Q-lattices revisited. (1-dim case)

Yesterday, I found out a friend of mine back home in Colorado died and have since not had the focus to do much work. I haven't done that review of algebra topics like I said I would. But it's important to keep good habits, so I thought I'd review and write a bit about $\mathbb{Q}$-lattices a bit more.

Connes and Marcolli's monographs define a $\mathbb{Q}$-lattice as a lattice $\Lambda \in \mathbb{R}^n$ together with a homomorphism $\phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q} \Lambda / \Lambda$.  We call two $\mathbb{Q}$-lattices $\Lambda_1$ and $\Lambda_2$ commensurable ($\Lambda_1 \sim \Lambda_2$) if:
1) $\mathbb{Q} \Lambda_1 =\mathbb{Q} \Lambda_2$ and
2) $\phi_1$ = $\phi_2$ mod $\Lambda_1 + \Lambda_2$.

Condition 1) means that there exists some $\mathbb{Q}$-span in $\mathbb{R}^n$ such that $\Lambda_1$ and $\Lambda_2$ both sit inside it.  For instance, both $\frac{1}{2} \mathbb{Z}$ and $\frac{1}{5} \mathbb{Z}$ sit inside $\mathbb{Q}$, so they meet condition 1.  But if we take $< (1,0), (0,\sqrt{2}) >$ and $<(\frac{1}{2},0), (0,0) >$ we cannot find any vectors in $\mathbb{R}^2$ that, when spanned over $\mathbb{Q}$, would contain both of those lattices.

Condition 2) is actually how I originally understood it.  The homomorphism must "label the same points".  More formally, Lets say $X =\mathbb{Q} \Lambda_1 =\mathbb{Q} \Lambda_2$.  For each $\Lambda_i$ we have a projection map $\pi_i : X/\Lambda_i \to X/(\Lambda_1 + \Lambda_2)$ by $r+\Lambda_i \mapsto r + (\Lambda_1 + \Lambda_2)$ (of course, $r$ may not be the best representative element in $\Lambda_1 + \Lambda_2$).  Condition two says that $\pi_1 \circ \phi_1 = \pi_2 \circ \phi_2$.

1-dim Q-lattices 


We can describe the Bost-Connes system with $\mathbb{Q}$-lattices in $\mathbb{R}$, and 1-dim $\mathbb{Q}$-lattices can be described by $\hat{\mathbb{Z}} \times \mathbb{R}_{+}^{*}$, where $\hat{\mathbb{Z}} = \varprojlim_{n} \mathbb{Z} / n\mathbb{Z}$.  First note any lattice $\Lambda \in \mathbb{R}$ can be described by $\lambda \mathbb{Z}$, where $\lambda \in \mathbb{R}_{+}^{*}$.  So the only data we need is the homomorphism $\phi : \mathbb{Q} / \mathbb{Z} \to \mathbb{Q} / \lambda \mathbb{Z}$.  But apparently $\hat{\mathbb{Z}} \cong Hom (\mathbb{Q} / \mathbb{Z},\mathbb{Q} / \mathbb{Z})$, that is, $\hat{\mathbb{Z}}$ is isomorphic to the set of all endomorphisms of $\mathbb{Q} / \mathbb{Z}$ (I'm not yet sure how! Something to do with Pontryagin duality?), so all that remains is a choice of $\rho \in \hat{\mathbb{Z}}$ and we have $(\Lambda,\phi) \mapsto (\rho,\lambda)$.  One should note that the inverse of this map is well defined, so we can identify the set of all 1-dim $\mathbb{Q}$-lattices with $\hat{\mathbb{Z}} \times \mathbb{R}_{+}^{*}$.

I'll continue reading about 1-dim $\mathbb{Q}$-lattices for the next several days, though I do still plan to take some time to catch up on some algebra (algebras over a field, group rings, group algebras, C* algebras, their representations, group C* algebras, Pontryagin duels, etale groupoids.)


Thanks to Professor Kim and the folks at mathoverlow for helping me understand this.
Also, for the next few days I will all but abandon Marcolli's monograph;  that text goes straight into a description of the Bost-Connes system via Shimura varieties, which is a bit above my level.  I do intent to understand the Shimura variety and the Hecke algebra description of Bost-Connes eventually, however.

KMS states and symmetries.

I'm still looking at Quantum Statistical Mechanical systems, which I'm treating as a C* algebra $\mathcal{A}$ together with a time evolution $\sigma_t$. Particularly, I'm interested in its equilibrium states, which we can characterize via the KMS condition (recall that a state on a QSM ($\mathcal{A}$,$\sigma_t$) is a positive [or zero] linear functional $\phi : \mathcal{A} \to \mathbb{C}$ with $\phi(1)=1$, or has operator norm 1 if $\mathcal{A}$ has no unit.)

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:

1. Its holomorphic on the strip $\{z \in \mathbb{C} : 0 < \Im(z) < \beta \}$




2. continuous on boundary and bounded




3. $F_{a,b}(t) = \phi(a \sigma_t(b))$ and $F_{a,b}(t+i\beta) = \phi(\sigma_t(b) a)$


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

We call such states $KMS_{\beta}$ states. We also have $KMS_{\infty}$ states, but rather than define them on half-plane (we call those ground states), we use a stronger condition (For reasons I neither know nor understand at this point): A state $\phi$ is a $KMS_{\infty}$ state if we have $KMS_{\beta}$ states $phi_{\beta}$ such that, for sufficiently large $\beta$, $\phi(a) = \lim_{\beta \to \infty} \phi_\beta (a)$ for all $a \in \mathcal{A}$. Mathematically, all I should need is the definitions, just to understand them and go from there. But I'm not quite satisfied with them. I'm wishing I took a course in statistical thermodynamics/mechanics at this point, as I have little intuition for how these equilibrium states relate to our phase transitions of the system (indeed, I'm still not sure what "phase transition" means for a C* dynamical system.) The $KMS_{\beta}$ states (we denote the set of them by $\Sigma_{\beta}$) are the equilibrium states [states that we can actually measure] for "temperatures" of $1/\beta$. At 0 temperature we have the $KMS_{\infty}$ states. Now, if $\phi_1$ and $\phi_2$ are two $KMS_{\beta}$ states, its not too hard to see that any linear combination of the two will also be, and from that its not too hard to see that $\Sigma_{\beta}$ is a convex set (there's a path between any two $KMS_{\beta}$ states). Connes goes further, and states that its compact under the weak topology (pointwise convergence?) and that its a Choquet Simplex. And I have no idea what that is. But for this blog's purposes, I don't think I need to. The important bit is that $KMS_{\beta}$ states are a convex set. Its not too hard to see that $\Sigma_{\infty}$ is a convex set too. Convex sets have extremal points (points that aren't in a line segment joining two other points.) We'll denote the set of extremal $KMS_{\beta}$ states by $\xi_{\beta}$. Connes has a proposition stating that the extremal $KMS_{\beta}$ states are the same as the states that have a factorial GNS representation (while I have an idea what a GNS representation is - the image of the state under a homomorphism to a Hilbert space, where the homomorphism has certain properties and is constructed in a certain way - I have no idea what it means for one to be factorial.) R. Haag in Local Quantum Physics has a similar statement, that $\xi_{\beta}$ is the same as the set of $KMS_{\beta}$ states that are primary. But I don't know what that means either.

So what's the structure and intuition behind $\xi_{\beta}$? How should I think of it? If I'm thinking just in terms of a physical thermodynamic system, what do the extremal equilibrium states represent? My guess is states of matter, or pure phase states (considering that a change in a state of matter is a "phase transition"). So does that mean there are "impure phase states", id est, states in a mixture of phases (so that $\Sigma_{\beta}$ is non-trivial). Back to the QSM: If we're working in a system with only one phase transition (I'm not sure what a phase transition means in this context) does that mean that there are only two extremal $KMS_{\beta}$ states?

Apart from the equilibrium states, we're also interested in symmetries of the QSM system. Rather frustratingly, I cannot talk much about the physics of QSM symmetries (but I resolve to be able to soon! as I do with statistical thermodynamics!), so I must stick to the mathematics. We're interested in both automorphisms (symmetries of the whole system) and endomorphisms (symmetries of only a part of the system....I think.) In either case, they must be compatible [commute] with our time evolution, id est, if g is our auto/endo-morphism, then $g \sigma_t = \sigma_t g$ for all t. for automorphism g, we say $g \in Aut(\mathcal{A}, \sigma_t)$.

A subgroup $G \subset Aut(\mathcal{A}, \sigma_t)$ we call a group of symmetries by automorphisms. It acts on KMS states via pullback, id est, $g^{*}(\phi) (a) = \phi ( g (a) ) $. Connes included a lemma stating that inner automorphisms (from a unitary state $u \in \mathcal{A}$, inner automorphisms acts by $a \mapsto u a u^{-1}$) act trivially on KMS states. Actually, the lemma stated something else, but that was in the proof. The proof essentially uses an analytic continuation of the map $t \mapsto \sigma_t(a)$ to the whole complex plane (I don't know how that's possible) and uses the fact that KMS states have the property that $\phi(b \sigma_{i \beta} (a)) = \phi (ab)$. It then follows that $\phi(u a u^{-1}) = \phi(a)$, and the action is trivial.

An endomorphism $\rho$ also has its pullback, $\rho^{*} (\phi) (a) = \frac{\phi(\rho(a))}{\phi(e)}$, assuming that $\phi(e) \neq 1$.
$e=\phi(1)$ (assuming $\mathcal{A}$ has unity, of course) and its not too hard to see that e is an idempotent ($e^2 = e$, because $e^2 = \phi(1)^2 = \phi (1^2) = e$). Inner endomorphisms also act trivially, by a similar proof.


Things I don't know this post:

Choquet Simplex
factorial GNS representation
primary KMS state.
The analytic continuation used in the proof of the triviality of the action of a group of inner automorphism.
Connes also talks a bit about multiplier algebras and essential ideas - I may look into those tomorrow, but as best I can tell I won't be needing those ideas any time soon. Marcolli doesn't mention them at all, for instance.
Connes also mentions that we can pushforward KSM states: I largely understood what I read there [or I think I did], but I choose not to write about it, largely because of time, and largely because I don't yet see how it relates.

melancholy mondays

Today started out with so much progress. I started early, made ample progress with programming and business contacts, bided on freelance projects, and was anxious to start a day of mathematics work. But when the time to do maths rolled around, I procrastinated, and procrastinated, and procrastinated. (Actually, for the past 3-4 days I've not done much maths work, for the same reason.)

I finally started doing some reading about 2 hours ago. It was vague, unfocused, distracted reading. More about states on a QMS, representations, KMS states, and quite a few things I don't understand (list at the end.) But I must keep good habits, so I read anyways, and am now writing something barely coherent about it.

Here's what I plan for this week:
1) KMS states, the structure of their sets, symmetries on QMS. 3-4 more hours and 1 more blog post.
2) Some algebra: group rings, group algebras, algebras over a field, et cetera. Just to get my definitions, examples, and intuition straight. 3-4 hours. 1 blog post should be enough, I think.
3) QMS from 1-dim Q lattices and the Bost-Connes system. However long it takes!

What I don't understand today:
Connes has a proposition (3.8) about $KMS_\beta$ states...they're extremal iff the GNS representation is factorial. I have an idea of what a GNS rep is, I have no idea what it means for one to be factorial

Same prob mentions that the set of $KMS_\beta$ states is a convex compact Choquet simplex. I've never seen "Choquet Simplex" before. I'll have to do some googling when I'm more awake.

"phase transition" is mentioned often. While I have an idea what this means for a classical thermodynamics system (changing states in matter) I have no idea what it means for a C* dynamical system. I did manage to find some stuff in R. Haag's Local Quantum Physics (around page 213) that I'll have to look at tomorrow.

I can barely keep my eyes open. Goodnight!

Quantum Statistical Mechanics, C* algebra setup, and the KMS Condition

(random tidbit: Connes & Marcolli's "Non-commutative geometry, quantum fields, and motives" is actually a lot more readable than Marcolli's "Lectures on Arithmetic Non-commutative geometry". Connes goes more in-depth on several topics, and explains more of the things that I have never seen before. I'm reading both side-by-side.)

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:

1. Its holomorphic on the strip $\{z \in \mathbb{C} : 0 < \Im(z) < \beta \}$




2. continuous on boundary and bounded




3. $F_{a,b}(t) = \phi(a \sigma_t(b))$ and $F_{a,b}(t+i\beta) = \phi(\sigma_t(b) a)$


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

(holomorphic and bounded.....constant?)As best as I understand things, this condition should ensure that the state is invariant with respect to the time evolution over all $a \in \mathcal{A}$.Connes had a great example of the KMS condition in the finite case, using a finite dimensional Hilbert space. I had planned to write about it, but I'm all out of time!May write more about KMS next time...or may move on! Things I don't understand from today's work:Connes makes several references to representations of C* algebras. I've seen this before while I was collecting literature on group C* algebras and C* algebras in general. More and more I am convinced that I need to spend a couple of weeks studying these things.

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.

the group C* algebra for $\mathcal{L}_n$

Not much progress this weekend. But I had originally written that Bost-Connes type systems where the C* algebra generated by $\mathcal{L}_n / \mathbb{R}^{*}_{+}$ (n=1 for Bost-Connes, n=2 for imaginary quadratic fields.) I got that from the notation $C^{*} (\mathcal{L}_n)$. I've learned a bit more about that symbol.

Its not the C* algebra generated by $\mathcal{L}_n$...its the group C* algebra of $\mathcal{L}_n$, which is apparently the C*-enveloping algebra of $L^1 (\mathcal{L}_n)$ From what my googling and textbook-glancing today has told me, its more related to harmonic analysis and the Haar measure stuff then to the material covered in Averson's Invitation to C* Algebras. The Haar measure is a Borel measure that we can find on any [locally] compact topological group G, according to a theorem in Functional/Harmonic analysis. The notation $L^1 (G)$ is then pretty obvious, its exactly what one would expect from the Lebesque spaces (I think, I need to look into this stuff more.) I have no idea what a C*-enveloping algebra is yet. I'll be looking into all these things from tomorrow onwards.

But its starting to look like $C^{*} (\mathcal{L}_n)$ is a pretty complex object. Its not even obvious to me that $\mathcal{L}_n$ is [locally?] compact! Then again, I'm not worried about that right now. I'm just trying to get the "big picture".