Notes for the Vanderbilt Subfactor Seminar, February 6th, 2009. Derivations on Group-measure Space Constructions

In this talk we will investigate the structure of a class of closable derivations on von Neumann algebras coming from group-measure space constructions. We will then show how to apply these results to obtain new examples of von Neumann algebras which do not arise as group-measure space constructions, for example the von Neumann algebra L(SL3(Z) ∗G) where G is any non-trivial group. Results in greater generality can be found in [27]. Throughout these notes all finite von Neumann algebras will be separable and come with a faithful normal trace which we will denote by τ . 1. The group-measure space construction of Murray and von Neumann [20] Let Γ be a countable discrete group and suppose we are given a measure preserving action σ : Γ → Aut(X,μ) of Γ on a probability space (X,μ). Consider the abelian von Neumann algebra A = L∞(X,μ) and for each γ ∈ Γ consider the mapping (which we still denote by σγ) σγ : A → A given by σγ(f)(x) = f(σγ−1(x)), ∀x ∈ X. The assignment γ 7→ σγ defines an action of Γ by integral preserving automorphisms of A. Let H = `(Γ, L(X,μ)) be the Hilbert space of square summable functions from Γ into L(X,μ). It will be convenient to view vectors in this Hilbert space as formal sums ξ = Σγ∈Γaγuγ where aγ ∈ L(X,μ) is the coefficient of the function ξ at γ, i.e. aγ = ξ(γ). In this setting the inner product on H then becomes 〈Σγ∈Γaγuγ,Σλ∈Γbλuλ〉 = Σγ∈Γ ∫ aγbγdμ. If ξ = Σγ∈Γaγuγ ∈ H, and η = Σλ∈Γbλuλ ∈ H we define the convolution of ξ and η to be the formal sum ξ · η = (Σγ∈Γaγuγ) · (Σλ∈Γbλuλ) = Σγ,λ∈Γaγσγ(bλ)uγλ = Σγ∈Γ(Σλ∈Γaγλ−1σγλ−1(bλ))uγ. Note that for each γ ∈ Γ we have that ‖Σλ∈Γaγλ−1σγλ−1(bλ)‖1 ≤ Σλ∈Γ‖aγλ−1σγλ−1(bλ)‖1 ≤ Σλ∈Γ‖aγλ−1‖2‖σγλ−1(bλ)‖2 ≤ (Σλ∈Γ‖aγλ−1‖2)(Σλ∈Γ‖bλ‖2) = ‖ξ‖2‖η‖2, hence ξ · η is well defined as a function in `∞(Γ, L(X,μ)), the space of bounded functions from Γ to L(X,μ).