Quasidiagonality was introduced by Halmos in the early 70’s in the context of single operator theory. An operator on a Hilbert space H is said to be quasidiagonal if, with respect to some basis, the operator can be written as a compact perturbation of a block diagonal operator with finite dimensional blocks. Similarly, a set of operators is quasidiagonal the operators in the set may be simultaneously quasidiagonalized.

Since the 80’s, quasidiagonality has become an important notion in C*-algebra theory — C*-algebras may be described concretely as certain well-behaved subalgebras of B(H), the algebra of bounded linear operators from a Hilbert space H to itself. Most notably, quasidiagonality has played an important role in the classification program for simple, amenable C*-algebras which has been an active area of research for the last 30 years and, in terms of the effort invested, parallels the classification of finite simple groups and 3-manifolds.

This classification program now has a complete solution under additional mild (and necessary) hypotheses. One of the last pieces to fall in place was the quasidiagonality theorem of Tikuisis, White, and Winter (Annals, 2017) given sufficient conditions for quasidiagonality. I will discuss this result and some elements of a simplified proof specializing to the case of groups. More precisely, for a group G, consider the Hilbert space H with basis indexed by G. Then G may be represented as operators on H using the action of G on itself by left translation. I will discuss the characterization of groups which are quasidiagonal in this representation.