Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · funct-an/9512004

Minimal E_0-semigroups

William Arveson
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 completely positive semigroup; however, it is unique (up to conjugacy) provided that certain conditions of {\it minimality} are met. Minimality is a subtle property, and it is often not obvious if it is satisfied for specific examples even in the simplest case where the von Neumann algebra is $\Cal B(H)$. In this paper we clarify these issues by giving a new characterization of minimality in terms projective cocycles and their limits. Our results are valid for semigroups of endomorphisms acting on arbitrary von Neumann algebras with separable predual.

Explore related subjects

funct-anmath.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

William Arveson. 1995-12-20. Minimal E_0-semigroups. https://arxiv.org/abs/funct-an/9512004

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2020

Isomorphism classes for quantum Heisenberg manifolds

We embed the quantum Heisenberg manifold in a crossed product algebra. This enables us to show that, in the irrational case, all tracial states on $\dc$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\dc$ and $D^c_{μ' ν'}$ are isomorphic if and only if the parameters $(μ,ν)$ and $(μ',ν')$ belong to the same orbit under the usual action of $GL_2(\ZZ)$ on the torus.

funct-an↗
arXiv2007

Quantum Mechanics and Operator algebras on the Hilbert ball

Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative $*$-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$. The Hilbert ball $B_{\cal H}$ is defined by the open unit ball in ${\cal H}$ and it is a Kähler manifold with $c<0$. We introduce the quantum mechanics on $B_{\cal H}$. As an application, we show the structure of the noncommutative function algebra on $B_{\cal H}$.

funct-an↗
arXiv2004

No More Than Mechanics. I

One can introduce so-called {\em Plain Mechanics} having an {\bf operator realization}. Then the set of one-dimension representations of this operator realization may be identified with the Classical Mechanics. Different irreducible infinite-dimension representations may be recognized as Quantum Mechanics for different $\hbar$ (the Planck constant). It can be done in the such manner that the following diagram will be commutative. Plain Mechanics / \ / \ / \ Quantum Mechanics --> Classical Mechanics h->0 Here the horizontal arrow is well known correspondence between Quantum and Classical Mechanics if Planck constant tensing to zero. A {\em realization} of this scheme for a particle in $n$-dimensional space by two-sided convolutions on the Heisenberg group is constructed. We also introduce the {\em motion equations} for observables in this realization. The left arrow of the given diagram carries this equation to the Heisenberg one and the right arrow maps it to the Hamilton equation.

funct-an↗