Extreme Points of the Convex set of Stochastic Maps on A C*-Algebra

1998 ◽  
Vol 01 (04) ◽  
pp. 599-609 ◽  
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Bounded Operators ◽  
Completely Positive ◽  
C Algebra ◽  
Permutation Matrices ◽  
Normal Maps

Let [Formula: see text] be a unital C*-subalgebra of the C*-algebra ℬ(ℋ) of all bounded operators on a complex separable Hilbert space ℋ. Let [Formula: see text] denote the convex set of all unital, linear, completely positive and normal maps of [Formula: see text] into itself. Using Stinespring's theorem, we present a criterion for an element [Formula: see text] to be extremal. When [Formula: see text], this criterion leads to an explicit description of the set of all extreme points of [Formula: see text]. We also obtain a quantum probabilistic analogue of the classical Birkhoff's theorem2 that every bistochastic matrix can be expressed as a convex combination of permutation matrices.

On the Extreme Rays of the Metric Cone

1980 ◽  
Vol 32 (1) ◽  
pp. 126-144 ◽  
Author(s):  
David Avis
Keyword(s):  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Convex Polyhedra ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Image Position ◽  
Polyhedral Convex Set ◽  
Extreme Rays

A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equationsBirkhoff [2] and Von Neuman have shown that the extreme points of this bounded polyhedron are just the n × n permutation matrices. The importance of this result for mathematical programming is that it tells us that the maximum of any linear form over P will occur for a permutation matrix X.

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.

Asymptotic normality of sums of Hilbert space valued random elements

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Alfredas Račkauskas
Keyword(s):  
Hilbert Space ◽  
Asymptotic Normality ◽  
Separable Hilbert Space ◽  
Linear Process ◽  
Weighted Sums ◽  
Bounded Operators ◽  
Linear Processes ◽  
Summation Methods ◽  
Random Elements

Abstract We investigate the asymptotic normality of distributions of the sequence {\sum_{k\in\mathbb{Z}}u_{n,k}X_{k}} , {n\in\mathbb{N}} , where {(X_{k},k\in\mathbb{Z})} either is a sequence of i.i.d. random elements or constitutes a linear process with i.i.d. innovations in a separable Hilbert space. The weights {(u_{n,k})} are in general a family of linear bounded operators. This model includes operator weighted sums of Hilbert space valued linear processes, operator-wise discounted sums in a Hilbert space as well some extensions of classical summation methods.

THE REPRESENTATIONS AND ENDOMORPHISMS OF A SEPARABLE NUCLEAR C*-ALGEBRA

2003 ◽  
Vol 14 (03) ◽  
pp. 313-326 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Hilbert Space ◽  
Irreducible Representation ◽  
Separable Hilbert Space ◽  
Type I ◽  
C Algebra

It is shown that if A is a separable, non-type I, nuclear simple C*-algebra and π is an irreducible representation of A, then for any representation ρ on a separable Hilbert space of A there is an endomorphism α of A such that ρ is unitarily equivalent to πα.

scholarly journals SPECTRUM, NUMERICAL RANGE AND DAVIS-WIELANDT SHELL OF A NORMAL OPERATOR

2009 ◽  
Vol 51 (1) ◽  
pp. 91-100 ◽  
Author(s):  
CHI-KWONG LI ◽  
YIU-TUNG POON
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Convex Set ◽  
Numerical Range ◽  
Normal Operator ◽  
Additional Information ◽  
Commuting Operators ◽  
Boundary Points ◽  
Reducing Subspace ◽  
Joint Numerical Range

AbstractWe denote the numerical range of the normal operator T by W(T). A characterization is given to the points in W(T) that lie on the boundary. The collection of such boundary points together with the interior of the the convex hull of the spectrum of T will then be the set W(T). Moreover, it is shown that such boundary points reveal a lot of information about the normal operator. For instance, such a boundary point always associates with an invariant (reducing) subspace of the normal operator. It follows that a normal operator acting on a separable Hilbert space cannot have a closed strictly convex set as its numerical range. Similar results are obtained for the Davis-Wielandt shell of a normal operator. One can deduce additional information of the normal operator by studying the boundary of its Davis-Wielandt shell. Further extension of the result to the joint numerical range of commuting operators is discussed.

scholarly journals Completely positive maps for imprimitive complex reflection groups

2021 ◽  
Vol 13 (2) ◽  
pp. 452-459
Author(s):  
H. Randriamaro
Keyword(s):  
Hilbert Space ◽  
Coxeter Groups ◽  
Fock Space ◽  
Inner Product ◽  
Reflection Groups ◽  
Bounded Operators ◽  
Completely Positive ◽  
Creation And Annihilation Operators ◽  
Complex Reflection ◽  
Complex Reflection Groups

In 1994, M. Bożejko and R. Speicher proved the existence of completely positive quasimultiplicative maps from the group algebra of Coxeter groups to the set of bounded operators. They used some of them to define an inner product associated to creation and annihilation operators on a direct sum of Hilbert space tensor powers called full Fock space. Afterwards, A. Mathas and R. Orellana defined in 2008 a length function on imprimitive complex reflection groups that allowed them to introduce an analogue to the descent algebra of Coxeter groups. In this article, we use the length function defined by A. Mathas and R. Orellana to extend the result of M. Bożejko and R. Speicher to imprimitive complex reflection groups, in other words to prove the existence of completely positive quasimultiplicative maps from the group algebra of imprimitive complex reflection groups to the set of bounded operators. Some of those maps are then used to define a more general inner product associated to creation and annihilation operators on the full Fock space. Recall that in quantum mechanics, the state of a physical system is represented by a vector in a Hilbert space, and the creation and annihilation operators act on a Fock state by respectively adding and removing a particle in the ascribed quantum state.

scholarly journals A Certain Subclass of Multivalent Analytic Functions with Negative Coefficients for Operator on Hilbert Space

2019 ◽  
pp. 355-364
Author(s):  
Asraa Abdul Jaleel Husien
Keyword(s):  
Hilbert Space ◽  
Analytic Functions ◽  
Convex Set ◽  
Extreme Points ◽  
Complex Hilbert Space ◽  
Coefficient Estimates ◽  
Geometric Properties

In the present work, we introduce and study a certain subclass for multivalent analytic functions with negative coefficients defined on complex Hilbert space. We establish a number of geometric properties, like, coefficient estimates, convex set, extreme points and radii of starlikeness and convexity.

scholarly journals New Family of Multivalent Analytic Functions Defined on Complex Hilbert Space

2020 ◽  
pp. 155-165 ◽  
Author(s):  
Abbas Kareem Wanas ◽  
S. R. Swamy
Keyword(s):  
Hilbert Space ◽  
Analytic Functions ◽  
Convex Set ◽  
Extreme Points ◽  
Complex Hilbert Space ◽  
Coefficient Estimates ◽  
Geometric Properties ◽  
New Class ◽  
New Family

In this article, we define a certain new class of multivalent analytic functions with negative coefficients on complex Hilbert space. We derive a number of important geometric properties, such as, coefficient estimates, radii of starlikeness and convexity, extreme points and convex set.

Products of idempotent linear transformations

1985 ◽  
Vol 100 (1-2) ◽  
pp. 123-138 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bounded Operators ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

SynopsisIn 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

A separable quasidiagonal C*-algebra with a non-quasidiagonal quotient by the compact operators

1991 ◽  
Vol 110 (1) ◽  
pp. 143-145 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Finite Rank ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
C Algebra ◽  
Alternative Characterization ◽  
Image Position ◽  
Algebra Of Operators

A C*-algebra A of operators on a separable Hilbert space H is said to be quasidiagonal if there is an increasing sequence E1, E2, … of finite-rank projections on H tending strongly to the identity and such thatas i → ∞ for T∈A. More generally a C*-algebra is quasidiagonal if there is a faithful *-representation π of A on a separable Hilbert space H such that π(A) is a quasidiagonal algebra of operators. When this is the case, there is a decomposition H = H1 ⊕ H2 ⊕ … where dim Hi < ∞ (i = 1, 2,…) such that each T∈π(A) can be written T = D + K where D= D1 ⊕ D2 ⊕ …, with Di∈L(Hi) (i = 1, 2,…), and K is a compact linear operator on H. As is well known (and readily seen), this is an alternative characterization of quasidiagonality.

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