Title: Some basics on the basic construction
Speaker: Daniel Wallick (The Ohio State University)
Abstract: Given a unital inclusion of tracial von Neumann algebras $A \subseteq B$, Vaughan Jones’ basic construction gives a von Neumann algebra $\langle B, e_A \rangle$ with $B \subseteq \langle B, e_A \rangle$ a unital inclusion. When $A$ and $B$ are $\mathrm{II}_1$ factors and $A \subseteq B$ is finite index, then $\langle B, e_A \rangle$ is a $\mathrm{II}_1$ factor, with the unique trace satisfying a Markov property. However, if $B$ is an arbitrary tracial von Neumann algebra, then there may not exist a trace on $\langle B, e_A \rangle$ satisfying this Markov property, even if $\langle B, e_A \rangle$ is finite. We will give a necessary and sufficient condition for there to exist a Markov trace on $\langle B, e_A \rangle$ when $B$ is finite-dimensional. The results in this talk are from Jones' seminal 1983 paper "Index for Subfactors."