On the Local Lifting Properties of Operator Spaces

2009 ◽  
Vol 61 (6) ◽  
pp. 1262-1278
Author(s):  
Z. Dong
Keyword(s):  
Operator Space ◽  
Ternary Ring ◽  
Operator Spaces ◽  
Local Reflexivity

Abstract In this paper, we mainly study operator spaces which have the locally lifting property (LLP). The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the LLP. Several homological characterizations of the LLP and weak expectation property are given. We also prove that for any operator space V, V** has the LLP if and only if V has the LLP and V* is exact.

scholarly journals The {OLLP} and $\scr T$-local reflexivity of operator spaces

2007 ◽  
Vol 51 (4) ◽  
pp. 1103-1122 ◽  
Author(s):  
Z. Dong
Keyword(s):  
Operator Spaces ◽  
Local Reflexivity

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 $B$-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 The Operator Space Projective Tensor Product: Embedding into the Second Dual and Ideal Structure

2013 ◽  
Vol 57 (2) ◽  
pp. 505-519 ◽  
Author(s):  
Ranjana Jain ◽  
Ajay Kumar
Keyword(s):  
Tensor Product ◽  
Bilinear Form ◽  
Operator Space ◽  
Ideal Structure ◽  
Operator Spaces ◽  
Canonical Embedding ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Second Dual ◽  
Continuous Inverse

AbstractWe prove that, for operator spaces V and W, the operator space V** ⊗hW** can be completely isometrically embedded into (V ⊗hW)**, ⊗h being the Haagerup tensor product. We also show that, for exact operator spaces V and W, a jointly completely bounded bilinear form on V × W can be extended uniquely to a separately w*-continuous jointly completely bounded bilinear form on V× W**. This paves the way to obtaining a canonical embedding of into with a continuous inverse, where is the operator space projective tensor product. Further, for C*-algebras A and B, we study the (closed) ideal structure of which, in particular, determines the lattice of closed ideals of completely.

scholarly journals Effective representations of the space of linear bounded operators

2003 ◽  
Vol 4 (1) ◽  
pp. 115 ◽  
Author(s):  
Vasco Brattka
Keyword(s):  
Data Structures ◽  
Point Of View ◽  
Operator Norm ◽  
Operator Space ◽  
Topological Spaces ◽  
Operator Spaces ◽  
Bounded Operators ◽  
Norm Topology ◽  
Finite Dimensionality ◽  
Surjective Mapping

<p>Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. We discuss a number of representations of the space of linear bounded operators on a Banach space. Since the operator norm topology of the operator space is nonseparable in typical cases, the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, such as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn.</p>

On the Duality of Operator Spaces

1995 ◽  
Vol 38 (3) ◽  
pp. 334-346 ◽  
Author(s):  
Christian Le Merdy
Keyword(s):  
Banach Space ◽  
Von Neumann Algebras ◽  
Related Result ◽  
Space Structure ◽  
Operator Space ◽  
Operator Spaces ◽  
Bounded Operators ◽  
Von Neumann ◽  
Operator Space Structure ◽  
Dual Banach Space

AbstractWe prove that given an operator space structure on a dual Banach space Y*, it is not necessarily the dual one of some operator space structure on Y. This allows us to show that Sakai's theorem providing the identification between C*-algebras having a predual and von Neumann algebras does not extend to the category of operator spaces. We also include a related result about completely bounded operators from B(ℓ2)* into the operator Hilbert space OH.

scholarly journals The “maximal” tensor product of operator spaces

1999 ◽  
Vol 42 (2) ◽  
pp. 267-284 ◽  
Author(s):  
Timur Oikhberg ◽  
Gilles Pisier
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
Tensor Products ◽  
Operator Space ◽  
Operator Spaces ◽  
Two Factors

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

On the analogues of integral mappings and local reflexivity for operator spaces

1997 ◽  
Vol 46 (4) ◽  
pp. 0-0 ◽  
Author(s):  
Edward G. Effros ◽  
Zhong-Jin Ruan
Keyword(s):  
Operator Spaces ◽  
Local Reflexivity

scholarly journals METRIC AND TOPOLOGICAL FREEDOM FOR SEQUENTIAL OPERATOR SPACES

2017 ◽  
Vol 20 (10) ◽  
pp. 55-67
Author(s):  
N.T. Nemesh ◽  
S.M. Shteiner
Keyword(s):  
Duality Theory ◽  
Operator Space ◽  
Space Theory ◽  
Operator Spaces ◽  
Normed Spaces ◽  
Full Characterization ◽  
Definition Of ◽  
Phd Thesis ◽  
Space Homology

In 2002 Anselm Lambert in his PhD thesis [1] introduced the definition of sequential operator space and managed to establish a considerable amount of analogs of corresponding results in operator space theory. Informally speaking, the category of sequential operator spaces is situated ”between” the categories of normed and operator spaces. This article aims to describe free and cofree objects for different versions of sequential operator space homology. First of all, we will show that duality theory in above-mentioned category is in many respects analogous to that in the category of normed spaces. Then, based on those results, we will give a full characterization of both metric and topological free and cofree objects.

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