algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
The theorem of (Alfsen-Shultz 78) says that two C*-algebras have isomorphic state spaces precisely if as Jordan algebras they are isomorphic.
This is to some extent a Jordan algebra-analog of the Gelfand-Naimark theorem.
Other theorems about the foundations and interpretation of quantum mechanics include:
Erik Alfsen, Frederic Shultz, A Gelfand Neumark theorem for Jordan algebras, Advances in Math., 28 (1978), 11-56.
Erik Alfsen, H. Hanche-Olsen, Frederic Shultz, State spaces of $C^\ast$-algebras, Acta Math., 144 (1980), 267-305.
See also