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 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

Type-based confinement

2005 ◽  
Vol 16 (1) ◽  
pp. 83-128 ◽  
Author(s):  
TIAN ZHAO ◽  
JENS PALSBERG ◽  
JAN VITEK
Keyword(s):  
Data Structures ◽  
Type System ◽  
Object Oriented ◽  
Point Of View ◽  
Generic Extension ◽  
Practical Point ◽  
Related Research ◽  
Language Based Security ◽  
Object Graphs ◽  
Object Oriented Systems

Confinement properties impose a structure on object graphs which can be used to enforce encapsulation properties. From a practical point of view, encapsulation is essential for building secure object-oriented systems as security requires that the interface between trusted and untrusted components of a system be clearly delineated and restricted to the smallest possible set of operations and data structures. This paper investigates the notion of package-level confinement and proposes a type system that enforces this notion for a call-by-value object calculus as well as a generic extension thereof. We give a proof of soundness of this type system, and establish links between this work and related research in language-based security.

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.

EXTENDING THE DOUBLY LINKED FACE LIST FOR THE REPRESENTATION OF 2-PSEUDOMANIFOLDS AND 2-MANIFOLDS WITH BOUNDARIES

2011 ◽  
Vol 21 (04) ◽  
pp. 467-494
Author(s):  
THIAGO R. DOS SANTOS ◽  
HANS-PETER MEINZER ◽  
LENA MAIER-HEIN
Keyword(s):  
Data Structure ◽  
Data Structures ◽  
Point Of View ◽  
Minimal Amount ◽  
Computer Memory ◽  
Memory Usage ◽  
Surgery Simulation ◽  
Practical Applications ◽  
Software Applications ◽  
Behavioral Observations

The Doubly Linked Face List (DLFL) is a data structure for mesh representation that always ensures topological 2-manifold consistency. Furthermore, it uses a minimal amount of computer memory and allows queries to be performed very efficiently. However, the use of the DLFL for the implementation of practical applications is very limited, mainly because of two drawbacks: (1) the DLFL is only able to represent 2-manifold objects; (2) its operators may be ambiguous, modifying the structure in an unexpected way from the user's point of view. In order to overcome these drawbacks, we present the Extended Doubly Linked Face List (XDLFL), which extends the DLFL for the representation of 2-pseudomanifolds and 2-manifolds with boundaries, increasing its applicability for practical software applications. Using these extensions, we also show how to avoid ambiguities in the original DLFL's operators. A new set of intuitive operators for the manipulation of the extensions and for the unambiguous manipulation of the data structure is also presented. The implementation of these extensions is straightforward, since the modifications to the DLFL are trivial and based on behavioral observations of the DLFL's operators. After integrating the extensions to the DLFL, memory usage increases very slightly, while is still smaller than the memory usage of other well-known data structures. Furthermore, queries related to the new extensions, such as whether an edge belongs to a boundary, may be performed very efficiently. The proposed extensions and their operators are very beneficial for applications such as surgery simulation softwares, where the interactions with the models, such as cutting or appending objects to each other, must be performed in an efficient and transparent manner.

scholarly journals GENERIC q-MARKOV SEMIGROUPS AND SPEED OF CONVERGENCE OF q-ALGORITHMS

2006 ◽  
Vol 09 (04) ◽  
pp. 567-594 ◽  
Author(s):  
LUIGI ACCARDI ◽  
FRANCO FAGNOLA ◽  
SKANDER HACHICHA
Keyword(s):  
Pure State ◽  
Point Of View ◽  
Convergence To Equilibrium ◽  
Speed Of Convergence ◽  
Markov Semigroups ◽  
Bounded Operators ◽  
Initial State ◽  
Order Of Magnitude ◽  
The Given ◽  
Generic System

We study a special class of generic quantum Markov semigroups, on the algebra of all bounded operators on a Hilbert space [Formula: see text], arising in the stochastic limit of a generic system interacting with a boson–Fock reservoir. This class depends on an orthonormal basis of [Formula: see text]. We obtain a new estimate for the trace distance of a state from a pure state and use this estimate to prove that, under the action of a semigroup of this class, states with finite support with respect to the given basis converge to equilibrium with a speed which is exponential, but with a polynomial correction which makes the convergence increasingly worse as the dimension of the support increases (Theorem 5.1). We interpret the semigroup as an algorithm, its initial state as input and, following Belavkin and Ohya,10 the dimension of the support of a state as a measure of complexity of the input. With this interpretation, the above results mean that the complexity of the input "slows down" the convergence of the algorithm. Even if the convergence is exponential and the slow down the polynomial, the constants involved may be such that the convergence times become unacceptable from a computational standpoint. This suggests that, in the absence of estimates of the constants involved, distinctions such as "exponentially fast" and "polynomially slow" may become meaningless from a constructive point of view. We also show that, for arbitray states, the speed of convergence to equilibrium is controlled by the rate of decoherence and the rate of purification (i.e. of concentration of the probability on a single pure state). We construct examples showing that the order of magnitude of these two decays can be quite different.

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 On almost-N-continuous functions

1995 ◽  
Vol 59 (1) ◽  
pp. 118-130 ◽  
Author(s):  
Ch. Konstadilaki-Savvopoulou ◽  
I. L. Reilly
Keyword(s):  
Continuous Functions ◽  
Point Of View ◽  
Topological Spaces

AbstractRecently the class of almost-N-continuous functions between topological spaces has been defined. This paper continues the study of such functions, especially from the point of view of changing the topology on the codomain.

scholarly journals Designing for Geometrical Symmetry Exploitation

2006 ◽  
Vol 14 (2) ◽  
pp. 61-80 ◽  
Author(s):  
André Yamba Yamba ◽  
Krister Ålander ◽  
Malin Ljungberg
Keyword(s):  
Data Structures ◽  
Numerical Data ◽  
General Purpose ◽  
Point Of View ◽  
Generalized Fourier Transform ◽  
Data Layout ◽  
Algorithms And Data Structures ◽  
Software Libraries ◽  
Practical Design ◽  
Performance Results

Symmetry-exploiting software based on the generalized Fourier transform (GFT) is presented from a practical design point of view. The algorithms and data structures map closely to the relevant mathematical abstractions, which primarily are based upon representation theory for groups. Particular care has been taken in the design of the data layout of the performance-sensitive numerical data structures. The use of a vanilla strategy is advocated for the design of flexible mathematical software libraries: An efficient general-purpose routine should be supplied, to obtain a practical and useful system, while the possibility to extend the library and replace the default routine with a special-purpose – even more optimized – routine should be supported. Compared with a direct approach, the performance results show the superiority of the GFT-based approach for so-called dense equivariant systems. The GFT application is found to be well suited for parallelism.

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