The modulus of a weakly compact operator

1987 ◽  
Vol 195 (4) ◽  
pp. 473-480 ◽  
Author(s):  
G. Groenewegen ◽  
A. van Rooij
Keyword(s):  
Compact Operator ◽  
Weakly Compact ◽  
Weakly Compact Operator

Properties of the Product of Two Derivations of a C*-Algebra

1989 ◽  
Vol 32 (4) ◽  
pp. 490-497 ◽  
Author(s):  
Martin Mathieu
Keyword(s):  
Compact Operator ◽  
Weakly Compact ◽  
C Algebra ◽  
Weakly Compact Operator

AbstractLet δ1,δ2 be two derivations of a C*-algebra. We characterize when δ1δ2 is a derivation, a compact, or a weakly compact operator

Weak Sequential Completeness of 𝑲(X,Y)

2013 ◽  
Vol 56 (3) ◽  
pp. 503-509 ◽  
Author(s):  
Qingying Bu
Keyword(s):  
Banach Spaces ◽  
Compact Operator ◽  
Approximation Property ◽  
Compact Operators ◽  
Weakly Compact ◽  
Sequential Completeness ◽  
Weakly Compact Operator ◽  
Compact Approximation

AbstractFor Banach spaces X and Y, we show that if X* and Y are weakly sequentially complete and every weakly compact operator from X to Y is compact, then the space of all compact operators from X to Y is weakly sequentially complete. The converse is also true if, in addition, either X* or Y has the bounded compact approximation property.

scholarly journals Compact actions on C*-algebras

1980 ◽  
Vol 21 (2) ◽  
pp. 143-149
Author(s):  
Charles A. Akemann ◽  
Steve Wright
Keyword(s):  
Banach Algebra ◽  
Compact Operator ◽  
Weakly Compact ◽  
C Algebra ◽  
C Algebras ◽  
Compact Perturbations

In Section 33 of [2], Bonsall and Duncan define an elementtof a Banach algebratoact compactlyonif the mapa→tatis a compact operator on. In this paper, the arguments and technique of [1] are used to study this question for C*-algebras (see also [10]). We determine the elementsbof a C*-algebrafor which the mapsa→ba,a→ab,a→ab+ba,a→babare compact (respectively weakly compact), determine the C*-algebras which are compact in the sense of Definition 9, of [2, p. 177] and give a characterization of the C*-automorphisms ofwhich are weakly compact perturbations of the identity.

scholarly journals Lipschitz integral operators represented by vector measures

2021 ◽  
Vol 22 (2) ◽  
pp. 367
Author(s):  
Elhadj Dahia ◽  
Khaled Hamidi
Keyword(s):  
Hausdorff Space ◽  
Integral Operators ◽  
Operator Ideals ◽  
Weakly Compact ◽  
Vector Measures ◽  
Factorization Theory ◽  
Weakly Compact Operator ◽  
Lipschitz Operator ◽  
The Relationship ◽  
Lipschitz Operators

<p>In this paper we introduce the concept of Lipschitz Pietsch-p-integral <br />mappings, (1≤p≤∞), between a metric space and a Banach space. We represent these mappings by an integral with respect to a vector<br />measure defined on a suitable compact Hausdorff space, obtaining in this way a rich factorization theory through the classical Banach spaces C(K), L_p(μ,K) and L_∞(μ,K). Also we show that this type of operators fits in the theory of composition Banach Lipschitz operator ideals. For p=∞, we characterize the Lipschitz Pietsch-∞-integral mappings by a factorization schema through a weakly compact operator. Finally, the relationship between these mappings and some well known Lipschitz operators is given.</p>

scholarly journals Compact actions on C*-algebras

1980 ◽  
Vol 21 (1) ◽  
pp. 143-149 ◽  
Author(s):  
Charles A. Akemann ◽  
Steve Wright
Keyword(s):  
Banach Algebra ◽  
Compact Operator ◽  
Weakly Compact ◽  
C Algebra ◽  
C Algebras ◽  
Compact Perturbations

In Section 33 of [2], Bonsall and Duncan define an elementtof a Banach algebratoact compactlyonif the mapa→tatis a compact operator on. In this paper, the arguments and technique of [1] are used to study this question for C*-algebras (see also [10]). We determine the elementsbof a C*-algebrafor which the mapsa→ba,a→ab,a→ab+ba,a→babare compact (respectively weakly compact), determine the C*-algebras which are compact in the sense of Definition 9, of [2, p. 177] and give a characterization of the *-automorphisms ofwhich are weakly compact perturbations of the identity.

Weakly Compact, Operator-Valued Derivations of Type I Von Neumann Algebras

1984 ◽  
Vol 36 (3) ◽  
pp. 436-457
Author(s):  
Steve Wright
Keyword(s):  
Von Neumann Algebras ◽  
Linear Operators ◽  
Type I ◽  
Weakly Compact ◽  
Bounded Linear ◽  
Von Neumann ◽  
Weakly Compact Operator ◽  
Weak Operator ◽  
Image Position ◽  
Compact Extension

In [18], the author initiated an investigation of compact, Banach-module-valued derivations of C*-algebras. In collaboration with C. A. Akemann [3] and S.-K. Tsui [16], he determined the structure of all compact and weakly compact, A-valued derivations of a C*-algebra A, and of all compact, B(H)-valued derivations of a C*-subalgebra of B(H), the algebra of bounded linear operators on a Hilbert space H. In this paper, we begin the study of weakly compact, B(H)-valued derivations of C*-subalgebras of B(H).Let R be a C*-subalgebra of B(H), δ:R → B(H) a weakly compact derivation, i.e., a weakly compact linear map which hasSince δ has a unique weakly compact extension to a derivation of the closure of R in the weak operator topology (WOT) on B(H) (consult the proof of Theorem 3.1 of [16]), we may assume with no loss of generality that R is a von Neumann subalgebra of B(H). In this paper, we determine in Lemma 4.1 and Theorems 4.3 and 4.10 the structure of δ when R is type I, using I. E. Segal's multiplicity theory [14] for type I von Neumann algebras and results of E. Christensen [6], [7] on B(H)-valued derivations of von Neumann algebras.

scholarly journals Uniformly factoring weakly compact operators and parametrised dualisation

2021 ◽  
Vol 9 ◽  
Author(s):  
L. Antunes ◽  
K. Beanland ◽  
B. M. Braga
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Borel Subset ◽  
Compact Operators ◽  
Schauder Basis ◽  
Bounded Operators ◽  
Weakly Compact ◽  
Weakly Compact Operator ◽  
Borel Map ◽  
Weakly Compact Operators

Abstract This article deals with the problem of when, given a collection $\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in $\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to $L_1[0,1]$ factors through Z. We also prove the following descriptive set theoretical result: Let $\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for $A \in \mathcal {B}$ , the assignment $A \to A^*$ can be realised by a Borel map $\mathcal {B}\to \mathcal {L}$ .

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