$C^*$-algebras of Toeplitz type associated with algebraic number fields

Abstract

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra ${\mathfrak T }[R]$. It is an extension of the ring C*-algebra ${\mathfrak A }[R]$ studied previously by the first named author in collaboration with X. Li. In contrast to ${\mathfrak A }[R]$, it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$-semigroup $R\rtimes R^\times$ on $\ell ^2 (R\rtimes R^\times )$. The algebra ${\mathfrak T }[R]$ carries a natural one-parameter automorphism group $(\sigma _t)_{t\in {\mathbb R }}$. We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The “partition functions” are partial Dedekind $\zeta$-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $\zeta$-functions, which we then use to show uniqueness of the $\beta$-KMS state for each inverse temperature $\beta \in (1,2]$.

Appendices

Appendix A: Asymptotics for partial $\zeta$-functions

As above let $R$ be the ring of algebraic integers in a number field $K$. Also let $P_1,P_2,\ldots$ be an enumeration of the prime ideals in $R$ such that $N (P_i) \le N (P_{i+1})$ for all $i \ge 1$ and let $\mathcal I _n$ be the semigroup generated by $P_1,P_2,\ldots ,P_n$. For each $\gamma$ in the class group $\Gamma$ of $K$ and each $0<\sigma \le 1$ set

$$\begin{aligned} \zeta ^{(n)}_\gamma (\sigma ) =\sum _{I\in \mathcal I _n\cap \gamma }N(I)^{-\sigma } \end{aligned}$$

Recall the statement of Theorem 6.6:   Let $0<\sigma \le 1$. Then for any two ideal classes $\gamma _1,\gamma _2$ we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\;\frac{\zeta ^{(n)}_{\gamma _1} (\sigma )}{\zeta ^{(n)}_{\gamma _2} (\sigma )}\;=1. \end{aligned}$$

Proof of Theorem 6.6

Let $\psi _\gamma :\Gamma \rightarrow \{0,1\}$ denote the characteristic function of the one-point set $\{\gamma \}$. For every character $\chi$ of the abelian group $\Gamma$ let $a_\gamma (\chi )= |\Gamma |^{-1} \overline{\chi (\gamma )}$ so that

$$\begin{aligned} \psi _\gamma =\sum _{\chi \in \hat{\Gamma }}a_\gamma (\chi )\chi \end{aligned}$$

In the following we also consider $\chi$ and $\psi _\gamma$ as functions on the set of non-zero integral ideals. We have

$$\begin{aligned} \zeta _\gamma ^{(n)}(\sigma )&= \sum _{I\in \mathcal I _n}\psi _{\gamma } (I)N(I)^{-\sigma }= \sum _{\chi \in \hat{\Gamma }}\left(a_\gamma (\chi )\sum _{I\in \mathcal I _n}\chi (I)N(I)^{-\sigma }\right)\nonumber \\&= \sum _{\chi \in \hat{\Gamma }}\left(a_\gamma (\chi ) \prod _{i=1}^n \left(1-\chi (P_i)N(P_i)^{-\sigma }\right)^{-1}\right) \end{aligned}$$

(13)

In order to study the asymptotics of $\prod _{i=1}^n \left(1-\chi (P_i)N(P_i)^{-\sigma }\right)^{-1}$ for $n\rightarrow \infty$ we consider

$$\begin{aligned} f_n(\chi ,\sigma )&= \log \prod _{i=1}^n \left(1-\chi (P_i)N(P_i)^{-\sigma }\right)^{-1}\nonumber \\&:= \sum _{\nu =1}^\infty \frac{1}{\nu }\sum _{i=1}^n \chi (P_i^\nu )N(P_i)^{-\nu \sigma } \end{aligned}$$

(14)

Up to finitely many terms the first sum is bounded by a constant which is independent of $n$:

$$\begin{aligned} \left| \sum _{\nu > 1/\sigma } \frac{1}{\nu } \sum ^n_{i=1} \chi (P^{\nu }_i) N (P_i)^{-\nu \sigma }\right|&\le \sum _{\nu > 1/\sigma } \frac{1}{\nu } \sum ^{\infty }_{i=1} N (P_i)^{-\nu \sigma } \\&= \sum ^{\infty }_{i=1} N(P_i)^{-\sigma [1/\sigma ]} \sum ^{\infty }_{\nu =1} \frac{N (P_i)^{-\nu \sigma }}{\nu + [1/ \sigma ]} \\&\le \sum ^{\infty }_{i=1} N (P_i)^{-\sigma [1/\sigma ]} \frac{N(P_i)^{-\sigma }}{1-N (P_i)^{-\sigma }} \\&\le \frac{1}{1-2^{-\sigma }} \sum ^{\infty }_{i=1} N (P_i)^{-\sigma (1 + [1/\sigma ])} \\&< \frac{1}{1-2^{-\sigma }} \zeta _K (\sigma (1 + [1/\sigma ]) < \infty . \end{aligned}$$

Therefore

$$\begin{aligned} f_n (\chi , \sigma ) =\sum _{1\le \nu \le 1/\sigma }\frac{1}{\nu }\sum _{i=1}^n \chi (P_i^\nu )N(P_i)^{-\nu \sigma }\,+\,O(1) \end{aligned}$$

(15)

where the $O$-constant depends on $\sigma$ but not on $n$ or $\chi$.

Let us now fix some $1 \le \nu \le 1/\sigma$. The values of $\chi$ are $h$th roots of unity where $h=|\Gamma |$ is the class number. We get

$$\begin{aligned} \sum ^n_{i=1} \chi (P^{\nu }_i) N(P_i)^{-\nu \sigma } = \sum _{\zeta ^h=1} \zeta ^{\nu } \sum _{\gamma \in \chi ^{-1} (\zeta )} \omega ^{(\nu \sigma )}_{\gamma } (n). \end{aligned}$$

(16)

Here for $\kappa \in \mathbb R , \gamma \in \Gamma$ and $n \ge 1$ we have set:

$$\begin{aligned} \omega ^{(\kappa )}_{\gamma } (n) = \sum \limits ^{n}_{{\small \begin{array}{c}i=1\\ P_i \in \gamma \end{array}}}N (P_i)^{-\kappa }. \end{aligned}$$

Lemma A.1

Fix some $0 \le \kappa \le 1$ and write $\omega _{\gamma } (n) = \omega ^{(\kappa )}_{\gamma } (n)$. Set

$$\begin{aligned} \omega (n) = \frac{1}{h} \sum ^n_{i=1} N (P_i)^{-\kappa }. \end{aligned}$$

Then $\omega (n) \rightarrow \infty$ as $n \rightarrow \infty$ and for arbitrary $\gamma \in \Gamma$ we have $\lim _{n\rightarrow \infty } \frac{\omega _{\gamma } (n)}{\omega (n)} = 1$.

The proof of the lemma is given below. For $1 \le \nu \le 1/\sigma$ we have $0 < \kappa = \nu \sigma \le 1$. Using (16) and the lemma, we get for $n \rightarrow \infty$:

$$\begin{aligned} \frac{1}{\omega (n)} \sum ^n_{i=1} \chi (P^{\nu }_i) N (P_i)^{-\nu \sigma } \rightarrow \sum _{\zeta ^h=1} \zeta ^{\nu } |\chi ^{-1} (\zeta )|. \end{aligned}$$

We have the identities

$$\begin{aligned} \sum _{\zeta ^h =1}\zeta ^\nu |\chi ^{-1}(\zeta )|= |\mathrm{Ker\,}(\chi )|\sum _{\zeta \in \mathrm{Im\,}\chi }\zeta ^\nu =\left\{ \begin{array}{l} h \quad \text{ if}\;|\mathrm{Im\,}\chi |\,\mid \,\nu \\ 0\quad \text{ if}\;|\mathrm{Im\,}\chi |\,\not \mid \,\nu \end{array}\right. \end{aligned}$$

Therefore, using (15) we get

$$\begin{aligned} \lim _{n\rightarrow \infty } \; \frac{1}{\omega (n)}f_n(\chi ,\sigma ) = \alpha (\chi ):= h\sum _{1\le \nu \le 1/\sigma ,\,|\mathrm{Im\,}\chi |\big |\nu }\frac{1}{\nu } \ge 0 \end{aligned}$$

(17)

Note that if $\chi$ is not the trivial character $\mathbf 1$, then $\alpha (\chi )<\alpha (\mathbf 1 )$.

Let

$$\begin{aligned} L_n(\chi ,\sigma )\mathrel {:=}\prod _{i=1}^n \left(1-\chi (P_i)N(P_i)^{-\sigma }\right)^{-1}=\exp f_n(\chi ,\sigma ) \end{aligned}$$

(18)

From (13) we get

$$\begin{aligned} \zeta ^{(n)}_\gamma (\sigma ) = \sum _\chi a_\gamma (\chi )L_n(\chi ,\sigma ) \end{aligned}$$

Because of (17) and (18) one knows that for $n\rightarrow \infty$

$$\begin{aligned} 0\,<\,L_n(\mathbf 1 ,\sigma ) = \prod _{i=1}^n \left(1-N(P_i)^{-\sigma } \right)^{-1}\longrightarrow \infty \end{aligned}$$

Also

$$\begin{aligned} \Big |\frac{L_n(\chi ,\sigma )}{L_n(\mathbf 1 ,\sigma )}\Big |\,=\,\exp \mathrm Re \left(f_n(\chi ,\sigma )-f_n(\mathbf 1 ,\sigma )\right) \end{aligned}$$

Now assume that $\chi \ne \mathbf 1$. Since $\omega (n)\rightarrow \infty$ and

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{1}{\omega (n)}\left(f_n(\chi ,\sigma )-f_n(\mathbf 1 ,\sigma )\right) = \alpha (\chi )-\alpha (\mathbf 1 )\,<\,0 \end{aligned}$$

by (17), we find that

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathrm Re \left(f_n(\chi ,\sigma )-f_n(\mathbf 1 ,\sigma )\right) =\,-\infty \end{aligned}$$

and thus

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{L_n(\chi ,\sigma )}{L_n(\mathbf 1 ,\sigma )}=\,0\quad \mathrm{for}\,\chi \ne \mathbf 1 \end{aligned}$$

This gives:

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\zeta ^{(n)}_\gamma (\sigma )}{L_n(\mathbf 1 ,\sigma )}=a_\gamma (\mathbf 1 ) =\frac{1}{h} \end{aligned}$$

and hence

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\zeta ^{(n)}_\gamma (\sigma )}{\zeta ^{(n)}_\eta (\sigma )}=\,1 \end{aligned}$$

for any two ideal classes $\gamma$ and $\eta$.

It remains to prove lemma A.1. For this we need a version of the prime number theorem for prime ideals in a given ideal class with a simple remainder term. For $x \ge 0$ let $\pi _K (\gamma , x)$ denote the number of prime ideals $P$ in $\gamma$ with $N (P) \le x$. Using the relation

$$\begin{aligned} \mathrm li \, (x) = \frac{x}{\log x} + O \left( \frac{x}{(\log x)^2} \right) \quad \text{ for} \; x \rightarrow \infty \end{aligned}$$

the corollary after lemma 7.6 in chap. 7, § 2 of [17] implies the following asymptotics:

$$\begin{aligned} \pi _K (\gamma , x) = \frac{1}{h} \frac{x}{\log x} + O \left( \frac{x}{(\log x)^2} \right) \end{aligned}$$

(19)

For $x \ge 0$ and $\kappa \le 1$ let us write:

$$\begin{aligned} \Omega _{\gamma } (x) = \Omega ^{(\kappa )}_{\gamma } (x) = {\mathop {\mathop {\sum \limits _{N (P) \le x}}\limits _{P \in \gamma }}} N (P)^{-\kappa } \end{aligned}$$

and

$$\begin{aligned} \Omega (x) = \Omega ^{(\kappa )} (x) = \frac{1}{h} \sum _{N (P) \le x} N (P)^{-\kappa }. \end{aligned}$$

We now use the following version of summation by parts: Consider a function $f$ on the integers $\nu \ge 1$ and a $C^1$-function $g$ on $[1,\infty )$. For $x \ge 1$ we set $M_f (x) = \sum _{\nu \le x} f (\nu )$. Then we have

$$\begin{aligned} \sum _{\nu \le x} f (\nu ) g (\nu ) = M_f (x) g (x) - \int ^x_1 M_f (t) g^{\prime } (t)\,dt. \end{aligned}$$

Setting $f (\nu ) = \big | \{ P \, | \,P \in \gamma$ and $N (P) = \nu \}\big |$ and $g (x) = x^{-\kappa }$ we have

$$\begin{aligned} \Omega _{\gamma } (x) = \sum _{\nu \le x} f (\nu ) g (\nu ) \quad \text{ and} \quad M_f (x) = \pi _K (\gamma , x). \end{aligned}$$

Hence using (19) we get for $x \rightarrow \infty$:

$$\begin{aligned} \Omega _{\gamma } (x)&= \pi _K (\gamma , x) x^{-\kappa } + \kappa \int ^x_2 \pi _K (\gamma , t) t^{-\kappa } \frac{dt}{t}\\&= \frac{1}{h} \frac{x^{1-\kappa }}{\log x} + \frac{\kappa }{h} \int ^x_2 \frac{t^{-\kappa }}{\log t}\,dt + O \left( \frac{x^{1-\kappa }}{(\log x)^2} \right) + O \left( \int ^x_2 \frac{t^{-\kappa }}{(\log t)^2}\,dt \right). \end{aligned}$$

For $\kappa < 1$ we have:

$$\begin{aligned} \int ^x_e \frac{t^{-\kappa }}{(\log t)^2}\,dt&= \int ^{\sqrt{x}}_e \frac{t^{-\kappa }}{(\log t)^2}\,dt + \int ^x_{\sqrt{x}} \frac{t^{-\kappa }}{(\log t)^2}\,dt \\&\le \int ^{\sqrt{x}}_e t^{-\kappa }\,dt + \frac{1}{(\log \sqrt{x})^2} \int ^x_{\sqrt{x}} t^{-\kappa }\,dt\\&= O \left( \frac{x^{1-\kappa }}{(\log x)^2} \right)\;. \end{aligned}$$

Hence we get for $\kappa < 1$:

$$\begin{aligned} \Omega _{\gamma } (x) = \frac{1}{h} \frac{x^{1-\kappa }}{\log x} + \frac{\kappa }{h} \int ^x_2 \frac{t^{-\kappa }}{\log t}\,dt + O \left( \frac{x^{1-\kappa }}{(\log x)^2} \right). \end{aligned}$$

(20)

For the case $\kappa = 1$ note that

$$\begin{aligned} \int ^x_2 \frac{t^{-1}}{(\log t)^2} \, dt = \frac{1}{\log 2} - \frac{1}{\log x} = O (1) \end{aligned}$$

and

$$\begin{aligned} \int ^x_2 \frac{t^{-1}}{\log t} \, dt = \log \log x + O (1). \end{aligned}$$

Thus for $\kappa = 1$ we get

$$\begin{aligned} \Omega _{\gamma } (x) = \frac{1}{h} \log \log x + O (1). \end{aligned}$$

(21)

Relations (20) and (21) also hold for $\Omega (x)$ instead of $\Omega _{\gamma } (x)$ since the right hand sides do not depend on $\gamma$ and $\Omega (x) = h^{-1} \sum _{\gamma \in \Gamma } \Omega _{\gamma } (x)$. It follows that for $\kappa \le 1$ we have $\Omega _{\gamma } (x) \sim \Omega (x)$. It remains to show that for $n \rightarrow \infty$ we have $\omega _{\gamma } (n) \sim \omega (n)$ as well. For a given prime number $p$ there are at most $(K: \mathbb Q )$ different prime ideals $P$ in $R$ with $P \, | \,p$. It follows that for every $\nu \ge 1$ the equation $N (P) = \nu$ has at most $(K: \mathbb Q )$ solutions in primes $P$ of $R$. Since $N (P_i) \le N (P_{i+1})$ for all $i$ we therefore get:

$$\begin{aligned} \omega _{\gamma } (n)&= \Omega _{\gamma } (N (P_n)) + O (N (P_n)^{-\kappa })\\&= \Omega _{\gamma } (N (P_n)) + O (1) \quad \text{ since} \; \kappa \ge 0 \end{aligned}$$

and analogously

$$\begin{aligned} \omega (n) = \Omega (N (P_n)) + O (1). \end{aligned}$$

This implies the result.$\square$

Appendix B: List of notations

For $u^x,s_a,e_I$ see Definition 2.1. The projections $f_I,\varepsilon _I$ are introduced before Lemma 2.4. The commutative subalgebra $\bar{\mathcal D }$ is introduced at the beginning of Sect.4. $\mathcal I _n,\mathcal D _n,\bar{\mathcal D }_n$ are introduced after Lemma 4.5. The representation $\mu$ of $\mathfrak T [R]$ is defined before Lemma 4.6. The minimal projections $\delta ^x_{I,n}$ in $\mathcal D _n$ are introduced in Lemma 4.7. $e_I^x$ is defined after 4.8. For $Y_R$ see Remark 4.10. The notation $\mathfrak T$ is introduced after Corollary 4.14. The automorphism $\sigma _t$ is defined at the beginning of Sect.6. $R^*$ denotes the group of units (invertible elements) in $R$.