Groups and Operator Algebras Seminar

Speaker: David Jekel (University of Copenhagen)

Title: Approximate freeness and C*-regularity in W*-bundles

Abstract: Introduced by Ozawa, W*-bundles can be understood as continuous fields of tracial von Neumann algebras over a topological space.  To study W*-bundles, we thus need to analyze von Neumann algebraic phenomena in a way that varies continuously.  This work proves several C*-algebraic regularity properties, such as real rank zero, stable rank one, and comparison of projections for a class of W*-bundles that includes all locally trivial bundles.  Namely, we consider W*-bundles which are selfless, or which admit a sequence of asymptotically free Haar unitaries, where the freeness occurs with respect to the trace in each fiber, uniformly over all fibers of the bundle.  The proofs use Baire-category arguments where the density of each open set is demonstrated using perturbations by an approximately free element.  For instance, to prove stable rank, or density of invertible elements, we consider perturbing a given x by a free circular operator, which will regularize it to something with trivial kernel, which we show admits a polar decomposition in the bundle.
 
This is based on joint work, to appear soon, with Stuart White and Maxwell Ryder.