A note on elementary equivalence of C(K) spaces

1987 ◽  
Vol 52 (2) ◽  
pp. 368-373 ◽  
Author(s):  
S. Heinrich ◽  
C. Ward Henson ◽  
L. C. Moore
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Supremum Norm ◽  
Closed Unit Ball ◽  
Elementary Equivalence ◽  
Bounded Number ◽  
Totally Disconnected

In this paper we give a closer analysis of the elementary properties of the Banach spaces C(K), where K is a totally disconnected, compact Hausdorff space, in terms of the Boolean algebra B(K) of clopen subsets of K. In particular we sharpen a result in [4] by showing that if B(K1) and B(K2) satisfy the same sentences with ≤ n alternations of quantifiers, then the same is true of C(K1) and C(K2). As a consequence we show that for each n there exist C(K) spaces which are elementarily equivalent for sentences with ≤ n quantifier alternations, but which are not elementary equivalent in the full sense. Thus the elementary properties of Banach spaces cannot be determined by looking at sentences with a bounded number of quantifier alternations.The notion of elementary equivalence for Banach spaces which is studied here was introduced by the second author [4] and is expressed using the language of positive bounded formulas in a first-order language for Banach spaces. As was shown in [4], two Banach spaces are elementarily equivalent in this sense if and only if they have isometrically isomorphic Banach space ultrapowers (or, equivalently, isometrically isomorphic nonstandard hulls.)We consider Banach spaces over the field of real numbers. If X is such a space, Bx will denote the closed unit ball of X, Bx = {x ϵ X∣ ∣∣x∣∣ ≤ 1}. Given a compact Hausdorff space K, we let C(K) denote the Banach space of all continuous real-valued functions on K, under the supremum norm. We will especially be concerned with such spaces when K is a totally disconnected compact Hausdorff space. In that case B(K) will denote the Boolean algebra of all clopen subsets of K. We adopt the standard notation from model theory and Banach space theory.

scholarly journals On Banach spaces of vector valued continuous functions

1983 ◽  
Vol 28 (2) ◽  
pp. 175-186 ◽  
Author(s):  
Pilar Cembranos
Keyword(s):  
Banach Space ◽  
Large Class ◽  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Vector Valued

Let K be a compact Hausdorff space and let E be a Banach space. We denote by C(K, E) the Banach space of all E-valued continuous functions defined on K, endowed with the supremum norm.Recently, Talagrand [Israel J. Math.44 (1983), 317–321] constructed a Banach space E having the Dunford-Pettis property such that C([0, 1], E) fails to have the Dunford-Pettis property. So he answered negatively a question which was posed some years ago.We prove in this paper that for a large class of compacts K (the scattered compacts), C(K, E) has either the Dunford-Pettis property, or the reciprocal Dunford-Pettis property, or the Dieudonné property, or property V if and only if E has the same property.Also some properties of the operators defined on C(K, E) are studied.

scholarly journals STRONGLY BOUNDED REPRESENTING MEASURES AND CONVERGENCE THEOREMS

2010 ◽  
Vol 52 (3) ◽  
pp. 435-445 ◽  
Author(s):  
IOANA GHENCIU ◽  
PAUL LEWIS
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Convergence Theorems

AbstractLet K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

scholarly journals Some extremal properties of section spaces of Banach bundles and their duals

2002 ◽  
Vol 29 (10) ◽  
pp. 563-572 ◽  
Author(s):  
D. A. Robbins
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Real Banach Space ◽  
Extremal Properties

WhenXis a compact Hausdorff space andEis a real Banach space there is a considerable literature on extremal properties of the spaceC(X,E)of continuousE-valued functions onX. What happens if the Banach spaces in which the functions onXtake their values vary overX? In this paper, we obtain some extremal results on the section spaceΓ(π)and its dualΓ(π)*of a real Banach bundleπ:ℰ→X(with possibly varying fibers), and point out the difficulties in arriving at totally satisfactory results.

scholarly journals Integral operators on the section space of a Banach bundle

1993 ◽  
Vol 16 (3) ◽  
pp. 449-458
Author(s):  
J. W. kitchen ◽  
D. A. Robbins
Keyword(s):  
Banach Space ◽  
Recent Result ◽  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Integral Operators

Letπ:E→Xandρ:F→Xbe bundles of Banach spaces, whereXis a compact Hausdorff space, and letVbe a Banach space. LetΓ(π)denote the space of sections of the bundleπ. We obtain two representations of integral operatorsT:Γ(π)→Vin terms of measures. The first generalizes a recent result of P. Saab, the second generalizes a theorem of Grothendieck. We also study integral operatorsT:Γ(π)→Γ(ρ)which areC(X)-linear.

Isometries of self-adjoint complex function spaces

1989 ◽  
Vol 105 (1) ◽  
pp. 133-138 ◽  
Author(s):  
A. J. Ellis
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Complex Function ◽  
Supremum Norm ◽  
Affine Functions ◽  
Continuous Complex ◽  
Image Position ◽  
Complex Valued ◽  
Uniformly Closed

By a complex function space A we will mean a uniformly closed linear space of continuous complex-valued functions on a compact Hausdorff space X, such that A contains constants and separates the points of X. We denote by S the state-spaceendowed with the w*-topology. If A is self-adjoint then it is well known (cf. [1]) that A is naturally isometrically isomorphic to , and re A is naturally isometrically isomorphic to A(S), where (respectively A(S)) denotes the Banach space of all complex-valued (respectively real-valued) continuous affine functions on S with the supremum norm.

Elementary equivalence of Cσ(K) spaces for totally disconnected, compact Hausdorff K

1986 ◽  
Vol 51 (1) ◽  
pp. 135-146
Author(s):  
S. Heinrich ◽  
C. Ward Henson ◽  
L. C. Moore
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Unique Fixed Point ◽  
Elementary Theory ◽  
Elementary Equivalence ◽  
Isolated Points ◽  
Isolated Point ◽  
Dense Set ◽  
Totally Disconnected ◽  
Elementary Theories

This paper is a continuation of the authors' paper [7]; in particular, we give a sharper and more useful criterion for the approximate elementary equivalence of Cσ(K) spaces, where K is a totally disconnected compact Hausdorff space. (See Theorem 2 below.) As an application, we obtain a complete description of the Banach spaces X which are approximately equivalent to c0. Namely, X ≡ Ac0 iff X = Cσ(K) where K is a totally disconnected compact Hausdorff space which has a dense set of isolated points and σ is an involutory homeomorphism on K which has a unique fixed point t, and t is not an isolated point. (See Theorem 7.)These results are derived from an analysis which we give of the elementary theories of structures (B, σ), where B is a Boolean algebra and σ is an involutory automorphism of B which leaves at most one nontrivial ultrafilter invariant. Let U(σ) denote this ultrafilter, if it exists; let U(σ) = B in case σ leaves no nontrivial ultrafilter invariant. We show that the elementary theory of (B, σ) is completely determined by the theory of (B, U(σ)) (and conversely, because U(σ) is definable in (B, σ)). This makes possible the use of the explicit invariants given by Éršov [4] for structures (B, U) where U is an ultrafilter on B. (These generalize the Tarski invariants for Boolean algebras [15].) We also use the Éršov invariants in the proof of our main result.

Parallelisability in Banach spaces: a parallelisable dynamical system with uniformly bounded trajectories

1988 ◽  
Vol 108 (3-4) ◽  
pp. 371-378
Author(s):  
B. M. Garay
Keyword(s):  
Dynamical System ◽  
Banach Space ◽  
Banach Spaces ◽  
Supremum Norm ◽  
Uniformly Bounded ◽  
Bounded Trajectories

SynopsisIn the Banach space of real sequences which converge to zero with the supremum norm, we construct a parallelisable dynamical system with uniformly-bounded trajectories.

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]):

scholarly journals Semi-embeddings of Banach spaces which are hereditarily c0

1983 ◽  
Vol 26 (2) ◽  
pp. 163-167 ◽  
Author(s):  
L. Drewnowski
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Closed Unit Ball ◽  
Continuous Linear ◽  
Scattered Space ◽  
Complete Set

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).

scholarly journals p-Summing operators on injective tensor products of spaces

1992 ◽  
Vol 120 (3-4) ◽  
pp. 283-296 ◽  
Author(s):  
Stephen Montgomery-Smith ◽  
Paulette Saab
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Bounded Linear Operator ◽  
Hausdorff Space ◽  
Tensor Products ◽  
Dimensional Subspace ◽  
Identity Operator ◽  
Bounded Linear

SynopsisLet X, Y and Z be Banach spaces, and let Πp (Y, Z) (1 ≦ p < ∞) denote the space of p-summing operators from Y to Z. We show that, if X is a ℒ∞-space, then a bounded linear operator is 1-summing if and only if a naturally associated operator T#: X → Πl (Y, Z) is 1-summing. This result need not be true if X is not a ℒ∞-space. For p > 1, several examples are given with X = C[0, 1] to show that T# can be p-summing without T being p-summing. Indeed, there is an operator T on whose associated operator T# is 2-summing, but for all N ∈ N, there exists an N-dimensional subspace U of such that T restricted to U is equivalent to the identity operator on . Finally, we show that there is a compact Hausdorff space K and a bounded linear operator for which T#: C(K) → Π1 (l1, l2) is not 2-summing.

Sign in / Sign up

Export Citation Format

Share Document