Non-tracial free graph von Neumann algebras

Abstract

Given a finite, directed, connected graph Γ equipped with a weighting µ on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional (M(Γ, µ), ϕ). When the weighting µ is instead on the vertices of Γ, the first author showed the isomorphism class of (M(Γ, µ), ϕ) depends only on the data (Γ, µ) and is an interpolated free group factor equipped with a scaling of its unique trace (possibly direct sum copies of C). Moreover, the free dimension of the interpolated free group factor is easily computed from µ. In this paper, we show for a weighting µ on the edges of Γ that the isomorphism class of (M(Γ, µ), ϕ) depends only on the data (Γ, µ), and is either as in the vertex weighting case or is a free Araki-Woods factor equipped with a scaling of its free quasi-free state (possibly direct sum copies of C). The latter occurs when the subgroup of R + generated by µ(e 1 ) • • • µ(e n ) for loops e 1 • • • e n in Γ is non-trivial, and in this case the point spectrum of the free quasi-free state will be precisely this subgroup. As an application, we give the isomorphism type of some infinite index subfactors considered previously by Jones and Penneys.

Classifying module categories for generalized Temperley–Lieb–Jones ∗-2-categories

“…It is easy to see that MW-type graphs together with all the information listed above constitute examples of balanced Γ-fair graphs. However these conditions are not exactly equivalent as we will see in the following proposition, which is based on the discussion on top of page 12 of [HN18]. Moreover, if there are no arrows from v to w, then dim(H e vw ) = 0.…”