I find this kind of stuff very interesting, but unfortunately I'm not yet at a level where I can fully understand it. (I'm currently studying the proofs of the spectral theorems of functional analysis). I would still be interested in seeing the definitions of the three "types" you're talking about.In my opinion one of the major advantages of C*-algebra approach is that it allows one to see a tower of probability theories. For every type of algebra one gets a probability theory:
Abelian: Discrete Kolmogorov probability
NonAbelian: Quantum Mechanics
Abelian: Statistical Mechanics
NonAbelian: Quantum Statistical Mechanics
NonAbelian: Quantum Field Theory
I don't know what other people think of this, but it's very interesting from a conceptual level.