What's the point with the Gelfand–Naimark theorem?

Question

Does anybody can explain me in plain english what's the real point with the Gelfand–Naimark Theorem. I know it's crucial, but I think I'm missing how much it's crucial.

Answer 1

The Gelfand-Naimark theorem is a "Cayley's theorem" for C*-algebras.

One way to think about Cayley's theorem is the following. You can define a "concrete group" to be a subgroup of the group of permutations of some set (this is how mathematicians implicitly used to work with groups, back in the day), and define an "abstract group" in the usual way. Cayley's theorem guarantees that every abstract group arises as a concrete group, so the abstract definition really captures the concrete phenomenon it was intended to.

Similarly, you can define a "concrete C*-algebra" to be a closed and star-closed subalgebra of $B(H)$ for some Hilbert space $H$, and an "abstract C*-algebra" in the usual way. Gelfand-Naimark guarantees that every abstract C*-algebra arises as a concrete C*-algebra, so the abstract definition again captures the concrete phenomenon it was intended to. (This is much less obvious than Cayley's theorem for groups, and in particular the C*-identity, which isn't obvious to write down, plays a crucial role.)

Various other theorems in mathematics also play this role; for example Whitney embedding is a Cayley's theorem for manifolds, and the PBW theorem implies a Cayley's theorem for Lie algebras.