Characterizing injective operator space $V$ for which $I_{11}(V) \cong B(H)$

2013 ◽  
Vol 82 (1) ◽  
pp. 21-30 ◽  
Author(s):  
ALI REZA MEDGHALCHI ◽  
HAMED NIKPEY
Keyword(s):  
Operator Space ◽  
Injective Operator

A New Approach to Operator Spaces

1991 ◽  
Vol 34 (3) ◽  
pp. 329-337 ◽  
Author(s):  
Edward G. Effros ◽  
Zhong-Jin Ruan
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Operator Space ◽  
Operator Spaces ◽  
New Approach ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Space Concepts ◽  
Injective Operator

AbstractThe authors previously observed that the space of completely bounded maps between two operator spaces can be realized as an operator space. In particular, with the appropriate matricial norms the dual of an operator space V is completely isometric to a linear space of operators. This approach to duality enables one to formulate new analogues of Banach space concepts and results. In particular, there is an operator space version ⊗μ of the Banach space projective tensor product , which satisfies the expected functorial properties. As is the case for Banach spaces, given an operator space V, the functor W |—> V ⊗μ W preserves inclusions if and only if is an injective operator space.

Injective matricial Hilbert spaces

1991 ◽  
Vol 110 (1) ◽  
pp. 183-190 ◽  
Author(s):  
A. Guyan Robertson
Keyword(s):  
Hilbert Spaces ◽  
Von Neumann Algebras ◽  
Linear Span ◽  
Bounded Operator ◽  
Operator Space ◽  
Operator Spaces ◽  
Von Neumann ◽  
Type Iv ◽  
Injective Operator ◽  
Cartan Factor

Injective matricial operator spaces have been classified up to Banach space isomorphism in [20]. The result is that every such space is isomorphic to l∞, l2, B(l2), or a direct sum of such spaces. A more natural project, given the matricial nature of the definitions involved, would be the classification of such spaces up to completely bounded isomorphism. This was done for injective von Neumann algebras in [6] and for injective operator systems (i.e. unital injective operator spaces) in [19]. It turns out that the spaces l∞ and B(l2) are in a natural way uniquely characterized up to completely bounded isomorphism. However, as shown in [20], a problem arises in the case of l2. For there are two injective operator spaces which are each isometrically isomorphic to l2 but not completely boundedly isomorphic to each other. We shall resolve this problem by showing that these are the only two possibilities, in the sense that any injective operator space which is isometric to l2 is completely isometric to one of them. (See Corollary 3 below.) The Hilbert spaces in von Neumann algebras investigated in [17], [13] turn out to be injective matricial operator spaces and are therefore completely isometric to one of our two examples. Another Hilbert space in B(l2) which has been much studied in operator theory, complex analysis and physics is the Cartan factor of type IV [10]. This is the complex linear span of a spin system and generates the Fermion C*-algebra ([3], §5·2). We show that a Cartan factor of type IV is not even completely boundedly isomorphic to an injective matricial operator space. One curious property of all the aforementioned Hilbert spaces is that every bounded operator on them is actually completely bounded, a fact that is crucial in our proofs.

scholarly journals Operator Space Theory: A Natural Framework for Bell Inequalities

2010 ◽  
Vol 104 (17) ◽  
Author(s):  
M. Junge ◽  
C. Palazuelos ◽  
D. Pérez-García ◽  
I. Villanueva ◽  
M. M. Wolf
Keyword(s):  
Bell Inequalities ◽  
Operator Space ◽  
Space Theory

scholarly journals Real time evolution at finite temperatures with operator space matrix product states

2014 ◽  
Vol 16 (7) ◽  
pp. 073007 ◽  
Author(s):  
Iztok Pižorn ◽  
Viktor Eisler ◽  
Sabine Andergassen ◽  
Matthias Troyer
Keyword(s):  
Real Time ◽  
Time Evolution ◽  
Operator Space ◽  
Matrix Product ◽  
Matrix Product States ◽  
Space Matrix

scholarly journals Capacity bounds via operator space methods

2018 ◽  
Vol 59 (12) ◽  
pp. 122202 ◽  
Author(s):  
Li Gao ◽  
Marius Junge ◽  
Nicholas LaRacuente
Keyword(s):  
Operator Space ◽  
Capacity Bounds

scholarly journals A counterexample to the density of the space generated by the composition operators

1977 ◽  
Vol 16 (1) ◽  
pp. 79-81
Author(s):  
Ronald Beattie
Keyword(s):  
Vector Space ◽  
Dual Space ◽  
Composition Operators ◽  
Operator Space ◽  
Natural Analogue ◽  
Convergence Space ◽  
Continuous Convergence ◽  
Continuous Mappings ◽  
Convergence Structure

It is known that, for an arbitrary convergence space X, the vector space generated by X is dense in LcCc (X) where both C(X) and its dual space carry the continuous convergence structure. In this note, a natural analogue formulated for the operator space L(Cc(X), Cc(X)) is considered, namely: is the vector space generated by the composition operators associated to the continuous mappings in C(X, X) dense in Lc (Cc (X), Cc (X)) ? This question is answered in the negative by a counterexample.

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