Free Graph von Neumann Algebras

Graphs provide a rich source of examples in C*-algebras through a variety of constructions, most of which can be obtained by way of a Toeplitz algebra. Moreover, these constructions have the nice feature that combinatorial properties of the graph beget algebraic and analytic properties of the C*-algebra. It turns out that one can obtain a similar sunny situation with von Neumann algebras by modifying Voiculescu's free Gaussian functor for the Toeplitz algebra. Given a pair $(\Gamma,\mu)$ consisting of a graph equipped with a weighting on its edges, this construction yields a pair $(\mathcal{M}(\Gamma,\mu),\vphi)$ consisting of a von Neumann algebra—called a free graph von Neumann algebra—equipped with a faithful, normal, positive, linear functional. These pairs provide interesting examples, particularly from the perspective of free probability theory. In this talk, I will guide you through the construction of these von Neumann algebras, paying careful attention to where and why it diverges from the C*-algebraic construction. I will also survey some of my joint work with Michael Hartglass on these algebras. Beyond the definition, very little familiarity with von Neumann algebras will be needed for the talk, and in particular no familiarity with free probability will be assumed.