After a brief introduction to uniform trace norms, I will define the uniform tracial completion of a C*-algebra and discuss the key examples and features of these objects. In the second half of the talk, I will discuss the trace problem. The trace problem asks whether every trace on the uniform tracial completion is the continuous extension of a trace on A. I will end the talk with an outline of the solution to the trace problem in the case of Z-stable C*-algebras.
