Σ-finite Von Neumann Algebra Research Articles

Coordinates for analytic operator algebras

Let M be a σ-finite von Neumann algebra and α = {αt}t∈A be a representation of a compact abelian group A as *-automorphisms of M. Let Γ be the dual group of A and suppose that Γ is totally ordered with a positive semigroup Σ⊆Γ. The analytic algebra associated with α and Σ iswhere spα(a) is Arveson's spectrum. These algebras were studied (also for A not necessarily compact) by several authors starting with Loebl and Muhly [10].

Maximality of analytic operator algebras

LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH ∞(α) be the associated analytic subalgebra; i.e.H ∞(α)={X ∈M: sp∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH ∞(α) isH ∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M α = Ci)H ∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH ∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α).

Algebras of Analytic Operators associated with a Periodic Flow on a Von Neumann Algebra

Let M be a σ-finite von Neumann algebra and {σt}t∊T be a σ-weakly continuous representation of the unit circle, T, as *-automorphisms of M. Let H∞(σ) be the set of all x ∊ M such thatThe structure of H∞(σ) was studied by several authors (see [2-13]).The main object of this paper is to study the σ-weakly closed subalgebras of M that contain H∞(σ). In [12] this was done for the special case where H∞(σ) is a nonselfadjoint crossed product.Let Mn, for n ∊ Z, be the set of all x ∊ M such that