I'm moving on from the group $C^{*}$-algebra $C^{*} (\mathcal{L}_1/\mathbb{R}^{*}_{+})$ and am continuing through the monographs of Connes and Marcolli. My goal is to get a general, high level understanding of the construction of the Bost-Connes system (1-dim $\mathbb{Q}$-lattice systems) and their use in the explicit class field theory of $\mathbb{Q}$. As I develop that "big picture", I'm keeping a list of all the things I need to explore in-depth. But not until I have that big picture will I then start chasing down things like group C* algebras. (Of course, I'll need to pick up a quick idea of these things as I go on.)

So that brings me to the section on Quantum Statistical Mechanics (QSM). QSM, as best I can tell, is actually a lot easier to work with than Classical Stat Mech, as I deal with operator and C* algebras, rather than trying to work out some delicate approximation here and there. In classical mechanics the observables are functions on the phase space. But in the quantum world, observables are treated as operators, and they form a C* algebra. So in fact, we can think of a QSM system as just a C* dynamical system, id est, a C* algebra $\mathcal{A}$ together with a time evolution $\sigma$. By "time evolution", I simple mean a family of automorphisms of $\mathcal{A}$ that can be expressed by a single parameter. A "state" on such a system is then defined as a linear functional $phi : \mathcal{A} \to \mathbb{C}$ with $\phi(1)=1$ and $\phi(a^{*}a) \geq 0$ (the first condition is changed to stating that the operator norm of $\phi$ is 1, if $\mathcal{A}$ doesn't have a unit.)

According to the monographs [and what follows from physical intuition] there's one particular class of states we want to look at: the equilibrium states, states that are invariant with respect to time evolution. According to QSM, the equilibrium states are the ones that meet the "KMS" condition at a specific "inverse temperature" $\beta$ of the system. I can't say where the KMS condition comes from or how we got it, but I can read it from the monographs and make some sense of it: A state $\phi$ satisfies the KMS condition at inverse temperature $\beta$ if for all $a,b \in \mathcal{A}$ we can find a function $F_{a,b} (z)$ with the following properties:

- Its holomorphic on the strip $\{z \in \mathbb{C} : 0 < \Im(z) < \beta \}$
- continuous on boundary and bounded
- $F_{a,b}(t) = \phi(a \sigma_t(b))$ and $F_{a,b}(t+i\beta) = \phi(\sigma_t(b) a)$

for all $t \in \mathbb{R}$

I'm not sure how the KMS condition implies the equilibrium (that states are invariant with respect to the time evolution.) Connes mentions that it's a consequence of the Liouville Theorem. I suspect its just some complex analysis that wouldn't take me too long to think about [I can kinda see it...I think...] but we're not worrying about it right now.

Additionally....I don't know where the KMS condition comes from. I have a couple books on "operator algebras and QSM" that I use to figure that out later, as well.

Its probably worth noting that $C^{*} (\mathcal{L}_1/\mathbb{R}^{*}_{+})$ is a C* dynamical system, though some work is required to get the time evolution. I think $C^{*} (\mathcal{L}_1/\mathbb{R}^{*}_{+})$ is the Bost-Connes system.

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