# KMS-states of Bost-Connes type system

I have some struggles with understanding theorem 25 in the paper "Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory".

More precisely, there is a construction of the universal C*-dynamic system $(A, \sigma_t)$, where the algebra of observables $A$ is generated by the elements $\mu_n, e_{\gamma}\;n \in \mathbb{N}_+, \gamma \in \mathbb{Q}/\mathbb{Z}$, subject to relation:

(a) $\mu^*_n\mu_n = 1\;\forall n$

(b) $\mu_n\mu_m = \mu_{nm}\;\forall n,m$

(c) $\mu_n\mu^*_m = \mu_m^*\mu_n\;$ if (m,n) = 1

(d) $e_{\gamma}^* = e_{-\gamma}, e_{\gamma_1 + \gamma_2} = e_{\gamma_1}e_{\gamma_2}\; \forall \gamma, \gamma_1, \gamma_2$

(e) $e_{\gamma}\mu_n = \mu_n e_{n\gamma}$

(f) $\mu_n e_{\gamma} \mu_n^* = \frac{1}{n} \sum_{n\delta = \gamma} e_{\delta}$,

and time evolution is uniquely defined by:

$\sigma_t(\mu_n) = n^{it}\mu_n,\; \sigma_t(e_{\gamma}) = e_{\gamma}$.

Consider the representation $\pi$ of this system on $l_2(\mathbb{N})$, given by the rule:

$\pi(\mu_n)\epsilon_k = \epsilon_{nk},\; \pi(e_{\gamma})\epsilon_k = exp(2\pi ik\gamma) \epsilon_k$.

There is a unbound operator $H$ (Hamiltonian) on $l_2(\mathbb{N}): H(\epsilon_k) = log(k)\epsilon_k.$

Consider the following KMS-state on $A$: $\phi_\beta(a) = \frac{Tr(\pi(a)e^{-\beta H})}{Tr(e^{-\beta H})}$, where $\beta > 1$ to ensure that corresponding operators are of Trace-class.

My question is the following: in the flow of the proof of theorem 25, there is a remark that this state is a factor state of type $I_{\infty}$, and this fact is somehow deduced from $\phi_\beta$ being a KMS-state and the representation $\pi$ being irreducible. As far as I am concerned state is a factor state by definition, if the Von Neumann algebra closure of the GNS representation for the state is a factor, but I do not see any obvious relations between representaion $\pi$ and GNS space for $\phi_\beta.$ What have I missed?

## 1 Answer

$\pi$ is an irreducible representation. So because of Schur lemma and the double commutant theorem the map $\pi:A \rightarrow B(H)$ is surjective.

Now the state $\phi$ is defined as a normale state on $B(H)$, and one easily see from this that this implies that his GNS representation factor through the map $\pi:A \rightarrow B(H)$ and the corresponding GNS representation of $B(H)$. But normale representations of B(H) are all type $I$, and as the map is surjective they are still type $I$ when seen as representation of $A$.

With a little bit of work you should be able to see that these GNS representations are isomorphic to a sum of copies of the representation $\pi$.