Abstract: The notion of (m,n)-pure C*-algebras was introduced by Winter in his seminal work on separable, simple, unital C*-algebras of finite nuclear dimension. Although a lot of effort has been put on understanding (0,0)-pureness (often simply called pureness), much less is known about the apparently weaker notion of (m,n)-pureness for m,n>0. This is especially the case in the non-simple setting.
I will begin the talk by recalling the Toms-Winter conjecture and the importance of pureness in its study. I will then discuss results from two different ongoing projects: Pure *-homomorphisms and their properties (joint with J. Bosa), and (m,n)-pure C*-algebras (joint with R. Antoine, F. Perera, and H. Thiel).