Boolean Algebra Of Projections Research Articles

Team games, hypergraph spaces, and projective Boolean algebras

We modify the game Fuchino, Koppelberg, and Shelah used to characterize the κ-Freese-Nation property for a given Boolean algebra A, replacing players I and II each with a team of n players with limited information. We show that A is tightly κ-filtered exactly when team II has a winning strategy for every finite team size.Case κ=ℵ0 characterizes projective Boolean algebras and, hence, Dugundji spaces. In terms of the open-open game of Daniels, Kunen, and Zhou, this characterization is a team version of “very I-favorable.” We similarly characterize Cohen algebras in terms of a team version of I-favorability.If A is the clopen algebra of the space of n-uniform hypergraphs on κ+n that avoid copies of [n+1]n, then team II has a winning strategy for our modified FKS game for team size n−1 but not n. For n≥3, this algebra also answers a question of Geschke when combined with a locally <κ-sized characterization of tightly κ-filtered Boolean algebras that we prove. Case κ=ℵ0 includes a locally finite characterization of projective Boolean algebras.

Derivations on ideals in commutative AW*-algebras

LetA be a commutativeAW*-algebra.We denote by S(A) the *-algebra of measurable operators that are affiliated with A. For an ideal I in A, let s(I) denote the support of I. Let Y be a solid linear subspace in S(A). We find necessary and sufficient conditions for existence of nonzero band preserving derivations from I to Y. We prove that no nonzero band preserving derivation from I to Y exists if either Y ⊂ Aor Y is a quasi-normed solid space. We also show that a nonzero band preserving derivation from I to S(A) exists if and only if the boolean algebra of projections in the AW*-algebra s(I)A is not σ-distributive.

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

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.

Projections in a Synaptic Algebra

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.

The number of openly generated Boolean algebras

AbstractThis article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly σ-filtered Boolean algebras.We show that for every uncountable regular cardinal κ there are 2κ pairwise non-isomorphic openly generated Boolean algebras of size κ &gt; ℵ1 provided there is an almost free non-free abelian group of size κ. The openly generated Boolean algebras constructed here are almost free.Moreover, for every infinite regular cardinal κ we construct 2κ pairwise non-isomorphic Boolean algebras of size κ that are tightly σ-filtered and c.c.c.These two results contrast nicely with Koppelberg's theorem in [12] that for every uncountable regular cardinal κ there are only 2κ isomorphism types of projective Boolean algebras of size κ.

Resolutions of the identity in Fr�chet spaces

It is a classical result that every Bade σ-complete Boolean algebra of (selfadjoint) projections in a separable Hilbert space coincides with the projections forming the resolution of the identity of some bounded selfadjoint operator. This result is extended to the setting of separable Frechet spaces. Namely, every Bade σ-complete Boolean algebra of projections in such a space coincides with the resolution of the identity of some (continuous) scalar-type spectral operator having spectrum a compact subset ofℝ.

The dual bade theorem in locally convex spaces and reflexivity of a closed unital subalgebra

The results presented in this paper extend a dual version of the reflexivity theorem of W. Bade to locally convex spaces. Dual versión of the Bade theorem in a Banach C(K)-module was firstly discovered in [1]. It is our aim to extend it to a locally convex C(K)-module. As a consequence, it is proven that each unital w* operator topology closed subalgebra of the w* operator topology closed algebra generated by a Boolean algebra of projections is reflexive.

The strong closure of $\sigma$-complete Boolean algebras of projections

A result of T. A. Gillespie implies that the strong operator closure of any abstractly $\sigma $ -complete Boolean algebra of projections in a Banach space X which does not contain a copy of c 0 is Bade complete. It is shown that the same conclusion is valid for another (extensive) class of Banach spaces X, namely those which are weakly compactly generated. As a consequence, it follows that a Boolean algebra of projections in a separable Banach space is abstractly $\sigma $ -complete iff it is abstractly complete. It is also shown that a Banach space X has the property that the strong closure of every abstractly complete Boolean algebra of projections in X is Bade complete iff X does not contain a copy of $\ell ^\infty \!$ .

Criteria for Closedness of Spectral Measures and Completeness of Boolean Algebras of Projections

The intimate connection between spectral measures and σ-complete Boolean algebras of projections in Banach spaces was intensively investigated by W. Bade in the 1950s, [“Linear Operators III: Spectral Operators,” Wiley-Interscience, New York, 1971; Chap. XVII]. It is well known that the most satisfactory situation occurs when the Boolean algebra is actually complete rather than just σ-complete. This makes it desirable to have available criteria (not just in Banach spaces but also in the nonnormable setting) which can be used to determine the completeness of Boolean algebras of projections. Such criteria, which should be effective and applicable in practice, as general as possible and apply to extensive classes of spaces, are presented in this article. A series of examples shows that these criteria are close to optimal.

On the bicommutant theorem for σ-complete Boolean algebras of projections in Banach spaces

On the bicommutant theorem for σ-complete Boolean algebras of projections in Banach spaces

Boolean algebras of projections and resolutions of the identity of scalar-type spectral operators

Let Μ be a Bade complete (or σ-complete) Boolean algebra of projections in a Banach space X. This paper is concerned with the following questions: When is Μ equal to the resolution of the identity (or the strong operator closure of the resolution of the identity) of some scalar-type spectral operator T (with σ(T) ⊆ ℝ) in X? It is shown that if X is separable, then Μ always coincides with such a resolution of the identity. For certain restrictions on Μ some positive results are established in non-separable spaces X. An example is given for which Μ is neither a resolution of the identity nor the strong operator closure of a resolution of the identity.

Boundedness Criteria for Boolean Algebras of Projections

Let E and F be two commuting bounded Boolean algebras of projections on a Banach spaceX. Various geometric conditions onXare established which ensure that the Boolean algebra of projections generated by E and F is also bounded. In particular, this is the case ifXis a Banach lattice or a subspace of ap-concave Banach lattice, where 1⩽p<∞. Several consequences of this boundedness property are also discussed.

The Sequential Closedness of σ-Complete Boolean Algebras of Projections

The Sequential Closedness of σ-Complete Boolean Algebras of Projections

Nonanalytic functional calculi and spectral maximal spaces

We are concerned with the algebraic representation of the spectral maximal spaces for certain classes of decomposable operators on Banach spaces. The main emphasis will be on those operators which admit a functional calculus on a suitable algebra of functions such as, for instance, differentiable functions, Lipschitz functions, or functions with absolutely convergent Fourier series. Using appropriate partitions of unity, we shall give a unified approach to various previous results in this area as well as to certain extensions and variants thereof. In the case of spectral operators, we shall obtain the optimal results without any assumption on the underlying Banach space or the Boolean algebra of projections corresponding to the spectral measure.

Operator algebras generated by Boolean algebras of projections in Montel spaces

It is shown that the classical result of W. Bade, stating that the uniformly closed operator algebra generated by a complete Boolean algebra of projections in a Banach space coincides with the weak operator closed algebra that it generates, is also valid (if suitably interpreted) in Montel spaces, Schwartz spaces and other “related” spaces.

Uniformly Closed Algebras Generated by Boolean Algebras of Projections in Locally Convex Spaces

The theory of operator algebras in Banach spaces generated by Boolean algebras of projections is by now well known. It is systematically exposed in the penetrating studies of W. Bade, [1], [2] and [6, Chapter XVII]. Many of these results, a priori independent on normability of the underlying space, have recently been extended to the setting of locally convex spaces; see [3], [4], [5], [11] and [15], for example.However, one of Bade's fundamental results, stating that the closed algebra generated by a complete Boolean algebra in the uniform operator topology is the same as the closed algebra that it generates in the weak operator topology, has remained remarkably resistant in attempts to extend it to locally convex spaces. Recently however, a class of Boolean algebras in non-normable spaces, called boundedly σ-complete Boolean algebras, was exhibited in which the analogue of Bade's result is valid, [14; Theorem 5.3].

On Boolean Algebras of Projections and Scalar-Type Spectral Operators

On Boolean Algebras of Projections and Scalar-Type Spectral Operators

Continuity properties of vector measures

This paper is a sequel to [ I 1. Using projections in the universal measure space of W. H. Graves [ 6 1, we study continuity and orthogonality properties of vector measures. To each strongly countably additive measure 4 on an algebra .d with range in a complete locally convex space we associate a projection P,, which may be thought of as the support of @ and as a projection-valued measure. Our main result is that, as measures, # and P, are mutually continuous. It follows that measures may be compared by comparing their projections. The projection P, induces a decomposition of vector measures which turns out to be Traynor’s general Lebesgue decomposition [8) relative to qs. Those measures $ for which algebraic d-continuity is equivalent to $continuity are characterized in terms of projections, while $-continuity is shown to be equivalent to &order-continuity whenever -d is a u-algebra. Using projections associated with nonnegative measures, we approximate every 4 uniformly on the algebra ,d by vector measures which have control measures. Last, we interpret our results algebraically. Under the equivalence relation of mutual continuity, and with the order induced by continuity, the collection of strongly countably additive measures on .d which take values in complete spaces forms a complete Boolean algebra 1 rV, which is isomorphic to the Boolean algebra of projections in the universal measure space. The homomorphic image of the lattice of nonnegative measures on .d is a dense u-ideal in 4. Using the results in this paper, the author and W. H. Graves have investigated the range of a vector measure [ 3 1 and characterized the closed measures of Kluvanek (21. All definitions and notation are as in [ 11. In particular, .d is a fixed algebra of subsets of a nonempty set X and L(.d) is the universal measure 268 0022-247X/82/030268-13$02.00/O

A representation theorem for a complete Boolean algebra of projections

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.

Boolean algebras of projections

Boolean algebras of projections