mp.mathematical physics - What's algebraic approach to QM good for?

If you want to consider quantum statistical physics properly this approach is necessary. The KMS condition gives a generalization of the quantum Gibbs postulate that allows for the treatment of phase transitions, coexistence of multiple phases, etc.

The functional-analytic approach is also important for other reasons (besides the rigged Hilbert space that is necessary even to understand a free particle): Fock space depends on it, as does axiomatic quantum field theory, and pages 12-13 of Bratteli and Robinson v.1 give some quick background to the KMS approach (both volumes are worth looking at). A book by Sewell called Quantum mechanics and its emergent macrophysics also gives quite a bit of relevant physical background.

BTW, the KMS condition is not as obscure as it might at first seem (the Wikipedia article is one of the few places I recall seeing that demystifies it). In the Heisenberg picture observables evolve under the time evolution map

$tau_t : A mapsto e^{iHt/ hbar}Ae^{-iHt/ hbar}.$

The appropriate generalization of the classical Gibbs rule is

$langle A rangle = Z^{-1}mbox{Tr}(e^{-beta H} A), quad Z := mbox{Tr}(e^{-beta H}).$

To see this, consider the projection observable $Pi_k := lvert k rangle langle k rvert$. We have that

$langle Pi_k rangle = Z^{-1}mbox{Tr}(e^{-beta E_k} lvert k rangle langle k rvert) = Z^{-1}e^{-beta E_k}$

in accordance with classical statistical physics. Now for generic observables $A$ and $C$, we have that

$left langle tau_t(A)C right rangle = Z^{-1}mbox{Tr}(e^{-beta H} e^{iHt/hbar}Ae^{-iHt/hbar}C)$

$= Z^{-1}mbox{Tr}(Ce^{iH(t+ihbarbeta)/hbar}Ae^{-iHt/hbar})$

$= Z^{-1}mbox{Tr}(Ce^{iH(t+ihbarbeta)/hbar}Ae^{-iH(t+ihbarbeta)/hbar}e^{-beta H})$

$= left langle Ctau_{t+ihbarbeta}(A) right rangle$

which gives the KMS condition:

$left langle tau_t(A)C right rangle = left langle Ctau_{t+ihbarbeta}(A) right rangle.$