scholarly journals The Infimum Norm of Completely Positive Maps

2022 ◽  
Vol 14 (1) ◽  
pp. 51
Author(s):  
Ching Yun Suen
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
Linear Map ◽  
C Algebra ◽  
Positive Maps ◽  
Positive Linear Map

Let A  be a unital C* -algebra, let L: A→B(H)  be a linear map, and let ∅: A→B(H)  be a completely positive linear map. We prove the property in the following:  is completely positive}=inf {||T*T+TT*||1/2:  L= V*TπV  which is a minimal commutant representation with isometry} . Moreover, if L=L* , then  is completely positive  . In the paper we also extend the result  is completely positive}=inf{||T||: L=V*TπV}  [3 , Corollary 3.12].

MODULES OVER MONOTONE COMPLETE C*-ALGEBRAS

1992 ◽  
Vol 03 (02) ◽  
pp. 185-204 ◽  
Author(s):  
MASAMICHI HAMANA
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
C Algebra ◽  
C Algebras ◽  
Positive Maps

The main result asserts that given two monotone complete C*-algebras A and B, B is faithfully represented as a monotone closed C*-subalgebra of the monotone complete C*-algebra End A(X) consisting of all bounded module endomorphisms of some self-dual Hilbert A-module X if and only if there are sufficiently many normal completely positive maps of B into A. The key to the proof is the fact that each pre-Hilbert A-module can be completed uniquely to a self-dual Hilbert A-module.

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 Completely positive maps into corona algebras

2009 ◽  
Vol 147 (2) ◽  
pp. 323-337 ◽  
Author(s):  
DAN Z. KUCEROVSKY ◽  
P. W. NG
Keyword(s):  
Decomposition Theorem ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
C Algebra ◽  
Positive Maps

AbstractWe prove a decomposition theorem similar to the well-known result of Voiculescu's: namely that completely positive maps A → (B)/B factor through a given homomorphism ι: A →(B)/B when the homomorphism ι has a certain infiniteness property.The algebra B is only assumed to be separable and nonunital; in particular, it is not assumed to be stable. The C*-algebra A is assumed to be separable, unital and nuclear.

On a dilation of a some class of completely positive maps

2019 ◽  
pp. 333-339
Author(s):  
Yakub V. ELSAEV
Keyword(s):  
Cartesian Product ◽  
Group Symmetry ◽  
Quantum Field ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Sesquilinear Forms ◽  
C Algebra ◽  
Covariant Quantum ◽  
Positive Maps ◽  
Minimal Representations

In this article we investigate sesquilinear forms defined on the Cartesian product of Hilbert C^*-module M over C^*-algebra B and taking values in B. The set of all such defined sesquilinear forms is denoted by S_B (M). We consider completely positive maps from locally C^*-algebra A to S_B (M). Moreover we assume that these completely positive maps are covariant with respect to actions of a group symmetry. This allow us to view these maps as generalizations covariant quantum instruments which are very important for the modern quantum mechanic and the quantum field theory. We analyze the dilation problem for these class of maps. In order to solve this problem we construct the minimal Stinespring representation and prove that every two minimal representations are unitarily equivalent.

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.

Dilations and Hahn Decompositions for Linear Maps

1981 ◽  
Vol 33 (4) ◽  
pp. 826-839 ◽  
Author(s):  
D. W. Hadwin
Keyword(s):  
Linear Operators ◽  
Completely Positive Maps ◽  
Linear Combinations ◽  
Completely Positive ◽  
Bounded Linear ◽  
Positive Maps ◽  
Linear Maps ◽  
Positive Linear Map ◽  
Finite Linear ◽  
Analogous Class

Suppose is a C*-algebra and H is a Hilbert space. Let denote the set of completely positive maps from into the set B(H) of (bounded linear) operators on H. This paper studies the vector space spanned by , i.e., the linear maps that are finite linear combinations of completely positive maps. From another viewpoint, a map ϕ is in precisely when it has a decomposition ϕ = (ϕ1 – ϕ2) + i(ϕ3 – ϕ4) with ϕ1, ϕ2, ϕ3, ϕ4 in CP ; this decomposition is analogous to the Hahn decomposition for measures [8, 111.4.10] (see also Theorem 20). The analogous class of maps with “completely positive” replaced by “positive” was studied by R. I. Loebl [11] and S.-K. Tsui [17], and when is commutative, this latter class coincides withi , since every positive linear map on a commutative C*-algebra is completely positive [16].

scholarly journals Quantum weakest preconditions

2006 ◽  
Vol 16 (3) ◽  
pp. 429-451 ◽  
Author(s):  
ELLIE D'HONDT ◽  
PRAKASH PANANGADEN
Keyword(s):  
Quantum Computation ◽  
Representation Theorem ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Weakest Precondition ◽  
Positive Maps ◽  
Weakest Preconditions ◽  
Predicate Transformer ◽  
Usual State ◽  
Stone Type

We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stone-type duality between the usual state-transformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum computation to probabilistic programming, we develop a notion that is directly taken from concepts used in quantum computation. The proof that weakest preconditions exist for completely positive maps follows immediately from the Kraus representation theorem. As an example, we give the semantics of Selinger's language in terms of our weakest preconditions. We also cover some specific situations and exhibit an interesting link with stabilisers.

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