$C^*$-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms

1.   Introduction

In [22] and [23], D. Ruelle has introduced the notion of Smale space. A Smale space is a hyperbolic dynamical system with local product structure. He has constructed groupoids and its operator algebras from the Smale spaces. After the Ruelle's initial study, I. Putnam in [14] (cf. [9], [15], [16], [17], [26], etc.) constructed various groupoids from Smale spaces and studied their $C^*$-algebras. The class of Smale spaces contain two important subclasses of topological dynamical systems as its typical examples. One is the class of shifts of finite type, which are sometimes called topological Markov shifts. The other one is the class of hyperbolic toral automorphisms. The study of the former class from the view point of $C^*$-algebras is closely related to the study of Cuntz-Krieger algebras as in [7], [8], [10], [11], [12], etc. That of the latter class is closely related to the study of the crossed product $C^*$-algebras of the homeomorphisms of the hyperbolic automorphisms on the torus.

In this paper, we will focus on the study of the latter class, the hyperbolic toral automorphisms from the view points of $C^*$-algebras constructed from the associated groupoids as Smale spaces. Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in {{{\operatorname{GL}}}} (2, \mathbb{Z})$ be a hyperbolic matrix. Let $q: \mathbb{R}^2 \longrightarrow \mathbb{R}^2/ \mathbb{Z}^2$ be the natural quotient map. We denote by $\mathbb{R}^2/ \mathbb{Z}^2$ the two-dimensional torus $\mathbb{T}^2$ with metric $d$ defined by

where $\| \cdot \|$ is the Euclid norm on $\mathbb{R}^2.$ Then the matrix $A$ defines a homeomorphism on $\mathbb{T}^2$ which is called a hyperbolic toral automorphism. It is a specific example of an Anosov diffeomorphism on a compact Riemannian manifold (see [4], [25], etc.). Let $\lambda_u, \lambda_s$ be the eigenvalues of $A$ such that $|\lambda_u| > 1 > |\lambda_s|.$ They are both real numbers. Let $v_u = (u_1, u_2), v_s = (s_1,s_2)$ be the normalized eigenvectors for $\lambda_u, \lambda_s$, respectively. The direction along $v_u$ expands by $A$, whereas the direction of $v_s$ expands by $A^{-1}$. These directions determine local product structure which makes $\mathbb{T}^2$ a Smale space. The groupoid $G_A^a$ introduced by D. Ruelle [22] of the asymptotic equivalence relation is defined by

with its unit space

The multiplication and the inverse operation on $G_A^a$ are defined by

As in [14], the groupoid $G_A^a$ has a natural topology defined by inductive limit topology, which makes $G_A^a$ étale. The étale groupoid $G_A^a$ is called the asymptotic groupoid for the hyperbolic toral automorphism $( \mathbb{T}^2, A)$. We will first see that the groupoid $G_A^a$ is realized as a transformation groupoid $\mathbb{T}^2\times_{\alpha^A} \mathbb{Z}^2$ by a certain action $\alpha^A: \mathbb{Z}^2 \longrightarrow {{{\operatorname{Homeo}}}}( \mathbb{T}^2)$ associated to $G_A^a$, so that the $C^*$-algebra $C^*(G_A^a)$ of the groupoid $G_A^a$ is isomorphic to the $C^*$-algebra of the crossed product $C( \mathbb{T}^2) \times_{\alpha^A} \mathbb{Z}^2$ by the induced action $\alpha^A: \mathbb{Z}^2\longrightarrow {{{\operatorname{Aut}}}}(C( \mathbb{T}^2))$. As the action $\alpha^A: \mathbb{Z}^2 \longrightarrow {{{\operatorname{Homeo}}}}( \mathbb{T}^2)$ is free and minimal having a unique invariant ergodic measure, a general theory of $C^*$-crossed product ensures that $C( \mathbb{T}^2) \times_{\alpha^A} \mathbb{Z}^2$ is a simple AT-algebra having a unique tracial state (cf. [13], [14], [17]).

Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in {{{\operatorname{GL}}}} (2, \mathbb{Z})$ be a hyperbolic matrix which satisfies ${{{\operatorname{det}}}}(A) = \pm 1,$ We denote by $\Delta(A) = (a+d)^2 - 4 (ad-bc)$ the discriminant of the characteristic polynomial of the matrix $A$, which is positive. We will show the following result.

Theorem 1.1 (Theorem 2.10 and Proposition 3.1). The $C^*$-algebra $C^*(G_A^a)$ of the étale groupoid $G_A^a$ for a hyperbolic matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is a simple AT-algebra with unique tracial state $\tau$ that is isomorphic to the four-dimensional non-commutative torus generated by four unitaries $U_1, U_2, V_1, V_2$ satisfying the following relations:

where

The range $\tau_*(K_0(C^*(G_A^a)))$ of the tracial state $\tau$ of the $K_0$-group $K_0(C^*(G_A^a))$ of the $C^*$-algebra $C^*(G_A^a)$ is

We note that the slopes $\theta_i, i = 1,2,3,4$ are determined by the formulas (10), (11) for the slopes of the eigenvectors $v_u = (u_1, u_2), v_s = (s_1, s_2).$

Since the étale groupoid $G_A^a$ is a flip conjugacy invariant and the $C^*$-algebra $C^*(G_A^a)$ has a unique tracial state written $\tau$, we know that the trace value $\tau_*(K_0(C^* (G_A^a)))$ is a flip conjugacy invariant of the hyperbolic toral automorphism $( \mathbb{T}^2,A).$

As commuting matrices have common eigenvectors, we know that if two matrices $A, B \in {{{\operatorname{GL}}}}(2, \mathbb{Z})$ commute with each other, then the $C^*$-algebras $C^*(G_A^a)$ and $C^*(G_B^a)$ are canonically isomorphic. Hence two matrices $\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$ and $\begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$ have the isomorphic $C^*$-algebras. On the other hand, as the range $\tau_*(K_0(C^*(G_A^a)))$ of the tracial state of the $K_0$-group $K_0(C^*(G_A^a))$ is invariant under isomorphism class of the algebra $C^*(G_A^a)$, the $C^*$-algebra $C^*(G_{A_1})$ is not isomorphic to $C^*(G_{A_2})$ for the matrices $A_1 = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$ and $A_2 = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix}$ (Proposition 4.2).

2.   The groupoid $G_A^a$ and its $C^*$-algebra $C^*(G_A^a)$

For a vector $(m,n) \in \mathbb{R}^2$, we write the vector $(m,n)^t$ as $\begin{bmatrix} m\\ n \end{bmatrix}$ and sometimes identify $(m,n)$ with $\begin{bmatrix} m\\ n \end{bmatrix}.$ A matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in {{{\operatorname{GL}}}} (2, \mathbb{Z})$ with ${{{\operatorname{det}}}}(A) = \pm 1$ is said to be hyperbolic if $A$ does not have eigenvalues of modulus $1$. Let $\lambda_u, \lambda_s$ be the eigenvalues of $A$ such that $|\lambda_u| > 1 > |\lambda_s|.$ They are eigenvalues for unstable direction, stable direction, respectively. We note that $b \ne 0, c \ne 0$ because of the conditions $ad-bc = \pm 1$ and $|\lambda_u| > 1 > |\lambda_s|.$ Take nonzero eigenvectors $v_u, v_s$ for the eigenvalues $\lambda_u, \lambda_s$ such that $\|v_u \| = \| v_s \| = 1.$ We set $v_u = (u_1, u_2), v_s = (s_1, s_2) \in \mathbb{T}^2$ as vectors. The numbers $\lambda_u, \lambda_s, u_1, u_2, s_1, s_2$ are all real numbers because of the hyperbolicity of the matrix $A$. It is easy to see that the slopes $\frac{u_1}{u_2},\frac{s_1}{s_2}$ are irrational. We set

Define two vectors

Lemma 2.1. For two vectors $x, z \in \mathbb{T}^2$, the following three conditions are equivalent.

(ⅰ) $(x,z) \in G_A^a.$

(ⅱ) $z = x + \frac{1}{1-r_A^2}\langle (m,n)|v_1\rangle v_u$ for some $m,n\in \mathbb{Z}.$

(ⅲ) $z = x + \frac{1}{1-r_A^2}\langle (m,n)|v_2\rangle v_s$ for some $m,n\in \mathbb{Z}.$

Proof. For two vectors $x, z \in \mathbb{T}^2$ regarding them as elements of $\mathbb{R}^2$ modulo $\mathbb{Z}^2,$ we have $(x,z) \in G_A^a$ if and only if

In this case, we see that $tv_u - s v_s = (m,n)$ for some $m,n\in \mathbb{Z}$ so that

and we have

This shows the implications (ⅰ) $\Longrightarrow$ (ⅱ) and (ⅲ).

Assume that (ⅱ) holds. By putting $s = \frac{1}{1-r_A^2}\langle (m,n)|v_2\rangle,$ we have the equalities both (5) and (6), so that $tv_u - s v_s = (m,n)$. Hence the equality (4) holds and we see that $(x,z)$ belongs to the groupoid $G_A^a.$ This shows that the implication (ⅱ) $\Longrightarrow$ (ⅰ) holds, and similarly (ⅲ) $\Longrightarrow$ (ⅰ) holds.

Let us define an action $\alpha^A: \mathbb{Z}^2 \longrightarrow {{{\operatorname{Homeo}}}}( \mathbb{T}^2)$ in the following way. We set

For a fixed $(m,n) \in \mathbb{Z}^2,$ the map $x \in \mathbb{T}^2 \longrightarrow \alpha^A_{(m,n)}(x) \in \mathbb{T}^2$ is the parallel transformation along the vector $\frac{1}{1-r_A^2}\langle (m,n)|v_1\rangle v_u.$ Hence $\alpha^A_{(m,n)}$ defines a homeomorphism on the torus $\mathbb{T}^2$. It is clear to see that $\alpha^A_{(m,n)}\circ \alpha^A_{(k,l)} = \alpha^A_{(m+k,n+l)}$ for $(m,n), (k,l) \in \mathbb{Z}^2$.

Lemma 2.2. Keep the above notation.

(ⅰ) If $\alpha^A_{(m,n)}(x) = x$ for some $x \in \mathbb{T}^2$, then $(m,n) = (0,0)$. Hence the action $\alpha^A: \mathbb{Z}^2 \longrightarrow {{{\operatorname{Homeo}}}}( \mathbb{T}^2)$ is free.

(ⅱ) For $x \in \mathbb{T}^2$, the set $\{\alpha^A_{(m,n)}(x) | \in (m,n) \in \mathbb{Z}^2\}$ is dense in $\mathbb{T}^2$. Hence the action $\alpha^A: \mathbb{Z}^2 \longrightarrow {{{\operatorname{Homeo}}}}( \mathbb{T}^2)$ is minimal.

Proof. (ⅰ) Suppose that $\alpha^A_{(m,n)}(x) = x$ for some $x \in \mathbb{T}^2,$ so that $\frac{1}{1-r_A^2}\langle (m,n)|v_1\rangle v_u = (k,l)$ for some $(k,l) \in \mathbb{Z}^2.$ As the slope of the vector $v_u$ is irrational, we have $(k,l) = (0,0)$ and hence $(m,n) = (0,0)$.

(ⅱ) Let $v_1 = (\gamma_1,\gamma_2).$ As the slope of $v_s$ is irrational and $\langle v_s | v_1\rangle = 0,$ the slope $\frac{\gamma_1}{\gamma_2}$ of $v_1$ is irrational, so that the set $\{ m\gamma_1 + n \gamma_2 | m,n \in \mathbb{Z}\}$ is dense in $\mathbb{R}.$ Since $\langle (m,n) | v_1\rangle v_u = (m\gamma_1 + n \gamma_2 )v_u$ and the set $\{ x + t v_u \in \mathbb{T}^2 | t \in \mathbb{R}\}$ is dense in $\mathbb{T}^2,$ we see that the set

is dense in $\mathbb{T}^2.$

The action $\alpha^A: \mathbb{Z}^2\longrightarrow {{{\operatorname{Homeo}}}}( \mathbb{T}^2)$ induces an action of $\mathbb{Z}^2$ to the automorphism group ${{{\operatorname{Aut}}}}(C( \mathbb{T}^2))$ of $C( \mathbb{T}^2)$ by $f \in C( \mathbb{T}^2) \longrightarrow f\circ\alpha^A_{(m,n)} \in C( \mathbb{T}^2).$ We write it still $\alpha^A$ without confusing.

If a discrete group $\Gamma$ acts freely on a compact Hausdorff space $X$ by an action $\alpha:\Gamma \longrightarrow {{{\operatorname{Homeo}}}}(X),$ the set $\{(x,\alpha_\gamma(x)) \in X\times X | x \in X, \gamma \in \Gamma\}$ has a groupoid structure in a natural way (cf. [2], [18], [19]). The groupoid is called a transformation groupoid written $X\times_\alpha\Gamma$.

Proposition 2.3. The étale groupoid $G_A^a$ is isomorphic to the transformation groupoid

defined by the action $\alpha^A: \mathbb{Z}^2\longrightarrow {{{\operatorname{Homeo}}}}( \mathbb{T}^2).$ Hence the $C^*$-algebra $C^*(G_A^a)$ of the groupoid $G_A^a$ is isomorphic to the crossed product $C( \mathbb{T}^2)\times_{\alpha^A} \mathbb{Z}^2$ of $C( \mathbb{T}^2)$ by the action $\alpha^A$ of $\mathbb{Z}^2.$

Proof. By the preceding discussions, a pair $(x,z) \in \mathbb{T}^2$ belongs to the groupoid $G_A^a$ if and only if $z = \alpha^A_{(m,n)}(x)$ for some $(m,n) \in \mathbb{Z}^2.$ Since the action $\alpha^A: \mathbb{Z}^2 \longrightarrow {{{\operatorname{Homeo}}}}( \mathbb{T}^2)$ is free, the groupoid $G_A^a$ is identified with the transformation groupoid $\mathbb{T}^2 \times_{\alpha^A} \mathbb{Z}^2$ in a natural way. By a general theory of the $C^*$-algebras of groupoids ([2], [18]), the $C^*$-algebra $C^*( \mathbb{T}^2 \times_{\alpha^A} \mathbb{Z}^2)$ of the groupoid $\mathbb{T}^2 \times_{\alpha^A} \mathbb{Z}^2$ is isomorphic to the crossed product $C^*( \mathbb{T}^2) \times_{\alpha^A} \mathbb{Z}^2.$

Remark 2.4. Define a map $\alpha_A: \mathbb{Z}^2 \longrightarrow \mathbb{T}^2$ by

It is easy to see that the étale groupoid $G_A^a$ may be written

as a transformation groupoid.

We set

Lemma 2.5. The real numbers $\theta_i, i = 1,2,3,4$ satisfy

Conversely, if real numbers $\zeta_i, i = 1,2,3,4$ satisfy

then we have $\zeta_i = \theta_i, i = 1,2,3,4.$

Proof. The identities (12) and (13) are immediate. Conversely, suppose that real numbers $\zeta_i, i = 1,2,3,4$ satisfy (14) and (15). As $\zeta_1 = \frac{u_2}{u_1}\zeta_2 = \frac{u_2}{u_1}( -\frac{s_2}{s_1}) \zeta_4,$ the equality (15) implies

so that

and hence

Proposition 2.6. For $x = (x_1, x_2) \in \mathbb{T}^2$, we have

and hence

Proof. We have

In particular, for $(m,n) = (1,0), (0,1)$, we have

We put $\xi_i = \frac{1}{1-r_A^2}(u_i -r_A s_i)$ for $i = 1,2$ so that

We then have

and similarly

Hence we have $\frac{\xi_1}{\xi_2} = - \frac{s_2}{s_1}.$ We also have

By Lemma 2.5, we have $\xi_1 u_1 = \theta_1, \, \xi_1 u_2 = \theta_2, \, \xi_2 u_1 = \theta_3, \, \xi_2 u_2 = \theta_4,$ proving the desired assertion from the identities (16) and (17).

We will next express $\theta_i, i = 1,2,3,4$ in terms of the matrix elements $a,b,c,d$ of $A$.

Lemma 2.7. The following identities hold.

(ⅰ)

and hence

(ⅱ)

and hence

Proof. By the identities

with $\theta_1 + \theta_4 = 1$, we see the desired assertions.

Lemma 2.8.

(ⅰ) $(a\theta_1 + b \theta_2) \theta_4 = (c\theta_3 + d \theta_4) \theta_1.$

(ⅱ) $(a\theta_3 - b \theta_1) \theta_2 = (-c\theta_4 + d \theta_2) \theta_3.$

Hence we have

Proof. (ⅰ) By the first and the fourth identities in Lemma 2.7 (ⅰ), we know the identity (ⅰ). The identities of (ⅱ) is similarly shown to those of (ⅰ). By (ⅰ) and (ⅱ) with the identity $\theta_1 \theta_4 = \theta_2 \theta_3$, we get $b \theta_2 = c \theta_3.$

Recall that $\Delta(A)$ denotes the discriminant $(a + d)^2 - 4(a d -b c)$ of the characteristic polynomial of the matrix $A$. The real number $\Delta(A)$ is positive because of the hyperbolicity of $A$. By elementary calculations, we see the following lemma.

Lemma 2.9. The identities

imply

We thus have the following theorem.

Theorem 2.10. The $C^*$-algebra $C^*(G_A^a)$ of the groupoid $G_A^a$ for a hyperbolic matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is isomorphic to the simple $C^*$-algebra generated by four unitaries $U_1, U_2, V_1, V_2$ satisfying the following relations:

where

Hence the $C^*$-algebra $C^*(G_A^a)$ is isomorphic to the four-dimensional non-commutative torus.

Proof. As in Lemma 2.2, the action $\alpha^A: \mathbb{Z}^2 \longrightarrow {{{\operatorname{Homeo}}}}( \mathbb{T}^2)$ is free and minimal, hence the $C^*$-crossed product $C( \mathbb{T}^2)\times_{\alpha^A} \mathbb{Z}^2$ is simple. The $C^*$-crossed product is canonically identified with the $C^*$-crossed product $((C( \mathbb{T})\otimes C( \mathbb{T})) \times_{\alpha^A_{(1,0)}} \mathbb{Z}) \times_{\alpha^A_{(0,1)}} \mathbb{Z}.$ Let $U_1, U_2$ be the unitaries in $C( \mathbb{T})\otimes C( \mathbb{T})$ defined by $U_1(t,s) = e^{2\pi i t}, U_2(t,s) = e^{2\pi i s}$. Let $V_1, V_2$ be the implementing unitaries corresponding to the automorphisms $\alpha^A_{(1,0)}, \alpha^A_{(0,1)},$ respectively. By Proposition 2.6, we know the commutation relations among the unitaries $U_1, U_2, V_1, V_2$ for the slopes $\theta_1, \theta_2, \theta_3, \theta_4$ satisfying (20). The second values of (20) go to the first of (20) by substituting $V_1, U_1$ with $V_2, U_2,$ respectively. The forth values of (20) go to the third of (20) by substituting $V_1, U_1$ with $V_1^*, U_1^*,$ respectively. When $a = d$, we have $\Delta(A) = 4 bc >0$ so that $\pm\sqrt{\frac{c}{b}} = \frac{c}{\sqrt{\Delta(A)}}, \pm\sqrt{\frac{b}{c}} = \frac{b}{\sqrt{\Delta(A)}}.$ Hence the first two of (20) include the second two of (20), so that we may unify (20) into (21).

Since the $C^*$-algebra $C^*(G_A^a)$ is isomorphic to a four-dimensional non-commutative torus, we know the following proposition by Slawny [24] (see also Putnam[14]).

Proposition 2.11 (Slawny [24], Putnam [14]). The $C^*$-algebra $C^*(G_A^a)$ has a unique tracial state.

Remark 2.12. (ⅰ) We note that the simplicity of the algebra $C^*(G_A^a)$ comes from a general theory of Smale space $C^*$-algebras as in [14], [17] as well as a unique existence of tracial state on it. It also follows from a general theory of crossed product $C^*$-algebras because the action $\alpha^A$ of $\mathbb{Z}^2$ to ${{{\operatorname{Homeo}}}}( \mathbb{T}^2)$ is free and minimal. It has been shown that a simple higher dimensional non-commutative torus is an AT-algebra by Phillips [13].

(ⅱ) Suppose that two hyperbolic matrices $A, B \in GL(2, \mathbb{Z})$ commute each other. By (8) and (9), the equality $\alpha_A( \mathbb{Z}^2) = \alpha_B( \mathbb{Z}^2)$ holds for the commuting matrices $A$ and $B$, because they have the same eigenvectors. Hence we know that $G_A^a = G_B^a$, so that the $C^*$-algebras $C^*(G_A^a)$ and $G_B^a)$ are isomorphic.

3.   The range $\tau_*(K_0(C^*(G_A^a)))$

In this section, we will describe the trace values $\tau_*(K_0(C^*(G_A^a)))$ of the $K_0$-group of the $C^*$-algebra $C^*(G_A^a)$ in terms of the hyperbolic matrix $A$.

In [20], M. A. Rieffel studied K-theory for irrational rotation $C^*$-algebras $A_\theta$ with irrational numbers $\theta$, which are called two-dimensional non-commutative tori, and proved that $\tau_*(K_0(A_\theta)) = \mathbb{Z} + \mathbb{Z}\theta$ in $\mathbb{R},$ where $\tau$ is the unique tracial state on $A_\theta$. In [6], G. A. Elliott (cf. [3], [13], [21], [24], etc.) initiated to study higher-dimensional non-commutative tori. It is well-known the $K$-groups of the four-dimensional non-commutative torus as in [6] which says

([6], cf. [24]). For $g = (a_1, b_1, a_2, b_2), h = (c_1, d_1, c_2, d_2) \in \mathbb{Z}^4,$ we define a wedge product $g\wedge h \in \mathbb{Z}^4$ by

where $\begin{vmatrix} x & y \\ z & w \end{vmatrix} = xw - yz.$ Let $\Theta = [\theta_{jk}]_{j,k = 1}^4$ be a $4\times 4$ skew symmetric matrix over $\mathbb{R}.$ We regard the matrix $\Theta$ as a linear map from $\mathbb{Z}^4\wedge \mathbb{Z}^4$ to $\mathbb{R}$ by $\Theta(x\wedge y) = \Theta x \cdot y.$ Then $\Theta\wedge\Theta:( \mathbb{Z}^4\wedge \mathbb{Z}^4)\wedge( \mathbb{Z}^4\wedge \mathbb{Z}^4) = \wedge^4 \mathbb{Z}^4 \longrightarrow \mathbb{R}$ is defined by

for $x_1, x_2, x_3, x_4 \in \mathbb{Z}^4.$ Although we may generally define $\wedge^n \Theta : \wedge^{2n} \mathbb{Z}^4 \longrightarrow \mathbb{R},$ the wedge product $\wedge^{2n} \mathbb{Z}^4 = 0$ for $n >3,$ so that

becomes

Let $A_{\Theta}$ be the $C^*$-algebra generated by four unitaries $u_j, j = 1,2,3,4$ satisfying the commutation relations $u_j u_k = e^{2 \pi i \theta_{jk}}u_k u_j,\, j,k = 1,2,3,4.$ The $C^*$-algebra $A_\Theta$ is called the four-dimensional non-commutative torus ([6]). If $\Theta$ is non-degenerate, the algebra $A_\Theta$ has a unique tracial state written $\tau.$ By Elliott's result in [6], there exists an isomorphism $h: K_0(A_{\Theta}) \longrightarrow \wedge^{\operatorname{even}} \mathbb{Z}^4$ such that ${{{\operatorname{exp}}}}_\wedge(\Theta) \circ h = \tau_*,$ so that we have

Proposition 3.1. Let $\tau$ be the unique tracial state on $C^*(G_A^a)$. Then we have

Proof. Take the unitaries $U_1, U_2, V_1, V_2$ and the real numbers $\theta_1, \theta_2, \theta_3, \theta_4$ as in Theorem 2.10. We set the real numbers $\theta_{jk}, j,k = 1,2,3,4$ such as $\theta_{jj} = \theta_{12} = \theta_{21} = \theta_{34} = \theta_{43} = 0$ for $j = 1,2,3,4$ and $\theta_{13} = \theta_4,\, \theta_{14} = \theta_3,\, \theta_{23} = \theta_2,\, \theta_{24} = \theta_1.$ Let $u_1 = V_2, u_2 = V_1, u_3 = U_2, u_4 = U_1$ so that we have the commutation relations

As $\theta_1 \cdot \theta_4 = \theta_2\cdot\theta_3,$ we have

By (22) or [6] (cf. [3,2.21], [13,Theorem 3.9]), we have

Remark 3.2. (ⅰ) It is straightforward to see that the skew symmetric matrix $\Theta = [\theta_{jk}]_{j,k = 1}^4$ in our setting above is non-degenerate.

(ⅱ) Suppose that two hyperbolic toral automorphisms $( \mathbb{T}^2, A)$ and $( \mathbb{T}^2,B)$ are topologically conjugate. We then know that both the $C^*$-algebras $C^*(G_A^a)$ and $C^*(G_B^a)$ are isomorphic. Since they have unique tracial states $\tau_A$ and $\tau_B$ respectively, we see that

We may also find a matrix $M \in GL(2, \mathbb{Z})$ such that $AM = MB$ by [1]. We then directly see that the ranges $\tau_{A*}(K_0(C^*(G_A^a)))$ and $\tau_{B*}(K_0(C^*(G_B^a)))$ coincide by using the formula (23). Similarly we may directly show that the equality $\tau_{A*}(K_0(C^*(G_A^a))) = \tau_{A^{-1}*}(K_0(C^*(G_{A^{-1}}^a)))$ by the formula (23).

4.   Examples

In this section, we will present some examples.

1. $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}.$ Since $a = b = c = 1, d = 0$, we have by Theorem 2.10,

By the formula (23), we have

Proposition 4.1. Let $A$ be the matrix $\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}.$ Put $\theta = \frac{1}{2}(1+\frac{1}{\sqrt{5}}).$ Then the $C^*$-algebra $C^*(G_A^a)$ is isomorphic to the tensor product $A_{\theta}\otimes A_{5\theta}$ between the irrational rotation $C^*$-algebras $A_\theta$ and $A_{5\theta}$ with its rotation angles $\theta$ and $5\theta$ respectively.

Proof. Let $U_1, U_2, V_1, V_2$ be the generating unitaries in Theorem 2.10. Since

by (24), we have

We set

It is straightforward to see that the following equalities hold

Since the $C^*$-algebra $C^*(u_1, u_2, v_1, v_2)$ generated by $u_1, u_2, v_1, v_2$ coincides with $C^*(G_A^a),$ we have

2. $A = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix}.$ Since $a = 3, b = d = 1, d = 2$, we have by Theorem 2.10,

and

Since $\theta_4 = 1 -\theta_1$, $\theta_2 = 2 \theta_3,$ $\theta_1 = \frac{1}{2} + \theta_3$, the formula (23) tells us

Proposition 4.2. Let $A_1 = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$ and $A_2 = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix}.$ Then the $C^*$-algebra $C^*(G_{A_1}^a)$ is not isomorphic to $C^*(G_{A_2}^a).$

Proof. Since the $C^*$-algebra $C^*(G_A^a)$ has the unique tracial state $\tau$, the range $\tau_*(K_0(C^*(G_A^a)))$ of $\tau$ of the $K_0$-group $K_0(C^*(G_A^a))$ is invariant under isomorphism class of the $C^*$-algebra. As

we see that $\tau_*(K_0(C^*(G_{A_1}^a))) \ne \tau_*(K_0(C^*(G_{A_2}^a))),$ so that the $C^*$-algebra $C^*(G_{A_1})$ is not isomorphic to $C^*(G_{A_2}).$

Acknowledgments

The author would like to deeply thank the referee for careful reading and lots of helpful advices in the presentation of the paper. This work was supported by JSPS KAKENHI Grant Numbers 15K04896, 19K03537.