scholarly journals The strong closure of Boolean algebras of projections in Banach spaces

2004 ◽  
Vol 77 (3) ◽  
pp. 365-370 ◽  
Author(s):  
J. Diestel ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Sequence Space ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Special Cases

AbstractThis note improves two previous results of the second author. They turn out to be special cases of our main theorem which states: A Banach space X has the property that the strong closure of every abstractly σ-complete Boolean algebra of projections in X is Bade complete if and only if X does not contain a copy of the sequence space ℓ∞.

scholarly journals Strongly closed bounded Boolean algebras of projections

1981 ◽  
Vol 22 (1) ◽  
pp. 73-75 ◽  
Author(s):  
T. A. Gillespie
Keyword(s):  
Banach Space ◽  
Scalar Field ◽  
Boolean Algebra ◽  
Sequence Space ◽  
Complex Space ◽  
Present Note ◽  
Boolean Algebras ◽  
Complete Boolean Algebra

It is known that every complete Boolean algebra of projections on a Banach space X is strongly closed and bounded and that, although the converse of this result fails in general, it is valid if X is weakly sequentially complete [1, XVII. 3, pp. 2194–2201]. In the present note it is shown that this converse is in fact valid precisely when X contains no subspace isomorphics to the sequence space c0. More explicitly, the following two results are proved. In both, X may be a real or complex space, but c0 will consist of the null sequences in the underlying scalar field.

M-structure in Banach spaces

1967 ◽  
Vol 63 (3) ◽  
pp. 613-629 ◽  
Author(s):  
F. Cunningham
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Measure Space ◽  
Complete Boolean Algebra ◽  
Ordinary Sense ◽  
One Dimensional ◽  
Algebra 2 ◽  
Image Position ◽  
Vector Valued

L-structure in a Banach space X was defined in (3) by L-projections, that is projections P satisfyingfor all x ∈ X. The significance of L-structure is shown by the following facts: (1) All L-projections on X commute and together form a complete Boolean algebra. (2) X can be isometrically represented as a vector-valued L1 on a measure space constructed from the Boolean algebra of its L-projections (2). (3) L1-spaces in the ordinary sense are characterized among Banach spaces by properties equivalent to having so many L-projections that the representation in (2) is everywhere one-dimensional.

scholarly journals Spectral measures on spaces not containing l∞

1981 ◽  
Vol 24 (1) ◽  
pp. 41-45 ◽  
Author(s):  
T. A. Gillespie
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Spectral Measure ◽  
Dual Space ◽  
Boolean Algebras ◽  
Special Role ◽  
Spectral Measures ◽  
Sequential Completeness ◽  
Algebra Of Sets

The property of weak sequential completeness plays a special role in the theory of Boolean algebras of projections and spectral measures on Banach spaces. For instance, if X is a weakly sequentially complete Banach space, then(i) every strongly closed bounded Boolean algebra of projections on X is complete (3, XVII.3.8, p. 2201); from which it follows easily that(ii) every spectral measure on X of arbitary class (Σ, Γ), where Σ is a σ-algebra of sets and Γ is a total subset of the dual space of X, is strongly countably additive; and hence that(iii) every prespectral operator on X is spectral.(See also (1, Theorem 6.11, p. 165) for (iii).)

scholarly journals Hermitian operators and isometries on sums of Banach spaces

1989 ◽  
Vol 32 (2) ◽  
pp. 169-191 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sequence Space ◽  
Reflexive Banach Space ◽  
Geometric Theory ◽  
Orlicz Sequence Space ◽  
Vector Sequence ◽  
Banach Sequence Space ◽  
Banach Sequence

Let E be a Banach sequence space with the property that if (αi) ∈ E and |βi|≦|αi| for all i then (βi) ∈ E and ‖(βi)‖E≦‖(αi)‖E. For example E could be co, lp or some Orlicz sequence space. If (Xn) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum of the Xn's and symbolize by ⊕EXn. Specifically, ⊕EXn = {(xn)|(xn)∈Xn and (‖xn‖)∈E}. The E sum is a Banach space with norm defined by: ‖(xn)‖ = ‖(‖xn‖)‖E. This type of space has long been the source of examples and counter-examples in the geometric theory of Banach spaces. For instance, Day [7] used E=lp and Xk=lqk, with appropriate choice of qk, to give an example of a reflexive Banach space not isomorphic to any uniformly conves Banach space. Recently VanDulst and Devalk [33] have considered Orlicz sums of Banach spaces in their studies of Kadec-Klee property.

Completions of Boolean algebras of projections and weak-star closures of C*-algebras on dual Banach spaces

2011 ◽  
Vol 54 (2) ◽  
pp. 515-529
Author(s):  
Philip G. Spain
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Boolean Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Weak Star ◽  
Weakly Closed ◽  
Operator Closure ◽  
Dual Banach Space ◽  
Completion Theorem

AbstractPalmer has shown that those hermitians in the weak-star operator closure of a commutative C*-algebra represented on a dual Banach space X that are known to commute with the initial C*-algebra form the real part of a weakly closed C*-algebra on X. Relying on a result of Murphy, it is shown in this paper that this last proviso may be dropped, and that the weak-star closure is even a W*-algebra.When the dual Banach space X is separable, one can prove a similar result for C*-equivalent algebras, via a ‘separable patch’ completion theorem for Boolean algebras of projections on such spaces.

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 Nonwandering operators in Banach space

2005 ◽  
Vol 2005 (24) ◽  
pp. 3895-3908 ◽  
Author(s):  
Lixin Tian ◽  
Jiangbo Zhou ◽  
Xun Liu ◽  
Guangsheng Zhong
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sequence Space ◽  
Separable Banach Space ◽  
Infinite Dimensional ◽  
Banach Sequence Space ◽  
Background Space ◽  
Physical Background ◽  
Structurally Stable ◽  
Banach Sequence

We introduce nonwandering operators in infinite-dimensional separable Banach space. They are new linear chaotic operators and are relative to hypercylic operators, but different from them. Firstly, we show some examples for nonwandering operators in some typical infinite-dimensional Banach spaces, including Banach sequence space and physical background space. Then we present some properties of nonwandering operators and the spectra decomposition of invertible nonwandering operators. Finally, we obtain that invertible nonwandering operators are locally structurally stable.

scholarly journals Boolean algebras of projections in (DF)-and (LF)-spaces

2003 ◽  
Vol 67 (2) ◽  
pp. 297-303 ◽  
Author(s):  
J. Bonet ◽  
W. J. Ricker
Keyword(s):  
Boolean Algebra ◽  
Spectral Measure ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Locally Convex Spaces ◽  
Important Class ◽  
Locally Convex ◽  
Convex Spaces

Conditions are presented which ensure that an abstractly σ-complete Boolean algebra of projections on a (DF)-space or on an (LF)-space is necessarily equicontinuous and/or the range of a spectral measure. This is an extension, to a large and important class of locally convex spaces, of similar and well known results due to W. Bade (respectively, B. Walsh) in the setting of normed (respectively metrisable) spaces.

scholarly journals Four games on Boolean algebras

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3389-3395
Author(s):  
Milos Kurilic ◽  
Boris Sobot
Keyword(s):  
Boolean Algebra ◽  
Winning Strategy ◽  
Zero Element ◽  
Boolean Algebras ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Other Hand

The games G2 and G3 are played on a complete Boolean algebra B in ?-many moves. At the beginning White picks a non-zero element p of B and, in the n-th move, White picks a positive pn < p and Black chooses an in ? {0,1}. White wins G2 iff lim inf pin,n = 0 and wins G3 iff W A?[?]? ? n?A pin,n = 0. It is shown that White has a winning strategy in the game G2 iff White has a winning strategy in the cut-and-choose game Gc&c introduced by Jech. Also, White has a winning strategy in the game G3 iff forcing by B produces a subset R of the tree <?2 containing either ??0 or ??1, for each ? ? <?2, and having unsupported intersection with each branch of the tree <?2 belonging to V. On the other hand, if forcing by B produces independent (splitting) reals then White has a winning strategy in the game G3 played on B. It is shown that ? implies the existence of an algebra on which these games are undetermined.

A representation theorem for a complete Boolean algebra of projections

1979 ◽  
Vol 83 (3-4) ◽  
pp. 225-237 ◽  
Author(s):  
H. R. Dowson ◽  
T. A. Gillespie
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Boolean Algebra ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
Complete Boolean Algebra ◽  
Norm Topology ◽  
Weak Operator ◽  
Bounded Sets ◽  
Algebra Of Operators

SynopsisLet B be a complete Boolean algebra of projections on a complex Banach space X and let (B) denote the closed algebra of operators generated by B in the norm topology. It is shown that there is a complex Hilbert space H, a complete Boolean algebra B0 of self-adjoint projections on H, and an algebraic isomorphism of B onto B. This isomorphism is bicontinuous when B and B are endowed with the norm topologies, the weak operator topologies or the ultraweak operator topologies. It is also bicontinuous on bounded sets with respect to the strong operator topologies on B and B. As an application, it is shown that the weak and ultraweak operator topologies in fact coincide on B.

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