scholarly journals A Completely Bounded Noncommutative Choquet Boundary for Operator Spaces

2017 ◽  
Vol 2019 (22) ◽  
pp. 6819-6886 ◽  
Author(s):  
Raphaël Clouâtre ◽  
Christopher Ramsey
Keyword(s):  
Structural Properties ◽  
Extension Property ◽  
Operator Space ◽  
Operator Spaces ◽  
Unique Extension ◽  
Bounded Linear ◽  
C Algebra ◽  
Choquet Boundary ◽  
Linear Maps ◽  
Contractive Maps

Abstract We develop a completely bounded counterpart to the noncommutative Choquet boundary of an operator space. We show how the class of completely bounded linear maps is too large to accommodate our purposes. To overcome this obstacle, we isolate the subset of completely bounded linear maps admitting a dilation of the same norm that is multiplicative on the associated C*-algebra. We view such maps as analogs of the familiar unital completely contractive maps, and we exhibit many of their structural properties. Of particular interest to us are those maps that are extremal with respect to a natural dilation order. We establish the existence of extremals and show that they have a certain unique extension property. In particular, they give rise to *-homomorphisms that we use to associate to any representation of an operator space an entire scale of C*-envelopes. We conjecture that these C*-envelopes are all *-isomorphic and verify this in some important cases.

scholarly journals A norm property for spaces of completely bounded maps between C*-algebras

1987 ◽  
Vol 29 (2) ◽  
pp. 181-184 ◽  
Author(s):  
Tadasi Huruya
Keyword(s):  
Banach Space ◽  
Complex Matrices ◽  
Bounded Linear ◽  
Linear Map ◽  
C Algebra ◽  
C Algebras ◽  
Linear Maps ◽  
Norm Property

Let Mn be the C*-algebra of n × n complex matrices. If A is a C*-algebra, let Mn(A) denote the C*-algebra of n × nmatrices a = [aij] with entries in A. For a linear map between C*-algebras, we define the multiplicity map by A linear map Ø is said to be completely bounded if Let B(A, B), CB(A, B) denote the Banach space of bounded linear maps, the set of completely bounded maps from A to B, respectively.

EXISTENCE OF FELLER COCYCLES ON A C*-ALGEBRA

2001 ◽  
Vol 33 (5) ◽  
pp. 613-621 ◽  
Author(s):  
J. MARTIN LINDSAY ◽  
STEPHEN J. WILLS
Keyword(s):  
Necessary Conditions ◽  
Invariance Condition ◽  
Quantum Stochastic Differential Equation ◽  
Completely Positive ◽  
Positive Contraction ◽  
Bounded Linear ◽  
C Algebra ◽  
Linear Maps ◽  
The Matrix ◽  
Dimension Space

The quantum stochastic differential equation dkt = kt ∘ θαβdΛβα(t) is considered on a unital C*-algebra, with separable noise dimension space. Necessary conditions on the matrix of bounded linear maps θ for the existence of a completely positive contractive solution are shown to be sufficient. It is known that for completely positive contraction processes, k satisfies such an equation if and only if k is a regular Markovian cocycle. ‘Feller’ refers to an invariance condition analogous to probabilistic terminology if the algebra is thought of as a non-commutative topological space.

scholarly journals Concrete Realizations of Quotients of Operator Spaces

2014 ◽  
Vol 114 (2) ◽  
pp. 205
Author(s):  
Marc A. Rieffel
Keyword(s):  
Unitary Operator ◽  
Closed Subspace ◽  
Operator Space ◽  
Operator Spaces ◽  
C Algebra ◽  
Contractive Map ◽  
Special Cases ◽  
Abstract Operator ◽  
Concrete Realization ◽  
Algebra Of Operators

Let $\mathscr{B}$ be a unital C*-subalgebra of a unital C*-algebra $\mathscr{A}$, so that $\mathscr{A}/\mathscr{B}$ is an abstract operator space. We show how to realize $\mathscr{A}/\mathscr{B}$ as a concrete operator space by means of a completely contractive map from $\mathscr{A}$ into the algebra of operators on a Hilbert space, of the form $A \mapsto [Z, A]$ where $Z$ is a Hermitian unitary operator. We do not use Ruan's theorem concerning concrete realization of abstract operator spaces. Along the way we obtain corresponding results for abstract operator spaces of the form $\mathscr{A}/\mathscr{V}$ where $\mathscr{V}$ is a closed subspace of $\mathscr{A}$, and then for the more special cases in which $\mathscr{V}$ is a $*$-subspace or an operator system.

scholarly journals On some “stability” properties of the full C*-algebra associated to the free group F∞

1998 ◽  
Vol 41 (1) ◽  
pp. 93-116 ◽  
Author(s):  
Asma Harcharras
Keyword(s):  
Tensor Product ◽  
Free Group ◽  
Extension Property ◽  
The Other ◽  
Operator Space ◽  
Complex Interpolation ◽  
Stability Properties ◽  
C Algebra ◽  
Other Hand

Let C*(F∞) be the full C*-algebra associated to the free group of countably many generators and SnC*(F∞) be the class of all n-dimensional operator subspaces of C*(F∞). In this paper, we study some stability properties of SnC*(F∞). More precisely, we will prove that for any E0, E1 in SnC*(F∞), the Haagerup tensor product E0⊗hE1 and the operator space obtained by complex interpolation Eθ are (1 + ∈)-contained in C*(F∞) for arbitrary ∈>0. On the other hand, we will show an extension property for WEPC*-algebras.

scholarly journals The spectrally bounded linear maps on operator algebras

2002 ◽  
Vol 150 (3) ◽  
pp. 261-271 ◽  
Author(s):  
Jianlian Cui ◽  
Jinchuan Hou
Keyword(s):  
Operator Algebras ◽  
Bounded Linear ◽  
Linear Maps ◽  
Spectrally Bounded

Preduals and Nuclear Operators Associated with Bounded, p-Convex, p-Concave and Positive p-Summing Operators

2007 ◽  
Vol 59 (3) ◽  
pp. 614-637 ◽  
Author(s):  
C. C. A. Labuschagne
Keyword(s):  
Banach Spaces ◽  
Positive Operators ◽  
Banach Lattices ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Nuclear Operators ◽  
Linear Maps

AbstractWe use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of p-convex, p-concave and positive p-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

scholarly journals Operators with a Single Spectrum

1966 ◽  
Vol 15 (1) ◽  
pp. 11-18 ◽  
Author(s):  
T. T. West
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Linear Operators ◽  
Complex Field ◽  
Unique Extension ◽  
Single Spectrum ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Resolvent Set ◽  
Image Position

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to beand the spectrum of T is the complement of ρ(T) in Z.

scholarly journals $B$-CONVEX OPERATOR SPACES

2003 ◽  
Vol 46 (3) ◽  
pp. 649-668 ◽  
Author(s):  
Javier Parcet
Keyword(s):  
Banach Spaces ◽  
A Priori ◽  
Operator Space ◽  
Subject Classification ◽  
Operator Spaces ◽  
Secondary 42C15 ◽  
Convex Operator ◽  
Primary 46L07

AbstractThe notion of $B$-convexity for operator spaces, which a priori depends on a set of parameters indexed by $\sSi$, is defined. Some of the classical characterizations of this geometric notion for Banach spaces are studied in this new context. For instance, an operator space is $B_{\sSi}$-convex if and only if it has $\sSi$-subtype. The class of uniformly non-$\mathcal{L}^1(\sSi)$ operator spaces, which is also the class of $B_{\sSi}$-convex operator spaces, is introduced. Moreover, an operator space having non-trivial $\sSi$-type is $B_{\sSi}$-convex. However, the converse is false. The row and column operator spaces are nice counterexamples of this fact, since both are Hilbertian. In particular, this result shows that a version of the Maurey–Pisier Theorem does not hold in our context. Some other examples of Hilbertian operator spaces will be considered. In the last part of this paper, the independence of $B_{\sSi}$-convexity with respect to $\sSi$ is studied. This provides some interesting problems, which will be posed.AMS 2000 Mathematics subject classification: Primary 46L07. Secondary 42C15

scholarly journals ORDER EMBEDDING OF A MATRIX ORDERED SPACE

2011 ◽  
Vol 84 (1) ◽  
pp. 10-18 ◽  
Author(s):  
ANIL K. KARN
Keyword(s):  
Adjoint Operator ◽  
Operator Space ◽  
Operator Spaces ◽  
Ordered Space

AbstractWe characterize certain properties in a matrix ordered space in order to embed it in a C*-algebra. Let such spaces be called C*-ordered operator spaces. We show that for every self-adjoint operator space there exists a matrix order (on it) to make it a C*-ordered operator space. However, the operator space dual of a (nontrivial) C*-ordered operator space cannot be embedded in any C*-algebra.

scholarly journals Propertiesandfor Bounded Linear Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
M. H. M. Rashid
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Extension Property ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Single Valued Extension Property ◽  
Bounded Linear ◽  
Weyl Type Theorems

We shall consider properties which are related to Weyl type theorem for bounded linear operators , defined on a complex Banach space . These properties, that we callproperty, means that the set of all poles of the resolvent of of finite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 and we callproperty, means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0. Properties and are related to a strong variants of classical Weyl’s theorem, the so-called property and property We shall characterize properties and in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties and in the frame of polaroid type operators.

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