When X is a locally compact Hausdorff space, continuous functions on X with compact support can approximate every continuous function in C_0(X). There is a natural notion of elements with compact supports for general, not necessarily commutative, C*-algebras and a result of Blackadar saying that in every separable C*-algebra one can choose from such elements an approximate unit (Blackadar calls it an almost idempotent approximate unit).

We address the issue of the existence of such an approximate unit for general, not necessarily separable C*-algebra and show that such approximate units exist in every C*-algebra of density omega_1, that they do not exist in some C*-algebras of density min{2^k: 2^k>continuum} and that their existence in all operator algebras acting on the separable Hilbert space is independent from ZFC. The infinitary combinatorics used involves CH, Canadian trees and Q-sets.

No knowledge of noncommutative mathematics beyond multiplication of 2x2 matrices will be assumed. These are the results of a joint research project with Tristan Bice available at arxiv.org/pdf/1707.09287.pdf .