I'm trying to come up with a popularization of the meaning of the Baum-Connes conjecture for people with only undergraduate math. One idea I had was that Group C*-algebras, when the group is commutative, are basically the algebra of function on the dual, so under the algebras-spaces equivalence, Group C*-algebra are attempting to generalize "the dual space of a group" to non-abelian groups, and then, the Baum-Connes conjecture is about algebraic-topologic invariants of that "noncommutative dual space" and maybe specifically relating it to the classifying space ?
Your framing is essentially right, and it's the standard heuristic. A few refinements will make it both accurate and more compelling.
What's right
- For a commutative group G (say a locally compact abelian one), Pontryagin duality plus Fourier transform identifies C*(G) with the algebra of continuous functions vanishing at infinity on the dual group Ĝ. That's the Gelfand-Naimark correspondence at work: commutative C*-algebras are spaces.