In linear partial differential equations, hypoellipticity is the condition that if Df=g, and if g is smooth, then f is necessarily smooth, too. The best-known hypoelliptic equations are the elliptic equations, which are characterized by an isotropy property that can be readily checked point-by-point. Various more general point-by-point sufficiency criteria for hypoellipticity have been studied, beginning with famous work of Lars Hormander in the 1960?s. Within the context of contact manifolds, Erik van Erp uses such a condition to formulate and prove an index theorem for certain hypoelliptic operators. His work involves C*-algebras, specifically the C*-algebras of Heisenberg Lie groups, in a simple but crucial way. I shall review this aspect of Van Erp?s work in my talk, and I shall also review an earlier result of Charles Rockland, which lies of index theory but which uses the same C*-algebras in an even more simple and even more crucial way. In recent work with a number of collaborators I have been attempting to use Rockland?s theorem to understand the analytic foundations of Bismut?s hypoelliptic Laplacian, and I shall explain something about this, too.