So we were working with the groupoid $\mathcal{U}_1 = \{(r,\rho) : r \in \mathbb{Q}^{*}_{+}, \rho \in \hat{\mathbb{Z}}\: \text{such that} \: r \rho \in \hat{\mathbb{Z}}\}$, where composition is defined by $(r_1,\rho_1) \circ (r_2,\rho_2) = (r_1 r_2, \rho_2)$ when $r_2 \rho_2 = \rho_1$. Much in the same way we can define a group algebra, we can also define a groupoid convolution algebra. To get some intuition, let us first talk about the group algebra. For some field $\mathbb{F}$ and [finite] group $G$, the group algebra $\mathbb{F}[G]$ is simply the $\mathbb{F}$-vector space with the group elements as it's basis. Since $G$ is the basis, for each $x \in \mathbb{F}[G]$, we can write
$\displaystyle x = \sum_{g \in G} x_g g \; \text{where} \; x_g \in \mathbb{F} $
The product is simply the group operation (acting on the basis, and thus all elements of the algebra) and is obviously only commutative if the underlying group is Abelian. To make it explicit, let $x$ be as above, $y = \sum_{g \in G} y_g g$ with $y_g \in \mathbb{F}$. Then$\displaystyle x y = \sum_{g, h \in G} (x_g y_h) gh$
Now, any $x \in \mathbb{F}[G]$ defines a function $f_x : G \rightarrow \mathbb{F}$ by $ g \mapsto x_g$. Conversely, any function $f : G \rightarrow \mathbb{F}$ defines an element in the group algebra $f \mapsto \sum_{g \in G} f(g) g$. So that we can also describe the group algebra $\mathbb{F}[G]$ as the set $ \{ f: G \rightarrow \mathbb{F} \} $. The product on this algebra of functions then looks like:$\displaystyle f_x \ast f_y (g) = \sum_{hk =g} x_h y_k = \sum_{hk=g} f_x(h) f_y(k)$
It's not hard to generalize this to a topological group, we simply need the sum $\sum_{hk=g} f_x(h) f_y(k)$ to converge in the field $\mathbb{F}$. Hence we're going to need to talk about the topology of $\mathbb{F}$. So for simplicity, let's just deal with the complex numbers $\mathbb{C}$ with the standard topology. Then $\mathbb{C}[G]$ almost makes sense, we just require that the functions $f \in \mathbb{C}[G]$ have finite support, id est, that they are nonzero only on a finite number of points.
Problem 1 Is the product on $\mathbb{C}[G]$ still finite for a topological group $G$ if we extend it to functions $f$ which have compact support?
Now, for an Etale groupoid $\mathcal{G}$, we can use the above to define its convolution algebra:
Definition 2 (Convolution Algebra of an Etale Groupoid) Let $\mathcal{G}$ be an Etale groupoid. The groupoid (or convolution) algebra $\mathbb{C}[\mathcal{G}]$ is the set$\displaystyle \{ f: \mathcal{G} \rightarrow \mathbb{C} \suchthat \text{f is continuous with compact support} \} $
With pointwise addition, scalar multiplication, and the product$\displaystyle f_1 \ast f_2 (g) = \sum_{g_1 \circ g_2=g} f_1(g_1)f_2(g_2)$
Problem 3 Why is continuity needed here? And what is it about Etale groupoids that allow finite sums when the functions only have compact, not finite, support?
Note that since $f \in \mathbb{C}[\mathcal{G}]$ is simply a function $\mathcal{G} \rightarrow \mathbb{C}$, we can define a $\ast$-operation by complex conjugation, id est, $f^{\ast}(g) = \overline{f(g)}$.
0.1. Completion into a $C^{\ast}$-algebra
According to Connes and Marcolli, we can complete $\mathbb{C}[\mathcal{G}]$ into a $C^{\ast}$-algebra via the following procedure, but first we need to define a few extra terms. Recall from my writeup on groupoids that $\mathcal{G}^0$ denotes set of objects of the groupoid, id est, $\mathcal{G}^0 = \{ \gamma \circ \gamma^{-1} \suchthat \gamma \in \mathcal{G}\}$. For all $y \in \mathcal{G}^0$, let $\mathcal{G}_y$ denote the set of all morphisms whose target and source is $y$. Then all elements in $\mathcal{G}_y$ can be composed with each other, and the category-theoretic definition of a groupoid gives us an identity and inverse, so $\mathcal{G}_y$ is a group.
Definition 4 (Isotropy Group) $\mathcal{G}_y$ is called the Isotropy Group at $y \in \mathcal{G}^0$.
Now for each isotropy group $\mathcal{G}_y$ (or any group, really), we can associate to it a Hilbert space
$\displaystyle \ell^2(\mathcal{G}_y) = \{ \alpha:\mathcal{G}_y \rightarrow \mathbb{C} \suchthat \sum_{g \in \mathcal{G}_y} |\alpha(g)|^2 < \infty \}$
With the inner product:$\displaystyle <\alpha_1,\alpha_2> = \sum_{g \in \mathcal{G}_y} \alpha_1(g)\overline{\alpha_2(g)}.$
Let $f \in \mathbb{C}[\mathcal{G}]$. For all $\alpha \in \ell^2(\mathcal{G}_y)$ we can define a map:$\displaystyle \beta_{\alpha} : \mathcal{G}_y \rightarrow \mathbb{C} \; \text{by} \; \beta(g) = \sum_{\substack{g_1 g_2 = g \\ g_1,g_2 \in \mathcal{G}_y}} f(g_1)\alpha(g_2)$
Problem 5 Show that $\beta_{\alpha} (g)$ converges, and show that $\beta_{\alpha} \in \ell^2(\mathcal{G}_y)$.
Problem 6 Hence we have a map $B_f:\ell^2(\mathcal{G}_y) \rightarrow \ell^2(\mathcal{G}_y)$ by $B(\alpha) = \beta_{\alpha}$. Show that $B_f \in \mathcal{B}(\ell^2(\mathcal{G}_y))$, id est, show that $B_f$ is a bounded [linear] operator.
Problem 7 Now we have a map $\pi_y : \mathbb{C}[\mathcal{G}] \rightarrow \mathcal{B}(\ell^2(\mathcal{G}_y))$ by $\pi_y(f) = B_f$ for all $y \in \mathcal{G}^0$. Show that this map is a representation of the groupoid convolution algebra, id est, that it's a homomorphism of algebras.
This representation allows us to define a norm on the groupoid convolution algebra. Specifically, we have:
$\displaystyle \|f\| = \sup_{y\in\mathcal{G}^0} \| \pi_y(f) \|_{\mathcal{B}(\ell^2(\mathcal{G}_y))$
This has all been a bit confusing. Let us recap briefly: we noted that at each object in the groupoid we can find an isotropy group. We built a Hilbert space on that isotropy group and then constructed a representation on $\mathbb{C}[\mathcal{G}]$ to that Hilbert space. We defined the norm of an $f \in \mathbb{C}[\mathcal{G}]$ in our groupoid convolution algebra as the supremum of the operator norms of all such representations of $f$.Connes and Marcolli state that if $\mathcal{G}^0$ is compact then $\mathbb{C}[\mathcal{G}]$ has a unit, while Khalkahali holds that $\mathbb{C}[\mathcal{G}]$ has a unit if and only if $\mathcal{G}^0$ is finite. Does compactness of the objects in an Etale groupoid imply that the objects are finite? In any case, since the algebra has a unit, we can complete it. Call its completion $C^{\ast}[\mathcal{G}]$.
Problem 8 Show that $\mathbb{C}[\mathcal{G}]$ is incomplete.
Problem 9 Prove that $C^{\ast}[\mathcal{G}]$ is a $C^{\ast}$-algebra, namely, show that the $C^{\ast}$-identity, $\| f f^{\ast} \| = \| f \|^2$, holds. We've established that it's a complete $\ast$-algebra.
Problem 10 Show that our groupoid $\mathcal{U}_1$ is unital.
In a previous post I described a time evolution on $C^{\ast}[\mathcal{U}_1]$ using the ratio of the covolumes of the underlying pair of commensurable $\mathbb{Q}$-lattices. Thus, except for the problems mentioned here, I've fully constructed the $C^{\ast}$-algebra Bost-Connes system.
0.2. What's next?
Considering my embarrassingly slow rate of progress, I think it's unreasonable for me to expect that I would progress beyond a detailed, in-depth description of the Bost-Connes system and it's interaction with the Class Field Theory of $\mathbb{Q}$. (I would very much appreciate advice from both of my supervisors on this point!) Which is unfortunate, as I would eventually like to discuss the two other systems mentioned in the Connes and Marcolli monograph (including the system that interacts with the Class Field Theory of an imaginary number field). Moreover, I'd especially like to visit the work of Ha and Paugam in their paper "Bost-Connes-Marcolli systems for Shimura varieties" which generalizes the construction of these systems in the way described in Connes' monograph. But again, I think it's unlikely that I would have time for that after finishing my study, and filling in the details, of the original Bost-Connes system. To that end, I still need to:
- State and prove the explicit presentation of the Bost-Connes System. I've been wholly unable to make sense of the proof given in the Connes and Marcolli monograph, and have spent some time staring at the Hecke algebra description (I still don't know what a "Hecke algebra" is.) given in the 1994 paper by Bost and Connes. Additionally, Dr Javier Lopez, who has graciously offered to assist me, suggested that I have a look through Fun with $\mathbb{F}_1$ to gather more intuition.
- Describe the ``arithmetic sub-algebra'' of the Bost-Connes System.
- Class Field Theory. I figure I need to understand at least enough CFT to be able to prove the Kronecker-Weber theorem and to state properly its interactions with the CFT of $\mathbb{Q}$, which I mentioned in in my talk at the Undergrad Maths Colloquium.
- Prove the statements about such interactions.
My next post (on 1.) should come soon.