Let $A$ and $B$ be Banach algebras with bounded approximate identities and let $\Phi:A\to B$ be a surjective continuous linear map which preserves two-sided zero products (i.e., $\Phi(a)\Phi(b)=\Phi(b)\Phi(a)=0$ whenever $ab=ba=0$). We show…

Let $\mathscr{A}$ be a unital $C^*$-algebra and let $\Phi: \mathscr{A} \to {\mathbb B}({\mathscr H})$ be a unital $n$-positive linear map between $C^*$-algebras for some $n \geq 3$. We show that $$\|\Phi(AB)-\Phi(A)\Phi(B)\| \leq…

We provide a characterization for operator valued completely bounded linear maps on Hilbert $C^*$-modules in terms of $\varphi$-maps. Also, we show that for every operator valued completely positive map $\varphi$ on a $C^*$-algebra…

We present an alternative (constructive) proof of the statement that for every completely positive, trace-preserving map $\Phi$ there exists an auxiliary Hilbert space $\mathcal K$ in a pure state $|\psi\rangle\langle\psi|$ as well as a…

Let $A$ be a Banach space, $p>1$, and $1/p+1/q=1$. If a sequence $a=(a_i)$ in $A$ has a finite $p$-sum, then the operator $\Lambda_a:\ell^q\to A$, defined by $\Lambda_a(\beta)=\sum_{i=1}^\infty \beta_i a_i, \beta=(\beta_i)\in \ell^q$, is…

Weighted shift operators $B$ in space $L^2(X,\mu)$ that are induced by Morse-Smale type of mappings are considered. A description of the properties of $B-\lambda I$ for $\lambda$ belonging to spectrum $\Sigma(B)$ is given. In particular,…

We show that each positive map from B(K) to B(H) with K and H finite dimensional Hilbert spaces is a scalar multiple of a map of the form $Tr - \psi$ with $\psi$ completely positive. This is used to give necessary and sufficient conditions…

We use induction and interpolation techniques to prove that a composition operator induced by a map $\phi$ is bounded on the weighted Bergman space $\A^2_\alpha(\mathbb{H})$ of the right half-plane if and only if $\phi$ fixes $\infty$…

This work develops a functional-analytic framework based on the transfinite iteration of a self-adjoint operator. Beginning with a densely defined self-adjoint operator $A$ on a Hilbert space $H$, a spectral-transform functor $\Phi$ is…

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 characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space…

For locally compact groups G and H let A(G) denote the Fourier algebra of G and B(H) the Fourier-Stieltjes algebra of H. Any continuous piecewise affine map alpha:Y -> G (where Y is an element of the open coset ring of H) induces a…

Uhlmann showed that there exists a positive, unital and trace-preserving map transforming a Hermitian matrix $A$ into another $B$ if and only if the vector of eigenvalues of $A$ majorizes that of $B$. In this work I characterize the…

Let $A=U|A|$ and $B=V|B|$ be the polar decompositions of $A\in \mathbb{B}(\mathscr{H}_1)$ and $B\in \mathbb{B}(\mathscr{H}_2)$ and let $Com(A,B)$ stand for the set of operators $X\in\mathbb{B}(\mathscr{H}_2,\mathscr{H}_1)$ such that…