No abstract

ABSTRACT. It is shown that every strongly closed algebra of operators acting on a separable Hubert space can be expressed as a direct integral of irreducible algebras. In particular, every reductive algebra is the direct integral of transitive algebras. This decomposition is used to study the relationship between the transitive and reductive algebra problems. The final section of the paper shows how to view direct integrals of algebras as measurable algebra-valued functions.

Abstract.It is shown that the usual processes of linear algebra (e.g., finding Jordan forms, eigenvalues, and eigenvectors) can be carried out in a Borel measurable fashion. These results follow easily from a variant of von Neumann's principle of measurable choice and can be applied to the study of Type I" von Neumann algebras.Introduction. Denote by Mn the algebra of n x n (complex) matrices, considered as operating on the Hilbert space C". We equip Mn with its usual (operator) norm, along with the subordinate topological and Borel structures.The main purpose of this paper is to show that many of the processes of linear algebra (e.g., finding Jordan canonical forms) are Borel measurable. The proofs, which turn out to be remarkably simple, depend only on a 'principle of measurable choice' presented as Theorem 1.The considerations of the preceding paragraph turn out to be important in the study of Type In von Neumann algebras. (This is because such algebras can always be represented as algebras of matrices whose entries are continuous functions.) Indeed, upon examining the papers listed in the bibliography, one finds many arguments which are essentially measuretheoretic in nature. The latter half of the present paper shows how such arguments can be replaced by applications of Theorem 1.

Let S be a linear manifold of bounded Hilbert space operators. An operator A belongs to the reflexive closure of S if Af belongs to the closure of S f for each vector f in the underlying Hilbert space. Two extreme possibilities are (1) S is reflexive in the sense that ref S=S, and (2) S is transitive in the sense that ref S includes all bounded operators on the underlying space. We show that every linear space B of Toeplitz operators which is closed in the ultraweak operator topology is either transitive or reflexive. No intermediate behavior is possible. The full space of all Toeplitz operators is transitive, but if B is properly contained in this space and contains all analytic Toeplitz operators, then B must be reflexive. In particular, the space of Toeplitz operators whose matrices have zeros on a fixed superdiagonal is reflexive.

ABSTRACT. Let T = fz T(S>) be a direct integral of Hilbert space operators, and equip the collection g of compact subsets of C with the Hausdorff metric topology. Consider the [set-valued] function sp which associates with each & e Z the spectrum of 7"(g). The main theorem of this paper states that sp is measurable.The relationship between