OA-seminar: Chris Schafhauser

Speaker: Chris Schafhauser (York University).

Title: Tracial Gluing in Z-stable C*-algebras


For a C*-algebra $B$ with unique trace and no finite dimensional representations, the GNS completion $\pi_\tau(B)''$ is a II_1-factor which is hyperfinite when $B$ is nuclear. Using this, $B$ admits a wealth of approximation structure with respect to the 2-norm induced by the trace: $\|b\|_2 = \mathrm{tr}(b^*b)^{1/2}$. Much of the recent progress in the classification and regularity programmes have come down to exploiting this simple observation and developing abstract methods for replacing 2-norm estimates with operator norm estimates. For C*-algebras with several traces, von Neumann algebraic techniques do not work directly as the 2-norm estimates must be obtained uniformly over all traces on the C*-algebra. Recent work of Castillejos, Evington, Tikuisis, White, and Winter provides a method for ``uniformizing'' point-wise 2-norm estimates for nuclear, Z-stable C*-algebras. In recent work with Evington, Carrion, Gabe, Tikuisis, White, we showed the nuclearity is not necessary. I will discuss these new tracial gluing methods and there connection with the recent progress on classification.