For a state $\omega$ on a C$^{*}$-algebra $A$, we characterize all states $\rho$ in the weak* closure of the set of all states of the form $\omega \circ \varphi$, where $\varphi$ is a map on $A$ of the form $\varphi (x)=\sum \nolimits _{i=1}^{n}a_i^{*}xa_i,$ $\sum \nolimits _{i=1}^{n}a_i^{*}a_i=1$ ($a_i\in A$, $n\in \mathbb {N}$). These are precisely the states $\rho$ that satisfy $\|\rho |J\|\leq \|\omega |J\|$ for each ideal $J$ of $A$. The corresponding question for normal states on a von Neumann algebra $\mathcal {R}$ (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form $\omega \circ \psi$, where $\psi$ is a quantum channel on $\mathcal {R}$ (that is, a map of the form $\psi (x)=\sum \nolimits _ja_j^{*}xa_j$, where $a_j\in \mathcal {R}$ are such that the sum $\sum \nolimits _ja_j^{*}a_j$ converge to $1$ in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C$^{*}$-algebra and for properly infinite von Neumann algebras the converse also holds.

We give a general method of extending unital completely positive maps to amalgamated free products of C*-algebras. As an application, we give a dilation theoretic proof of Boca's Theorem.

It is shown that the complexification of a positive linear map on a real C*-algebra need not be positive whereas the complexification of a completely positive linear map is completely positive. It is further shown that a real C*-algebra is amenable if and only if its complexification is amenable and that a completely positive linear map is amenable if and only if its complexification is. Finally, a real version of the Choi–Effros lifting theorem is established.

We study conjugacy invariants for completely positive maps that are inspired by the concept of curvature introduced for commuting d-tuples of contractions by Arveson.

We show that the representations of the Cuntz $C^*$-algebras $\mathcal{O}_n$ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work.

AMS 2000 Mathematics subject classification: Primary 46L45; 47A20; 46L60; 42C40; 81P15