Motivation and advantage of C$^*$-algebra formulation of quantum statistical mechanics

I am studying C$^*$-algebras and the formulation of quantum statistical mechanics by them, mostly from the book by Bratteli and Robinson Operator Algebras and Quantum Statistical Mechanics. I could make this question really long, but what I am essentially interested in is how to best motivate such a formulation of quantum statistical mechanics, what is the advantage of the algebraic approach? I understand that C$^*$-algebras (the CAR and CCR algebras) arise naturally when studying creation and annihilation operators on the Fock space, and these algebras are uniquely determined by the form of the (anti)commutation relation. Thus, we can take the abstract features of such a C$^*$-algebra and its states, and do statistical mechanics using these, without referring to the specific representation etc. I like the aesthetics of such a more abstract approach, however I do not understand the motivation for it. Or to be more precise, I do not understand if there is a physical motivation for it. Does it allow us to calculate more things? On the math-side, is it rigorous, while the usual (Hilbert space, density matrix, self-adjoint operators as observables, etc) approach is not, e.g. there are things in quantum statistical mechanics that do not behave well, like divergences in quantum field theory? How can I justify studying it to a hypothetical friend that studies theoretical physics, both from the perspective of mathematical rigor, and from a more physical one?