The algebraic formulation of quantum mechanics (and related stuff, like quantum thermodynamics & dynamical systems etc.) via C*-algebras provides a viewpoint based mostly on abstract functional analysis. However, I haven't really seen a working application of this approach, i.e. an example of a (preferably physical) problem which is difficult to solve or even to formulate in the standard formalism, while considerably easier to tackle with all this algebraic stuff. Any ideas? Without such concrete examples the whole field seems to be interesting mathematically, perhaps, but lacking any physical substance (even if it's sometimes masqueraded as having connections with physics).
edit: by "standard formalism" I mean "observables, i.e. operators, acting on a Hilbert space of physical states", either in Schroedinger picture (time dependence of states) or Heisenberg (time dependence of operators). C*-algebraic approach starts from a C*-algebra and defines states as positive functionals on the elements of algebra, time evolution as a *-automorphism etc. (for an introduction to this formalism, see http://hal.archives-ouvertes.fr/docs/00/12/88/67/PDF/qds.pdf)
This might be of less interest to non-physicists than a typical MO post, but I hope it's still relevant.