Related papers: E-dilation of strongly commuting CP-semigroups (th…

We prove that every strongly commuting pair of CP_0-semigroups has a minimal E_0-dilation. This is achieved in two major steps, interesting in themselves: 1: we show that a strongly commuting pair of CP_0-semigroups can be represented via a…

We construct a weak dilation of a not necessarily unital CP-semigroup to an E-semigroup acting on the adjointable operators of a Hilbert module with a unit vector. We construct the dilation in such a way that the dilating E-semigroup has a…

Using ideas due to Jean-Luc Sauvageot, we prove the existence of a continuous unital dilation of a CP$_0$-semigroup on a separable W$^*$-algebra. This paper presents the material in the author's Ph. D. thesis (arXiv.org:1304.0134.pdf) with…

We study dilations of finite tuples of normal, completely positive and completely contractive maps (which we call CP-maps) acting on a von Neumann algebra, and commuting according to a graph G. We show that if G is acyclic, then a tuple…

Dilations of completely positive semigroups to endomorphism semigroups have been studied by numerous authors. Most existing dilation theorems involve a non-unital embedding, corresponding to the embedding of $B(H)$ as a corner of $B(K)$ for…

We study completely contractive representations of product systems of $C^*$-correspondences over semigroups. For a product system of $C^*$-correspondences over the semigroup $\mathbb{N}^2$, we prove that every such representation can be…

We apply Hilbert module methods to show that normal completely positive maps admit weak tensor dilations. Appealing to a duality between weak tensor dilations and extensions of CP-maps, we get an existence proof for certain extensions. We…

Consider $d$ commuting $C_{0}$-semigroups (or equivalently: $d$-parameter $C_{0}$-semigroups) over a Hilbert space for $d \in \mathbb{N}$. In the literature (\textit{cf.} [29, 26, 27, 23, 18, 25]), conditions are provided to classify the…

It is known that every semigroup of normal completely positive maps $P = {P_t: t\geq 0}$ of $B(H)$, satisfying $P_t(1) = 1$ for every $t\geq 0$, has a minimal dilation to an E_0-semigroup acting on $B(K)$ for some Hilbert space K containing…

This is a continuation of the study of the theory of quantum stochastic dilation of completely positive semigroups on a von Neumann or $C^*$ algebra, here with unbounded generators. The additional assumption of symmetry with respect to a…

Motivated by Ball, Li, Timotin and Trent's Schur-Agler class version of commutant lifting theorem, we introduce a class, denoted by $\mathcal{P}_n(\mathcal{H})$, of $n$-tuples of commuting contractions on a Hilbert space $\mathcal{H}$. We…

It is known that every semigroup of normal completely positive maps of a von Neumann can be ``dilated" in a particular way to an E_0-semigroup acting on a larger von Neumann algebra. The E_0-semigroup is not uniquely determined by the…

We prove that if $\{\phi_t\}_{t \geq 0}$ is a CP-semigroup acting on a von Neumann algebra $M \subseteq B(H)$, then for every $A\in M$ and $\xi \in H$, the map $t \mapsto \phi_t(A)\xi$ is norm-continuous. We discuss the implications of this…

Let H be a separable Hilbert space. Given two strongly commuting CP_0-semigroups $\phi$ and $\theta$ on B(H), there is a Hilbert space K containing H and two (strongly) commuting E_0-semigroups $\alpha$ and $\beta$ such that $\phi_s \circ…

We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of factorizable Markov maps acting on a von Neumann algebra $M$ equipped with a normal faithful state can be dilated by a group of Markov $*$-automorphisms analogous to the…

Let $\varphi$ be a normal semi-finite faithful weight on a von Neumann algebra $A$,let $(\sigma^\varphi_r)_{r\in{\mathbb R}}$ denote the modular automorphism group of $\varphi$, and let $T\colon A\to A$ be a linear map. We say that $T$…

We consider characterisations of unitary dilations and approximations of irreversible classical dynamical systems on a Hilbert space. In the commutative case, building on the work in [9], one can express well known approximants (e.g. Hille-…

It is well-known that a commuting family of contractions possesses a regular unitary dilation if and only if it satisfies Brehmer's positivity condition. We extend this theorem to any family $\mathcal T$ of $q$-commuting contractions with…

Let B be a sigma-unital C*-algebra. We show that every strongly continuous E_0-semigroup on the algebra of adjointable operators on a full Hilbert B-module E gives rise to a full continuous product system of correspondences over B. We show…

We generalise a technique of Bhat and Skeide (2015) to interpolate commuting families $\{S_{i}\}_{i \in \mathcal{I}}$ of contractions on a Hilbert space $\mathcal{H}$, to commuting families $\{T_{i}\}_{i \in \mathcal{I}}$ of contractive…