A Reduction Theory For Non-Self-Adjoint Operator Algebras

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.

On a generalization of the Jordan canonical form theorem on separable Hilbert spaces

“…Theorem 2.2. For a fixed n, assume that T ∈ L (H ) is a direct integral of upper triangular strongly irreducible operators and the corresponding measure space is {Λ n , µ n , {Λ n , m φn }} as in (3). If there is a unitary operator U such that both…”

“…In [14], we proved that an operator A on H is similar to a direct integral of strongly irreducible operators if and only if its commutant {A} ′ contains a bounded maximal abelian set of idempotents. Related concepts about direct integrals can be found in [3,4].…”

A general view of reflexivity

“…In this section we develop analogues in our general setting of direct integrals, and we prove general versions of reflexivity results concerning direct integrals [AFG,HN1,HN2]. We also prove a new result in our general setting that translates in the Hilbert space case to the theorem that a direct integral of algebras is hyperreflexive if and only if almost every integrand is hyperreflexive and the constants of hyperreflexivity are essentially bounded.…”

Some Recent Results on Invariant Subspaces

“…, [23], [9]). Every operator has an invariant subspace if and only if every reductive operator is normal 6.3.…”

“…Every operator has an invariant subspace if and only if every reductive operator is normal 6.3. COROLLARY ( [9]). The following are equivalent:…”