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.

GENERALIZED p-FRAME IN SEPARABLE COMPLEX BANACH SPACES

2010 ◽  
Vol 08 (01) ◽  
pp. 133-148 ◽  
Author(s):  
XIANG-CHUN XIAO ◽  
YU-CAN ZHU ◽  
XIAO-MING ZENG
Keyword(s):  
Banach Space ◽  
Functional Analysis ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Riesz Basis ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
The Stability

The concept of g-frame and g-Riesz basis in a complex Hilbert space was introduced by Sun.18 In this paper, we generalize the g-frame and g-Riesz basis in a complex Hilbert space to a complex Banach space. Using operators theory and methods of functional analysis, we give some characterizations of a g-frame or a g-Riesz basis in a complex Banach space. We also give a result about the stability of g-frame in a complex Banach space.

scholarly journals Boolean algebras of projections

1975 ◽  
Vol 19 (3) ◽  
pp. 287-289
Author(s):  
P. G. Spain
Keyword(s):  
Hilbert Space ◽  
Boolean Algebra ◽  
Von Neumann Algebra ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Von Neumann ◽  
Relative Compactness ◽  
Weakly Closed ◽  
Result Theorem ◽  
Bounded Sets

Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a σ-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a σ-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets (8): Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of σ-completeness to weak relative compactness; indeed, a Boolean algebra of projections has σ-completion if and only if it is weakly relatively compact (Theorem 1). Then, following the derivation of the theorem of Edwards and Ionescu Tulcea from the Vidav characterisation of abstract C*-algebras (see (9)), I give a result (Theorem 2) which, with its corollary, includes (1: 2.7, 2.8, 2.9, 2.10, 3.2, 3.3, 4.5).

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.

Banach spaces whose algebras of operators have a large group of unitary elements

2008 ◽  
Vol 144 (1) ◽  
pp. 97-108 ◽  
Author(s):  
JULIO BECERRA GUERRERO ◽  
MARÍA BURGOS ◽  
EL AMIN KAIDI ◽  
ÁNGEL RODRÍGUEZ PALACIOS
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Linear Algebra ◽  
Real Banach Space ◽  
Complex Banach Space ◽  
Linear Isometry ◽  
Bounded Linear ◽  
Weak Operator Topology ◽  
Weak Operator

AbstractWe prove that a complex Banach space X is a Hilbert space if (and only if) the Banach algebra $\mathcal L (X)$ (of all bounded linear operator on X) is unitary and there exists a conjugate-linear algebra involution • on $\mathcal L (X)$ satisfying T• = T−1 for every surjective linear isometry T on X. Appropriate variants for real spaces of the result just quoted are also proven. Moreover, we show that a real Banach space X is a Hilbert space if and only if it is a real JB*-triple and $\mathcal L (X)$ is $w_{op}'$-unitary, where $w'_{op}$ stands for the dual weak-operator topology.

scholarly journals On a Numerical Radius Preserving Onto Isometry onL(X)

2016 ◽  
Vol 2016 ◽  
pp. 1-3
Author(s):  
Sun Kwang Kim
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Uniform Convexity ◽  
Numerical Radius ◽  
Uniform Smoothness

We study a numerical radius preserving onto isometry onL(X). As a main result, whenXis a complex Banach space having both uniform smoothness and uniform convexity, we show that an onto isometryTonL(X)is numerical radius preserving if and only if there exists a scalarcTof modulus 1 such thatcTTis numerical range preserving. The examples of such spaces are Hilbert space andLpspaces for1<p<∞.

Commutators in Factors of Type III

1966 ◽  
Vol 18 ◽  
pp. 1152-1160 ◽  
Author(s):  
Arlen Brown ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Complex Hilbert Space ◽  
Identity Operator ◽  
Von Neumann ◽  
Type Iii ◽  
Underlying Space ◽  
Special Cases ◽  
Weakly Closed ◽  
Algebra Of Operators

Let denote a separable, complex Hilbert space, and let R be a von Neumann algebra acting on . (A von Neumann algebra is a weakly closed, self-adjoint algebra of operators that contains the identity operator on its underlying space.) An element A of R is a commutator in R if there exist operators B and C in R such that A = BC — CB. The problem of specifying exactly which operators are commutators in R has been solved in certain special cases; e.g. if R is an algebra of type In (n < ∞) (2), and if R is a factor of type I∞ (1). It is the purpose of this note to treat the same problem in case R is a factor of type III. Our main result is the following theorem.

scholarly journals On commutative V*-algebras

1970 ◽  
Vol 17 (2) ◽  
pp. 173-180 ◽  
Author(s):  
P. G. Spain
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Boolean Algebra ◽  
Unitary Groups ◽  
Easy Proof ◽  
Complete Extension ◽  
Normal Operators ◽  
Similar Extension ◽  
Stone’S Theorem

We shall use results of Palmer (10, 11) and of Edwards and Ionescu Tulcea (6) to show that a commutative V*-algebra (with identity) of operators on a weakly complete Banach space is isomorphic to such an algebra on a Hilbert space, the isomorphism extending to the weak closures of the algebras. This result leads to an extension of Stone's theorem on unitary groups (a similar extension is proved by different methods in (2, p. 350) and of Nagy's theorems on semigroups of normal operators. The same technique yields an easy proof of Dunford's theorem on the existence of a σ-complete extension of a bounded Boolean algebra of projections on a weakly complete Banach space. We are indebted to H. R. Dowson for suggesting this topic and for help and guidance in pursuing it.

scholarly journals Bounded sets in the range of anX∗∗-valued measure with bounded variation

2000 ◽  
Vol 23 (1) ◽  
pp. 21-30 ◽  
Author(s):  
B. Marchena ◽  
C. Piñeiro
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Bounded Variation ◽  
Necessary Conditions ◽  
Bounded Set ◽  
Sufficient And Necessary Conditions ◽  
Bounded Sets

LetXbe a Banach space andA⊂Xan absolutely convex, closed, and bounded set. We give some sufficient and necessary conditions in order thatAlies in the range of a measure valued in the bidual spaceX∗∗and having bounded variation. Among other results, we prove thatX∗is a G. T.-space if and only ifAlies inside the range of someX∗∗-valued measure with bounded variation wheneverXAis isomorphic to a Hilbert space.

scholarly journals Biconcave-function characterisations of UMD and Hilbert spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 297-306 ◽  
Author(s):  
Jinsik Mok Lee
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Complex Banach Space ◽  
Unit Interval ◽  
Martingale Differences ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Image Position ◽  
Umd Banach Spaces

Suppose that X is a real or complex Banach space with norm |·|. Then X is a Hilbert space if and only iffor all x in X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and |x − Y| ≤ 2 almost everywhere. This leads to the following biconcave-function characterisation: A Banach space X is a Hilbert space if and only if there is a biconcave function η: {(x, y) ∈ X × X: |x − y| ≤ 2} → R such that η(0, 0) = 2 andIf the condition η(0, 0) = 2 is eliminated, then the existence of such a function η characterises the class UMD (Banach spaces with the unconditionally property for martingale differences).

scholarly journals On Classical Representations of Convex Descriptions

1993 ◽  
Vol 48 (3) ◽  
pp. 469-470 ◽  
Author(s):  
Slawomir Bugajski
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Vector Lattice ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Classical Representation ◽  
Base Norm

Abstract It is demonstrated that if V* is not a vector lattice, where V is a base norm Banach space, then there is no commutative observable providing a classical representation for V. This observation generalizes a similar result of Busch and Lahti, obtained for V - the trace class of operators on a separable complex Hilbert space.

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