On Approximately Finite-Dimensional Von Neuman Algebras, II

1978 ◽  
Vol 21 (4) ◽  
pp. 415-418 ◽  
Author(s):  
George A. Elliott
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Von Neumann ◽  
Intrinsic Characterization ◽  
C Algebras ◽  
Finite Dimensional ◽  
Positive Maps ◽  
Injective Von Neumann Algebra

AbstractAn intrinsic characterization is given of those von Neumann algebras which are injective objects in the category of C*-algebras with completely positive maps. For countably generated von Neumann algebras several such characterizations have been given, so it is in fact enough to observe that an injective von Neumann algebra is generated by an upward directed collection of injective countably generated sub von Neumann algebras. The present work also shows that three of the intrinsic characterizations known in the countably generated case hold in general.

scholarly journals BURES DISTANCE FOR COMPLETELY POSITIVE MAPS

2013 ◽  
Vol 16 (04) ◽  
pp. 1350031 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
K. SUMESH
Keyword(s):  
Von Neumann Algebra ◽  
Rigidity Theorem ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Von Neumann ◽  
C Algebras ◽  
Positive Maps ◽  
Normal States ◽  
Bures Distance

Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between C*-algebras by Kretschmann, Schlingemann and Werner. We present a Hilbert C*-module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples. We also prove a rigidity theorem, showing that representation modules of completely positive maps which are close to the identity map contain a copy of the original algebra.

On Completely Positive Maps Defined by an Irreducible Correspondence

1990 ◽  
Vol 33 (4) ◽  
pp. 434-441 ◽  
Author(s):  
C. Anantharaman-Delaroche
Keyword(s):  
Von Neumann Algebras ◽  
Convex Set ◽  
Extreme Points ◽  
Completely Positive Maps ◽  
Complex Matrices ◽  
Completely Positive ◽  
Von Neumann ◽  
Positive Maps

AbstractCompletely positive maps defined by an irreducible correspondence between two von Neumann algebras M and N are introduced. We give results about their structure and characterize, among them, those which are extreme points in the convex set of all unital completely positive maps from M to N. As particular cases we obtain known results of M. D. Choi [4] on completely positive maps between complex matrices and of J. A. Mingo [8] on inner completely positive maps.

scholarly journals ASYMPTOTIC STABILITY I: COMPLETELY POSITIVE MAPS

2004 ◽  
Vol 15 (03) ◽  
pp. 289-312 ◽  
Author(s):  
WILLIAM ARVESON
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Von Neumann Algebras ◽  
Completely Positive Map ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Locally Finite ◽  
Von Neumann ◽  
C Algebra ◽  
Positive Maps

We show that for every "locally finite" unit-preserving completely positive map P acting on a C*, there is a corresponding *-automorphism α of another unital C*-algebra such that the two sequences P, P2, P3, … and α, α2, α3, … have the same asymptotic behavior. The automorphism α is uniquely determined by P up to conjugacy. Similar results hold for normal completely positive maps on von Neumann algebras, as well as for one-parameter semigroups. These results are operator algebraic counterparts of the classical theory of Perron and Frobenius on the structure of square matrices with nonnegative entries.

scholarly journals When do pieces determine the whole? Extreme marginals of a completely positive map

2014 ◽  
Vol 26 (02) ◽  
pp. 1450002 ◽  
Author(s):  
E. Haapasalo ◽  
T. Heinosaari ◽  
J.-P. Pellonpää
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebras ◽  
Common Source ◽  
Completely Positive Map ◽  
Bounded Operators ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Von Neumann ◽  
Positive Maps ◽  
The Common

We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extreme point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extreme, then the whole map is extreme. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.

scholarly journals Decomposability of Bimodule Maps

2016 ◽  
Vol 119 (2) ◽  
pp. 283
Author(s):  
Christian Le Merdy ◽  
Lina Oliveira
Keyword(s):  
Linear Combination ◽  
Von Neumann Algebra ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Von Neumann ◽  
C Algebra ◽  
Positive Maps

Consider a unital $C^*$-algebra $A$, a von Neumann algebra $M$, a unital sub-$C^*$-algebra $C\subset A$ and a unital $*$-homomorphism $\pi\colon C\to M$. Let $u\colon A\to M$ be a decomposable map (i.e. a linear combination of completely positive maps) which is a $C$-bimodule map with respect to $\pi$. We show that $u$ is a linear combination of $C$-bimodule completely positive maps if and only if there exists a projection $e\in \pi(C)'$ such that $u$ is valued in $\mathit{e\mkern0.5muMe}$ and $e\pi({\cdot})e$ has a completely positive extension $A\to \mathit{e\mkern0.5muMe}$. We also show that this condition is always fulfilled when $C$ has the weak expectation property.

scholarly journals Normal Completely Positive Maps on the Space of Quantum Operations

2013 ◽  
Vol 20 (01) ◽  
pp. 1350003 ◽  
Author(s):  
Giulio Chiribella ◽  
Alessandro Toigo ◽  
Veronica Umanità
Keyword(s):  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Higher Order ◽  
Quantum Circuits ◽  
Bounded Operators ◽  
Completely Positive Maps ◽  
Von Neumann ◽  
Quantum Operations ◽  
Finite Dimensional ◽  
Positive Maps

Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite-dimensional setting, to the case of higher-order maps transforming quantum operations with input in a separable von Neumann algebra and output in the algebra of the bounded operators on a given separable Hilbert space. In this setting we prove two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. Finally, we consider the case of quantum superinstruments, namely measures with values in the set of quantum supermaps, and derive a dilation theorem for them that is analogue to Ozawa's theorem for quantum instruments. The three dilation theorems presented here show that all the supermaps defined in this paper can be implemented by connecting devices in quantum circuits.

Sign in / Sign up

Export Citation Format

Share Document