Σ-complete Boolean Algebras Research Articles

Spectral measures, boundedly 𝜎-complete Boolean algebras and applications to operator theory

A systematic study is made of spectral measures in locally convex spaces which are countably additive for the topology of uniform convergence on bounded sets, briefly, the bounded convergence topology. Even though this topology is not compatible for the duality with respect to the pointwise convergence topology it turns out, somewhat surprisingly, that the corresponding L 1 {L^1} -spaces for the spectral measure are isomorphic as vector spaces. This fact, together with I. Kluvanek’s notion of closed vector measure (suitably developed in our particular setting) makes it possible to extend to the setting of locally convex spaces a classical result of W. Bade. Namely, it is shown that if B B is a Boolean algebra which is complete (with respect to the bounded convergence topology) in Bade’s sense, then the closed operator algebras generated by B B with respect to the bounded convergence topology and the pointwise convergence topology coincide.

Cyclic Banach spaces and reflexive operator algebras

SynopsisLet X be a Banach space and let ℬ be a σ-complete Boolean algebra of projections on X with a cyclic vector. It is shown that there exists a normed Köthe space Lρ, the norm of which has the Fatou property, such that X is linearly homeomorphic to the subspace of Lρ consisting of those functions of absolutely continuous norm and such that, under this homeomorphism, the projections ℬ correspond to operators consisting of multiplication by characteristic functions. This representation theorem for X is used to show that certain operator algebras associated with ℬ are reflexive. As an immediate corollary of the reflexivity result, it is shown that, if T is a scalar type spectral operator whose resolution of the identity has a cyclic vector, then T is reflexive.

Homomorphic Images of σ-Complete Boolean Algebras

Homomorphic Images of σ-Complete Boolean Algebras

Boolean algebras of projections

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).

Homomorphic images of 𝜎-complete Boolean algebras

It is a well-known theorem of R. S. Pierce that, for every infinite cardinal α , α ℵ 0 = α \alpha ,{\alpha ^{{\aleph _0}}} = \alpha if and only if there is a complete Boolean algebra B B s.t. card B = α B = \alpha (see [3, Theorem 25.4]). Recently, Comfort and Hager proved [1] that, for every infinite σ \sigma -complete Boolean algebra B , ( card ⁡ B ) ℵ 0 = card ⁡ B B,{(\operatorname {card} B)^{{\aleph _0}}} = \operatorname {card} B . We extend this result to the class of homomorphic images of σ \sigma -complete algebras, following closely Comfort’s and Hager’s proof. As a corollary, an improvement of Shelah’s theorem on the cardinality of ultraproducts of finite sets [2] is derived (Theorem 2). 1 ^{1}

Boolean powers of simple groups

We prove that factor groups of cartesian powers of finite non-abelian simple groups cannot be countably infinite. Thisis not our main result, but it had been our original aim. The proof is based on a similar fact concerning σ-complete Boolean algebras, and on a representation of certain subcartesian powers of a group in its group ring over a Boolean ring. This representation, to which we give the name “Boolean power”, will be our central theme, and we begin with it.

On weak convergence of measures and σ-complete Boolean algebras

On weak convergence of measures and σ-complete Boolean algebras

On the representation of 𝜎-complete Boolean algebras

On the representation of 𝜎-complete Boolean algebras