Uniform tracial completions of C*-algebra have recently played an important role in the regularity and classification theories of nuclear stably finite C*-algebras, beginning with the work on Matui and Sato on the Toms-Winter conjecture. These uniform tracial completions belong to a new class of operators algebras called tracially complete C*-algebras which lie somewhere in between C*-algebras and von Neumann algebras and generalize Ozawa's notion of continuous W*-bundles. I will define this new class of operator algebras and discuss some basic results (such as a uniform version of Connes's theorem) along with several open questions.

This is based on joint work with Jose Carrion, Jorge Castillejos, Sam Evington, Jamie Gabe, Aaron Tikuisis, and Stuart White.