scholarly journals A remark on a theorem of Caradus

1972 ◽  
Vol 6 (3) ◽  
pp. 355-356
Author(s):  
J.A. Johnson
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Weak Topology ◽  
Approximation Property ◽  
Approximation Problem

It is shown how a result of S.R. Caradus on the approximation problem can be obtained from known theorems.Terms used here are standard (see [1] or [3]).Let X denote a Banach space, S its unit ball in the weak topology, and X* the dual of X. In [1], the following theorem is proved: (I) If X is reflexive and X* (considered as a subspaoe of the continuous scalar-valued functions C(S) in the canonical way) is complemented in C(S), then X has the approximation property.It is our purpose to point out that (I) is a corollary to some known theorems that yield the stronger conclusion (II) below.

scholarly journals Strongly Extreme Points and Approximation Properties

2018 ◽  
Vol 61 (3) ◽  
pp. 449-457
Author(s):  
Trond A. Abrahamsen ◽  
Petr Hájek ◽  
Olav Nygaard ◽  
Stanimir L. Troyanski
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Extreme Point ◽  
Convex Subset ◽  
Approximation Property ◽  
Extreme Points ◽  
Local Approximation ◽  
Approximation Properties ◽  
Compact Approximation ◽  
Approximation Of The Identity

AbstractWe show that if x is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at x, then x is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the suõcient conditions mentioned.

scholarly journals The approximation problem for compact operators

1969 ◽  
Vol 1 (3) ◽  
pp. 397-401 ◽  
Author(s):  
S.R. Caradus
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Finite Rank ◽  
Reflexive Banach Space ◽  
Approximation Problem ◽  
Compact Operators ◽  
Sufficient Condition

The following sufficient condition is obtained for the uniform approximability of compact operators on a reflexive Banach space by operators of finite rank: if S is the unit ball of X and ø: X* → C(S) is the imbedding ø(x*)x = x*(x) then we require ø(X*) to be complemented in C(S).

scholarly journals Ideals of compact operators

2004 ◽  
Vol 77 (1) ◽  
pp. 91-110 ◽  
Author(s):  
Åsvald Lima ◽  
Eve Oja
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Approximation Property ◽  
Closed Subspace ◽  
Reflexive Banach Space ◽  
Compact Operators ◽  
Compact Approximation

AbstractWe give an example of a Banach space X such that K (X, X) is not an ideal in K (X, X**). We prove that if z* is a weak* denting point in the unit ball of Z* and if X is a closed subspace of a Banach space Y, then the set of norm-preserving extensions H B(x* ⊗ z*) ⊆ (Z*, Y)* of a functional x* ⊗ Z* ∈ (Z ⊗ X)* is equal to the set H B(x*) ⊗ {z*}. Using this result, we show that if X is an M-ideal in Y and Z is a reflexive Banach space, then K (Z, X) is an M-ideal in K(Z, Y) whenever K (Z, X) is an ideal in K (Z, Y). We also show that K (Z, X) is an ideal (respectively, an M-ideal) in K (Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, an M-ideal) in Y and X * has the compact approximation property with conjugate operators.

Weak-Star Continuous Analytic Functions

1995 ◽  
Vol 47 (4) ◽  
pp. 673-683 ◽  
Author(s):  
R. M. Aron ◽  
B. J. Cole ◽  
T. W. Gamelin
Keyword(s):  
Unit Ball ◽  
Finite Type ◽  
Analytic Functions ◽  
Approximation Property ◽  
Entire Functions ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Open Unit Ball ◽  
Weakly Continuous ◽  
Weak Star

AbstractLet 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

scholarly journals Homological characterizations of the approximation property for Banach spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 229-239 ◽  
Author(s):  
Yu. V. Selivanov
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Cohomology Group ◽  
Approximation Property ◽  
Three Dimensional ◽  
Nuclear Operators ◽  
Finite Dimensionality

Let E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:(i) biprojectivity of N(E);(ii) biflatness of N(E);(iii) homological finite-dimensionality of N(E);(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).

Extreme Points in H1(R)

1967 ◽  
Vol 19 ◽  
pp. 312-320 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Convex Set ◽  
Complete Solution ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disk ◽  
Open Riemann Surface ◽  
Harmonic Majorant ◽  
Closed Convex Set

Let R be an open Riemann surface. ƒ belongs to H1(R) if ƒ is holomorphic on R and if the subharmonic function |ƒ| has a harmonie majorant on R. Let p be in R and define ||ƒ|| to be the value at p of the least harmonic majorant of |ƒ|. ||ƒ|| is a norm on the linear space H1(R), and with this norm H1(R) is a Banach space (7). The unit ball of H1(R) is the closed convex set of all ƒ in H1(R) with ||ƒ|| ⩽ 1. Problem: What are the extreme points of the unit ball of H1(R)? de Leeuw and Rudin have given a complete solution to this problem where R is the open unit disk (1).

On Additive Operators

1971 ◽  
Vol 23 (3) ◽  
pp. 468-480 ◽  
Author(s):  
N. A. Friedman ◽  
A. E. Tong
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Additive Functional ◽  
Representation Theorems ◽  
Image Position ◽  
Vector Valued ◽  
Kernel Representation

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

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