scholarly journals On a Boolean algebra of projections constructed by Dieudonné

1969 ◽  
Vol 16 (3) ◽  
pp. 259-262 ◽  
Author(s):  
H. R. Dowson
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Simple Proof ◽  
Direct Proof ◽  
Complete Boolean Algebra ◽  
Direct Sum ◽  
Algebra Of Operators

Dieudonné (4) has constructed an example of a Banach space X and a complete Boolean algebra B̃ of projections on X such that B̃ has uniform multiplicity two, but for no choice of x1, x2 in X and non-zero E in B̃ is EX the direct sum of the cyclic subspaces clm {Ex1:E∈B̃} and clm {Ex2:E∈B̃}. Tzafriri observed that it could be deduced from Corollary 4 (9, p. 221) that the commutant B̃′ of B̃ is equal to A(B̃), the algebra of operators generated by B̃ in the uniform operator topology. A study of (3) suggested the direct proof of the second property given in this note. From this there follows a simple proof that B̃ has the first property.

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.

scholarly journals The Center Of A Generalized Effect Algebra

2014 ◽  
Vol 47 (1) ◽  
pp. 1-21 ◽  
Author(s):  
D. J. Foulis ◽  
S. Pulmannová
Keyword(s):  
Boolean Algebra ◽  
Direct Summand ◽  
Effect Algebra ◽  
Complete Boolean Algebra ◽  
Cover System ◽  
Direct Sum ◽  
Generalized Effect Algebra

AbstractIn this article, we study the center of a generalized effect algebra (GEA), relate it to the exocenter, and in case the GEA is centrally orthocomplete (a COGEA), relate it to the exocentral cover system. Our main results are that the center of a COGEA is a complete boolean algebra and that a COGEA decomposes uniquely as the direct sum of an effect algebra (EA) that contains the center of the COGEA and a complementary direct summand in which no nonzero direct summand is an EA.

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 Scalar operators and integration

1988 ◽  
Vol 45 (3) ◽  
pp. 401-420 ◽  
Author(s):  
Igor Kluvánek
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Algebra ◽  
Operator Norm ◽  
Linear Combinations ◽  
Multiplicative Operator ◽  
Scalar Operator ◽  
Set Function ◽  
Algebra Of Sets ◽  
Algebra Of Operators

AbstractThe notion of a scalar operator on a Banach space, in the sense of N. Dunford, is widened so as to cover those operators which can be approximated in the operator norm by linear combinations of disjoint values of an additive and multiplicative operator valued set function, P, on an algebra of sets in a space Ω such that P(Ω) = I, subject to some conditions guaranteeing that this definition is unambiguous. An operator T turns out to be scalar in this sense, if and only if, there exists a (not necessarily bounded) Boolean algebra of bounded projections such that the Banach algebra of operators it generates is semisimple and contains T.

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.

Ε‎-Fr ´Echet Differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Simple Proof ◽  
Common Point ◽  
Lipschitz Functions ◽  
Closed Convex Hull ◽  
Uniformly Smooth ◽  
Fréchet Differentiability ◽  
Finite Set ◽  
Frechet Differentiability

This chapter treats results on ε‎-Fréchet differentiability of Lipschitz functions in asymptotically smooth spaces. These results are highly exceptional in the sense that they prove almost Frechet differentiability in some situations when we know that the closed convex hull of all (even almost) Fréchet derivatives may be strictly smaller than the closed convex hull of the Gâteaux derivatives. The chapter first presents a simple proof of an almost differentiability result for Lipschitz functions in asymptotically uniformly smooth spaces before discussing the notion of asymptotic uniform smoothness. It then proves that in an asymptotically smooth Banach space X, any finite set of real-valued Lipschitz functions on X has, for every ε‎ > 0, a common point of ε‎-Fréchet differentiability.

Normal Operators on the Banach Space Lp(-∞,∞). Part I

1961 ◽  
Vol 13 ◽  
pp. 505-518 ◽  
Author(s):  
Gregers L. Krabbe
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Abelian Subalgebra ◽  
The Family ◽  
Normal Operators ◽  
Image Position

Let be the Boolean algebra of all finite unions of subcells of the plane. Denote by εpthe algebra of all linear bounded transformations of Lp(— ∞, ∞) into itself. Suppose for a moment that p = 2, and let Rp be an involutive abelian subalgebra of εp if Rp is also a Banach space and if Tp ∈ Rp, then:(i) The family of all homomorphic mappings of into the algebra Rp contains a member EPT such that(1)

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