Tensor Products of Operator Spaces II

1991 ◽  
Vol 44 (1) ◽  
pp. 75-90 ◽  
Author(s):  
David P. Blecher
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Elementary Theory ◽  
Tensor Products ◽  
Space Theory ◽  
Operator Spaces ◽  
Tensor Norm ◽  
Tensor Norms

AbstractTogether with Vern Paulsen we were able to show that the elementary theory of tensor norms of Banach spaces carries over to operator spaces. We suggested that the Grothendieck tensor norm program, which was of course enormously important in the development of Banach space theory, be carried out for operator spaces. Some of this has been done by the authors mentioned above, and by Effros and Ruan. We give alternative developments of some of this work, and otherwise continue the tensor norm program.

scholarly journals Approximation properties of tensor norms and operator ideals for Banach spaces

2020 ◽  
Vol 18 (1) ◽  
pp. 1698-1708
Author(s):  
Ju Myung Kim
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Approximation Property ◽  
Approximation Properties ◽  
Operator Ideals ◽  
Finitely Generated ◽  
Tensor Norm ◽  
Tensor Norms

Abstract For a finitely generated tensor norm α \alpha , we investigate the α \alpha -approximation property ( α \alpha -AP) and the bounded α \alpha -approximation property (bounded α \alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ \lambda -bounded α p , q {\alpha }_{p,q} -AP ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ) (1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the λ \lambda -bounded g p {g}_{p} -AP. As a consequence, it follows that if a Banach space X has the λ \lambda -bounded g p {g}_{p} -AP, then X has the λ \lambda -bounded w p {w}_{p} -AP.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Auerbach's theorem and tensor products of Banach spaces

1962 ◽  
Vol 58 (3) ◽  
pp. 476-480 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Tensor Products ◽  
Finite Dimensional ◽  
Orthogonal Systems

In his well-known treatise, Banach mentions that there exist complete normed bi-orthogonal systems in any finite-dimensional (real) Banach space—a result he attributes to H. Auerbach ((1), p. 238). The purpose of this note is to present a proof of this result (valid for complex as well as real Banach spaces), and to give some applications to the theory of tensor products of Banach spaces.

scholarly journals On Banach spaces with Mazur's property

1991 ◽  
Vol 33 (1) ◽  
pp. 51-54 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functional ◽  
The Stability

A Banach space E is said to have Mazur's property if every weak* sequentially continuous functional in E” is weak* continuous, i.e. belongs to E. Such spaces were investigated in [5] and [9] where they were called d-complete and μB-spaces respectively. The class of Banach spaces with Mazur's property includes the WCG spaces and, more generally, the Banach spaces with weak* angelic dual balls [4, p. 564]. Also, it is easy to see that Mazur's property is inherited by closed subspaces. The main goal of this paper is to present generalizations of some results of [5] concerning the stability of Mazur's property with respect to forming some tensor products of Banach spaces. In particular, we show in Sections 2 and 3 that the spaces E ⊗εF and Lp(μ, E) inherit Mazur's property from E andF under some conditions. In Section 4, we will also show the stability of Mazur's property under the formation of Schauder decompositions and some unconditional sums of Banach spaces.

scholarly journals A remark on the tensor product of two maximal operator spaces

1997 ◽  
Vol 40 (2) ◽  
pp. 375-381
Author(s):  
Christian Le Merdy
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Maximal Operator ◽  
Space Structure ◽  
Operator Space ◽  
Operator Spaces ◽  
Operator Space Structure

Given a Banach space E, let us denote by Max(E) the largest operator space structure on E. Recently Paulsen-Pisier and, independently, Junge proved that for any Banach spaces E, F, isomorphically where and respectively denote the Haagerup tensor product and the spatial tensor product of operator spaces. In this paper we show that, in general, this equality does not hold completely isomorphically.

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.

The Reconstruction Property in Banach Spaces and a Perturbation Theorem

2008 ◽  
Vol 51 (3) ◽  
pp. 348-358 ◽  
Author(s):  
Peter G. Casazza ◽  
Ole Christensen
Keyword(s):  
Banach Space ◽  
Perturbation Theory ◽  
Banach Spaces ◽  
Space Theory ◽  
Perturbation Theorem ◽  
General Perturbation ◽  
The Given

AbstractPerturbation theory is a fundamental tool in Banach space theory. However, the applications of the classical results are limited by the fact that they force the perturbed sequence to be equivalent to the given sequence. We will develop amore general perturbation theory that does not force equivalence of the sequences.

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