PRIMITIVE IDEAL SPACE OF $C^*(R_+)\rtimes R^\times $

Abstract

For an integral domain R satisfying certain conditions, we characterise the primitive ideal space and its Jacobson topology for the semigroup crossed product $C^*(R_+) \rtimes R^\times $. We illustrate the result by the example $R=\mathbb {Z}[\sqrt {-3}]$.

1 Introduction

Motivated by the pioneering paper of Bost and Connes [2], Cuntz in [8] constructed the first ring $C^*$ -algebra. Cuntz and Li [11] generalised the work of [8] to an integral domain with finite quotients. Eventually, Li [18] generalised the work of [8] to arbitrary rings. There is more than one way of studying $C^*$ -algebras associated to rings. Hirshberg [12], Larsen and Li [17], and Kaliszewski et al. [13] independently investigated $C^*$ -algebras from p-adic rings. Li [19] defined the notion of semigroup $C^*$ -algebras and proved that the $ax+b$ -semigroup $C^*$ -algebra of a ring is an extension of the ring $C^*$ -algebra. When the ring is the ring of integers of a field, Li [19] proved that the $ax+b$ -semigroup $C^*$ -algebra is isomorphic to another construction due to Cuntz et al. [9]. Very recent work due to Bruce and Li [5, 6] and Bruce et al. [4] on algebraic dynamical systems and their associated $C^*$ -algebras solves quite a few open problems.

For an integral domain R, denote by $R_+$ the additive group $(R,+)$ and by $R^\times $ the multiplicative semigroup $(R\setminus \{0\},\cdot )$ . There is a natural unital and injective action of $R^\times $ on $C^*(R_+)$ by multiplication. Thus, we obtain a semigroup crossed product $C^*(R_+) \rtimes R^\times $ . We characterise the primitive ideal space and its Jacobson topology for the semigroup crossed product $C^*(R_+) \rtimes R^\times $ under certain conditions. Our main example is $R=\mathbb {Z}[\sqrt {-3}]$ . The semigroup crossed product $C^*(R_+) \rtimes R^\times $ is closely related to other constructions. In the Appendix, we show that $C^*(R_+) \rtimes R^\times $ is an extension of the boundary quotient of the opposite semigroup of the $ax+b$ -semigroup of the ring and that when the ring is a greatest common divisor (GCD) domain, $C^*(R_+) \rtimes R^\times $ is isomorphic to the boundary quotient of the opposite semigroup of the $ax+b$ -semigroup of the ring. There are only a few investigations of the opposite semigroup $C^*$ -algebra of the $ax+b$ -semigroup of a ring (see for example [10, 20, 21]).

Standing assumptions

Throughout the paper, any semigroup is assumed to be discrete, countable, unital and left cancellative; any group is assumed to be discrete and countable; any subsemigroup of a semigroup is assumed to inherit the unit of the semigroup; any ring is assumed to be countable and unital with $0 \neq 1$ ; and any topological space is assumed to be second countable.

2 Laca’s dilation theorem revisited

Laca [14] proved an important theorem which dilates a semigroup dynamical system $(A,P,\alpha )$ to a $C^*$ -dynamical system $(B,G,\beta )$ so that the semigroup crossed product $A \rtimes _{\alpha }^{e} P$ is Morita equivalent to the crossed product $B \rtimes _{\beta } G$ . In this section, we revisit Laca’s theorem when A is a unital commutative $C^*$ -algebra.

Notation 2.1. Let P be a subsemigroup of a group G satisfying $G=P^{-1}P$ . For ${p,q \in P}$ , define $p \leq q$ if $qp^{-1} \in P$ . Then, $\leq $ is a reflexive, transitive and directed relation on P.

Theorem 2.2 (See [14, Theorem 2.1])

Let P be a subsemigroup of a group G satisfying $G=P^{-1}P$ , let $A=C(X)$ , where X is a compact Hausdorff space, and let $\alpha :P \to \mathrm {End}(A)$ be a semigroup homomorphism such that $\alpha _p$ is unital and injective for all $p \in P$ . Then, there exists a dynamical system $(X_\infty ,G,\gamma )$ (where $X_\infty $ is compact Hausdorff) such that $A \rtimes _{\alpha }^{e} P$ is Morita equivalent to $C(X_\infty ) \rtimes _\gamma G$ .

Proof. By [14, Theorem 2.1], there exists a $C^*$ -dynamical system $(A_\infty ,G,\beta )$ such that $A \rtimes _{\alpha }^{e} P$ is Morita equivalent to $A_\infty \rtimes _\beta G$ . We cite the proof of [14, Theorem 2.1] to sketch the construction of $A_\infty $ and the definition of $\beta $ .

For $p \in P$ , define $A_p:=A$ . For $p,q \in P$ with $p \leq q$ , define $\alpha _{p,q}:A_p \to A_q$ to be $\alpha _{qp^{-1}}$ . Then, $\{(A_p,\alpha _{p,q}):p,q \in P,p \leq q\}$ is an inductive system. Let ${A_\infty :=\lim _{p}(A_p,\alpha _{p,q})}$ , let $\alpha ^{p}:A_p \to A_\infty $ be the natural unital embedding for all $p \in P$ and let $\beta :G \to \mathrm {Aut}(A_\infty )$ be the homomorphism satisfying $\beta _{p_0}\circ \alpha ^{pp_0}=\alpha ^p$ for all $p_0,p\in P$ .

For $p \in P$ , denote by $f_p:X \to X$ the unique surjective continuous map induced from $\alpha _p$ and set $X_p:=X$ . For $p,q \in P$ with $p \leq q$ , denote by $f_{q,p}:X_q \to X_p$ the unique surjective continuous map induced from $\alpha _{p,q}$ . Since $\alpha _{p,q}=\alpha _{qp^{-1}}$ , we have ${f_{q,p}=f_{qp^{-1}}}$ . Then, $\{(X_p,f_{q,p}):p,q \in P,p \leq q\}$ is an inverse system. Set

(2.1) $$ \begin{align} X_\infty :=\bigg\{(x_p)_{p \in P} \in \prod _{p \in P}X_p:f_{q,p}(x_q)=x_p\mbox { for all } p \leq q\bigg\}, \end{align} $$

which is the inverse limit of the inverse system. By [1, Example II.8.2.2(i)], ${A_\infty \cong C(X_\infty )}$ . For $p \in P$ , denote by $f^p:X_\infty \to X_p$ the unique projection induced from $\alpha ^p$ . Then, $f_{q,p}\circ f^q=f^p$ for all $p,q \in P,p \leq q$ . For  $p,p_0 \in P,f \in C(X_\infty )$ , denote by $\gamma _{p_0}:X_\infty \to X_\infty $ the unique homeomorphism such that $\beta _{p_0}(f)=f\circ \gamma _{p_0}^{-1}$ .

From this construction, $(X_\infty ,G,\gamma )$ is a dynamical system with $C(X_\infty ) \rtimes _\gamma G \cong A_\infty \rtimes _\beta G$ . Hence, $A \rtimes _{\alpha }^{e} P$ is Morita equivalent to $C(X_\infty ) \rtimes _\gamma G$ .

Notation 2.3. We give an explicit description of $X_\infty $ and the action of G on $X_\infty $ given in Theorem 2.2. We start with the definition of $X_\infty$ in (2.1). Then, for $p_0,p,q \in P$ with $q \geq p_0,p$ , and for $(x_p)_{p \in P} \in X_\infty $ , we have

$$ \begin{align*}(p_0\cdot (x_p))(p)=x_{pp_0},\quad (p_0^{-1}\cdot (x_p))(p)=f_{q,p}(x_{qp_0^{-1}}).\end{align*} $$

In particular, when G is abelian, we have a simpler form of the group action given by

$$ \begin{align*}\frac{p_0}{q_0} \cdot (x_p)=(f_{q_0}(x_{pp_0})).\end{align*} $$

Our goal is to apply Theorem 2.2 to characterise the primitive ideal space of the semigroup crossed product $C^{*}(R_+)\rtimes R^{\times }$ of an integral domain. Since $R^{\times }$ is abelian, we will need the following version of Williams’ theorem.

Definition 2.4. Let G be an abelian group, let X be a locally compact Hausdorff space and let $\alpha :G \to \mathrm {Homeo}(X)$ be a homomorphism. For $x,y\in X$ , define $x \sim y$ if $\overline {G\cdot x} =\overline {G \cdot y}$ . Then, $\sim $ is an equivalence relation on X. For $x \in X$ , define $[x] := \overline {G\cdot x}$ , called the quasi-orbit of x. The quotient space $Q(X /G)$ by the relation $\sim $ is called the quasi-orbit space. For $x \in X$ , define $G_{x}:=\{ g \in G : g \cdot x = x\}$ , called the isotropy group (or stability group) at x. For $([x],\phi ), ([y],\psi ) \in Q(X/ G)\times \widehat {G}$ , define $([x],\phi )\approx ([y],\psi )$ if $[x] = [y]$ and $\phi \vert _{G_{x}}= \psi \vert _{G_{x}}$ . Then, $\approx $ is an equivalence relation on $Q(X /G) \times \widehat {G}$ .

Theorem 2.5 [16, Theorem 1.1]

Let G be an abelian group, let X be a locally compact Hausdorff space and let $\alpha : G \to \mathrm {Homeo}(X)$ be a homomorphism. Then, ${\mathrm {Prim}(C_{0}(X)\rtimes _\alpha G)\cong (Q(X /G)\times \widehat {G})/\approx} $ .

3 Primitive ideal structure of $C^*(R_+)\rtimes R^\times $

In this section, we characterise the primitive ideal space and its Jacobson topology for the semigroup crossed product $C^*(R_+) \rtimes R^\times $ under certain conditions.

Notation 3.1. Let R be an integral domain. Denote by Q the field of fractions of R, by $R_+$ the additive group $(R,+)$ , by $\widehat {R_+}$ the dual group of $R_+$ , by $R^\times $ the multiplicative semigroup $(R\setminus \{0\},\cdot )$ , by $Q^\times $ the enveloping group $(Q\setminus \{0\},\cdot )$ of $R^\times $ , by $\{u_r\}_{r \in R_+}$ the family of unitaries generating $C^*(R_+)$ and by $\alpha :R^\times \to \mathrm {End}(C^*(R_+))$ the homomorphism such that $\alpha _{p}(u_r)=u_{pr}$ for all $p \in R^\times ,r \in R_+$ . Observe that for any $p \in R^\times , \alpha _p$ is unital and injective, and the map $f_p:\widehat {R_+} \to \widehat {R_+},\phi \mapsto \phi (p \cdot )$ is the unique surjective continuous map induced from $\alpha _p$ . Denote by

$$ \begin{align*}X_\infty(R):=\bigg\{\phi=(\phi_p)_{p\in R^\times}\in\prod_{p\in R^\times}\widehat{R_+}:\phi_q\bigg(\frac{q}{p}\cdot \bigg)=\phi_p,\text{ whenever } p \mid q\bigg\}.\end{align*} $$

Then, $({p_0}/{q_0})\cdot (\phi _p)=(\phi _{pp_0}(q_0\cdot ))$ .

Lemma 3.2. Let R be an integral domain. Fix $(\phi _p)_{p\in R^\times } \in X_\infty (R)$ . If $(\phi _p)_{p\in R^\times }\neq (1)_{p \in R^\times }$ , then $Q^{\times }_{\phi }=\{1_R\}$ . If $(\phi _p)_{p\in R^\times }= (1)_{p\in R^\times }$ , then $Q^{\times }_{\phi }=Q^{\times }$ .

Proof. To prove the first statement, suppose for a contradiction that there exists $p_0/q_0 \in Q^\times$ with $p_0/q_0 \ne 1$ and such that $({p_0}/{q_0}) \cdot \phi =\phi $ . Since $(\phi _p)_{p\in R^\times }\neq (1)_{p \in R^\times }$ , there exists ${p_1 \in R^\times }$ such that $\phi _{p_1} \neq 1$ . Then, $\phi _p=\phi _{pp_0}(q_0 \cdot )$ for any $p \in R^{\times }$ . Since $\phi _{pp_0}(p_0\cdot )=\phi _p$ for any $p \in R^\times $ , we deduce that $\phi _{pp_0}(p_0\cdot )=\phi _{pp_0}(q_0 \cdot )$ for all $p \in R^\times $ . So ${\phi _{pp_0}((p_0-q_0)\cdot )= 1}$ for any $p \in R^\times $ . Hence, $\phi _{pp_0}((p_0-q_0)p_0\cdot )= 1$ for any ${p \in R^\times }$ . When $p=p_1(p_0-q_0)$ , we get $\phi _{p_1}=\phi _{p_1(p_0-q_0)p_0}(((p_0-q_0)p_0\cdot )= 1$ , which is a contradiction. Therefore, ${Q^{\times }_{\phi }=\{1_R\}}$ .

To prove the second statement, suppose that $(\phi _p)_{p\in R^\times }= (1)_{p\in R^\times }$ . For ${{p_0}/{q_0} \in Q^{\times }}$ , we have $({p_0}/{q_0})\cdot (1)_{p\in R^\times }\kern1.3pt{=}\kern1.3pt({p_0}/{q_0})\cdot (\phi _p)_{p\in R^\times }\kern1.3pt{=}\kern1.3pt(\phi _{pp_0}(q_0\cdot ))_{p\in R^\times }\kern1.3pt{=}\kern1.3pt(1)_{p\in R^\times }$ . So $Q^{\times }_{\phi }=Q^{\times }$ .

Lemma 3.3. Let R be an integral domain. Suppose that for $\epsilon>0$ , $(1)_{p \in R^\times }\neq (\phi _p)_{p\in R^\times } \in X_\infty (R)$ , $\pi \in \widehat {R_+}$ , $P \in R^\times $ and $r_1,r_2,\ldots , r_n \in R_+$ , there exist $p,q \in R^\times $ with $P \mid p$ such that $\vert \phi _{p}(qr_i)-\pi (r_i)\vert <\epsilon ,i=1,2,\ldots ,n$ . Then, $Q(X_\infty (R)/Q^\times )$ consists of only two points with the only nontrivial closed subset $\{[(1)_{p \in R^\times }]\}$ .

Proof. Since $\overline {Q^{\times } \cdot (1)_{p \in R^\times }}=\overline {(1)_{p \in R^\times }}=(1)_{p \in R^\times }$ , we have $[(\phi _p)_{p\in R^\times }] \neq [(1)_{p \in R^\times }]$ whenever $(1)_{p \in R^\times }\neq (\phi _p)_{p\in R^\times } \in X_\infty (R)$ .

Fix $(\phi _p)_{p\in R^\times }, (\psi _p)_{p\in R^\times } \in X_\infty (R)$ such that $(\phi _p)_{p\in R^\times }, (\psi _p)_{p\in R^\times } \neq (1)_{p \in R^\times }$ . We aim to show that $[(\phi _p)_{p\in R^\times }]=[(\psi _p)_{p\in R^\times }]$ . It suffices to show that $(\psi _p)_{p\in R^\times } \in \overline {Q^{\times } \cdot (\phi _p)_{p\in R^\times }}$ since $(\phi _p)_{p\in R^\times } \in \overline {Q^{\times } \cdot (\psi _p)_{p\in R^\times }}$ follows from the same argument. Fix $\epsilon>0$ , $p_1,p_2,\ldots ,p_n \in R^\times $ and $r_1,r_2,\ldots ,r_n \in R$ . By the condition imposed in the lemma, there exist $p_0,q_0 \in R^{\times }$ such that

$$ \begin{align*}\vert \phi_{p_1p_2\cdots p_np_0}(q_0p_1\cdots p_{i-1}p_{i+1}\cdots p_nr_j)-\psi_{p_1p_2\cdots p_n}(p_1\cdots p_{i-1}p_{i+1}\cdots p_nr_j)\vert<\epsilon\end{align*} $$

for $1\leq i,j\leq n$ . So $\vert \phi _{p_ip_0}(q_0r_j)-\psi _{p_i}(r_j)\vert <\epsilon $ for $1\leq i,j\leq n$ . Hence, $(\psi _p)_{p\in R^\times } \in \overline {Q^{\times } \cdot (\phi _p)_{p\in R^\times }}$ . Therefore, $[(\phi _p)_{p\in R^\times }]=[(\psi _p)_{p\in R^\times }]$ .

We conclude that $Q(X_\infty (R)/Q^\times )$ consists of only two points. For any $(1)_{p \in R^\times }\neq (\phi _p)_{p\in R^\times } \in X_\infty (R)$ , $\overline {Q^{\times } \cdot (\phi _p)_{p\in R^\times }}=X_\infty (R) \setminus \{(1)_{p \in R^\times }\}$ is open but not closed. Finally, we deduce that $\{[(1)_{p \in R^\times }]\}$ is the only nontrivial closed subset of $Q(X_\infty (R)/Q^\times )$ .

Theorem 3.4. Let R be an integral domain satisfying the condition of Lemma 3.3. Take an arbitrary element $(\phi _p)_{p\in R^\times } \in X_\infty (R)$ with $(1)_{p \in R^\times } \neq (\phi _p)_{p\in R^\times }$ . Then, we have $\mathrm {Prim}(C^*(R_+) \rtimes R^\times ) \cong \{[(\phi _p)_{p\in R^\times }]\} \amalg \{[(1)_{p \in R^\times }]\} \times \widehat {Q^\times }$ , and the open sets of ${\mathrm {Prim}(C^*(R_+) \rtimes R^\times )}$ comprise $\{[(\phi _p)_{p\in R^\times }]\} \amalg \{[(1)_{p \in R^\times }]\} \times N$ , where N is an open subset of $\widehat {Q^\times }$ .

Proof. By Theorem 2.2, $(C^*(R_+) \rtimes R^\times )$ is Morita equivalent to $C(X_\infty (R)) \rtimes Q^\times $ . So $\mathrm {Prim}(C^*(R_+) \rtimes R^\times ) \cong \mathrm {Prim}(C(X_\infty (R)) \rtimes Q^\times )$ . By Theorem 2.5 and Lemma 3.3, $\mathrm {Prim}(C(X_\infty (R)) \rtimes Q^\times ) \cong \{[(\phi _p)_{p\in R^\times }],[(1)_{p \in R^\times }]\} \times \widehat {Q^\times } /\approx $ . By Lemma 3.2, $Q^{\times }_{(\phi _p)_{p\in R^\times }}=\{1_R\}$ and $Q^{\times }_{(1)_{p \in R^\times }}=Q^{\times }$ . So, $\mathrm {Prim}(C(X_\infty (R)) \rtimes Q^\times ) \cong \{[(\phi _p)_{p\in R^\times }]\} \amalg \{[(1)_{p \in R^\times }]\} \times \widehat {Q^\times }$ . Hence, $\mathrm {Prim}(C^*(R_+) \rtimes R^\times ) \cong \{[(\phi _p)_{p\in R^\times }]\} \amalg \{[(1)_{p \in R^\times }]\} \times \widehat {Q^\times }$ , and the open sets of $\mathrm {Prim}(C^*(R_+) \rtimes R^\times )$ are $\{[(\phi _p)_{p\in R^\times }]\} \amalg \{[(1)_{p \in R^\times }]\} \times N$ , where N is an open subset of  $\widehat {Q^\times }$ .

Example 3.5. Let $R=\mathbb {Z}$ . Then, $\widehat {R_+}=\mathbb {T}$ . Fix $\epsilon>0$ , $(1)_{p \in \mathbb {Z}^\times }\neq (\phi _p)_{p\in \mathbb {Z}^\times } \in X_\infty (\mathbb {Z})$ , $\pi \in \mathbb {T}$ , $P \in \mathbb {Z}^\times $ and $r_1,r_2,\ldots , r_n \in \mathbb {Z}_+$ . Take an arbitrary $p_0 \in \mathbb {Z}^\times $ such that $P\mid p_0$ and let $\phi _{p_0}=e^{2\pi i \theta }$ for some $\theta \in (0,1)$ .

Case 1: $\theta $ is rational. Then, $\phi _{p_0}^{\mathbb {Z}}=\{e^{{2\pi i k}/{n}}\}_{k=0}^{n-1}$ for some $n \geq 1$ . Since $\phi _{pp_0}^p=\phi _{p_0}$ for any $p \geq 1$ , we get $\phi _{pp_0}^{\mathbb {Z}}=\{e^{{2\pi i k}/{pn}} \}_{k=0}^{pn-1}$ . Choose $p_1 \geq 1$ such that $\vert e^{{2\pi i}/{p_1n}}-1 \vert <\epsilon /\sum _{i=1}^{n}\vert r_i \vert $ . Then, there exists $q_0 \in \mathbb {Z}^\times $ such that $\vert \phi _{p_1p_0}^{q_0}-\pi \vert <\epsilon /\sum _{i=1}^{n}\vert r_i \vert $ .

Case 2: $\theta $ is irrational. Then, by the properties of an irrational rotation, $\{\phi _{p_0}^z\}_{z \in \mathbb {Z}}$ is a dense subset of $\mathbb {T}$ . So, there exists $q_0 \in \mathbb {Z}^\times $ such that $\vert \phi _{p_0}^{q_0}-\pi \vert <\epsilon /\sum _{i=1}^{n}\vert r_i \vert $ .

In both cases, there exist $p,q \in \mathbb {Z}^\times $ with $P \mid p$ such that $\vert \phi _{p}^{q}-\pi \vert <\epsilon /\sum _{i=1}^{n}\vert r_i \vert $ . For $1\leq i\leq n$ , we may assume that $r_i \geq 0$ and we calculate that

$$ \begin{align*} \vert\phi_{p}(qr_i)-\pi(r_i)\vert=\vert\phi_{p}^{qr_i}-\pi^{r_i}\vert =\vert \phi_{p}^{q}-\pi\vert\,\bigg\vert\sum_{j=0}^{r_i-1}\phi_{p}^{q(r_i-1-j)}\pi^{j}\bigg\vert &\leq\vert \phi_{p}^{q}-\pi\vert\sum_{j=0}^{r_i-1}\vert\phi_{p}^{q(r_i-1-j)}\pi^{j}\vert \\&<\epsilon r_i\Big/\sum_{i=1}^{n}\vert r_i \vert <\epsilon. \end{align*} $$

So, $\mathbb {Z}$ satisfies the condition of Lemma 3.3.

Example 3.6. Let $R=\mathbb {Z}[\sqrt {-3}]$ . Then, $\mathbb {Z}[\sqrt {-3}]_+\cong \mathbb {Z}^2$ and $\widehat {\mathbb {Z}[\sqrt {-3}]_+}\cong \mathbb {T}^2$ . Fix $\epsilon>0$ , $((1,1))_{p \in R^\times }\kern1.3pt{\neq}\kern1.3pt ((a_p,b_p))_{p\in R^\times } \kern1.3pt{\in}\kern1.3pt X_\infty (\mathbb {Z}[\sqrt {-3}])$ , $(\pi ,\rho ) \kern1.3pt{\in}\kern1.3pt \mathbb {T}^2$ , $P \kern1.4pt{\in}\kern1.4pt R^\times $ and $r_i\kern1.4pt{+}\kern1.4pt s_i\sqrt{-3} \kern1.4pt{\in}\kern1.4pt \mathbb{Z}[\sqrt{-3}]_+ \mathrm{for}\ i=1,2\ldots,n$ . Take an arbitrary $P \mid p_0 \in R^\times $ such that $(a_{p_0},b_{p_0}) \neq (1,1)$ . There exist $p,q=q_1+q_2\sqrt {-3} \in R^\times $ with $P \mid p$ such that $\vert a_{p}^{q_1}b_p^{q_2}-\pi \vert ,\vert a_p^{-3q_2}b_p^{q_1}-\rho \vert <{\epsilon }/{\sum _{i=1}^{n}(\vert r_i \vert +\vert s_i \vert )}$ . For $1\leq i\leq n$ , we may assume that $r_i \geq 0$ and we estimate

$$ \begin{align*} &\vert(a_p,b_p)(q(r_i+s_i\sqrt{-3}))-(\pi,\rho)(r_i+s_i\sqrt{-3})\vert \\&\quad=\vert (a_{p}^{q_1}b_p^{q_2})^{r_i}(a_p^{-3q_2}b_p^{q_1})^{s_i}-\pi^{r_i}\rho^{s_i}\vert \\&\quad=\vert ((a_{p}^{q_1}b_p^{q_2})^{r_i}-\pi^{r_i})(a_p^{-3q_2}b_p^{q_1})^{s_i}+\pi^{r_i}((a_p^{-3q_2}b_p^{q_1})^{s_i}-\rho^{s_i})\vert \\&\quad\leq\vert (a_{p}^{q_1}b_p^{q_2})^{r_i}-\pi^{r_i}\vert+\vert(a_p^{-3q_2}b_p^{q_1})^{s_i}-\rho^{s_i}\vert \\&\quad< \frac{\epsilon\vert r_i\vert}{\sum_{i=1}^{n}\vert r_i \vert+\vert s_i \vert}+\frac{\epsilon \vert s_i \vert}{\sum_{i=1}^{n}\vert r_i \vert+\vert s_i \vert} \leq\epsilon. \end{align*} $$

So, $\mathbb {Z}[\sqrt {-3}]$ satisfies the condition of Lemma 3.3.

By a similar argument to this example, we conclude that any (concrete) order of a number field satisfies the condition of Lemma 3.3. (For the background about number fields, one may refer to [22].)

Appendix The relationship between $C^*(R_+) \rtimes R^\times $ and semigroup $C^*$ -algebras

In this appendix, we show that $C^*(R_+) \rtimes R^\times $ is an extension of the boundary quotient of the opposite semigroup of the $ax+b$ -semigroup of the ring and that when the ring is a GCD domain, $C^*(R_+) \rtimes R^\times $ is isomorphic to the boundary quotient of the opposite semigroup of the $ax+b$ -semigroup of the ring.

Definition A.1 ([15, Section 2], [19, Definition 2.13])

Let P be a semigroup, A be a unital $C^*$ -algebra and $\alpha :P \to \mathrm {End}(A)$ be a semigroup homomorphism such that $\alpha _p$ is injective for all $p \in P$ . Define the semigroup crossed product $A \rtimes _{\alpha } P$ to be the universal unital $C^*$ -algebra generated by the image of a unital homomorphism ${i_A:A \to A \rtimes _{\alpha } P}$ and a semigroup homomorphism $i_P:P \to \mathrm {Isom}(A \rtimes _{\alpha } P)$ satisfying the following conditions:

1. (1) $i_P(p)i_A(a)i_P(p)^*=i_A(\alpha _p(a))$ for all $p \in P,a \in A$ ;

2. (2) for any unital $C^*$ -algebra B, unital homomorphism $j_A:A \to B$ and semigroup homomorphism $j_P:P \to \mathrm {Isom}(B)$ satisfying $j_P(p)j_A(a)j_P(p)^*=j_A(\alpha _p(a))$ , there exists a unique unital homomorphism $\Phi :A \rtimes _{\alpha } P \to B$ such that $\Phi \circ i_A=j_A$ and $\Phi \circ i_P=j_P$ .

Remark A.2. We have $i_A(1_A)=i_P(1_P) = \mbox { the unit of } A \rtimes _{\alpha } P$ .

If $\alpha _p$ is unital for all $p \in P$ , then $i_P(p)$ is a unitary for any $p \in P$ . To see this, we calculate that $i_P(p)i_P(p)^*=i_P(p)i_A(1_A)i_P(p)^*=i_A(\alpha _p(1_A))=i_A(1_A)$ .

Notation A.3 [3, 19]

Let P be a semigroup. For $p \in P$ , we also denote by p the left multiplication map $q \mapsto pq$ . The set of constructible right ideals is defined to be

$$ \begin{align*} \mathcal{J}(P):=\{p_1^{-1}q_1\cdots p_n^{-1}q_nP:n \geq 1, p_1,q_1,\ldots, p_n,q_n \in P\} \cup \{\emptyset\}. \end{align*} $$

A finite subset $F \subset \mathcal {J}(P)$ is called a foundation set if for any nonempty $X \in \mathcal {J}(P)$ , there exists $Y \in F$ such that $X \cap Y \neq \emptyset $ .

Definition A.4 ([3, Remark 5.5], [19, Definition 2.2])

Let P be a semigroup. Define the full semigroup $C^*$ -algebra $C^*(P)$ of P to be the universal unital $C^*$ -algebra generated by a family of isometries $\{v_p\}_{p \in P}$ and a family of projections $\{e_X\}_{X \in \mathcal {J}(P)}$ satisfying the following relations:

1. (1) $v_p v_q=v_{pq}$ for all $p,q \in P$ ;

2. (2) $v_p e_X v_p^*=e_{pX}$ for all $p \in P,X \in \mathcal {J}(P)$ ;

3. (3) $e_\emptyset =0$ and $e_P=1$ ;

4. (4) $e_X e_Y=e_{X \cap Y}$ for all $X,Y \in \mathcal {J}(P)$ .

Define the boundary quotient $\mathcal {Q}(P)$ of $C^*(P)$ to be the universal unital $C^*$ -algebra generated by a family of isometries $\{v_p\}_{p \in P}$ and a family of projections $\{e_X\}_{X \in \mathcal {J}(P)}$ satisfying conditions (1)–(4) and $\prod _{X \in F}(1-e_X)=0$ for any foundation set $F \subset \mathcal {J}(P)$ .

Definition A.5 ([3, Definition 2.1], [23, Definition 2.17])

Let P be a semigroup. Then, P is said to be right LCM (or to satisfy the Clifford condition) if the intersection of two principal right ideals is either empty or a principal right ideal.

Notation A.6. Let P be a semigroup. Denote by $P^{\mathrm {op}}$ the opposite semigroup of P. Let R be an integral domain. Denote by $R_+ \rtimes R^\times $ the $ax+b$ -semigroup of R. Denote by $\times $ the multiplication of $(R_+ \rtimes R^\times )^{\mathrm {op}}$ , that is, $(r_1,p_1)\times (r_2,p_2)=(r_2,p_2)(r_1,p_1)=(r_2+p_2r_1,p_1p_2)$ .

Remark A.7. Let R be an integral domain. We claim that any nonempty element of $\mathcal {J}((R_+ \rtimes R^\times )^{\mathrm {op}})$ is a foundation set of $(R_+ \rtimes R^\times )^{\mathrm {op}}$ . To see this, for any $(r_1,p_1),(r_2,p_2) \in (R_+ \rtimes R^\times )^{\mathrm {op}}$ , we compute

$$ \begin{align*} (r_1,p_1) \times (p_1r_2,p_2)=(p_1r_2,p_2)(r_1,p_1) &=(p_1r_2+p_2r_1,p_1p_2) \\ &=(p_2r_1,p_1)(r_2,p_2) =(r_2,p_2)\times (p_2r_1,p_1). \end{align*} $$

Theorem A.8. Let R be an integral domain. Then, the crossed product $C^*(R_+) \rtimes R^\times $ is an extension of $\mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}})$ . Moreover, if R is a GCD domain (see [7]), then we have $C^*(R_+) \rtimes R^\times \cong \mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}})$ .

Proof. Denote by $i_A:C^*(R_+) \to C^*(R_+) \rtimes R^\times $ and $i_P:R^{\times } \to \mathrm {Isom}(C^*(R_+) \rtimes R^\times )$ the canonical homomorphisms generating $C^*(R_+) \rtimes R^\times $ . Let $\{v_{(r,p)}:(r,p) \in (R_+ \rtimes R^\times )^{\mathrm {op}}\}$ be the family of isometries and $\{e_X:X \in \mathcal {J}((R_+ \rtimes R^\times )^{\mathrm {op}})\}$ be the family of projections generating $\mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}})$ .

For any $(r,p)\in (R_+ \rtimes R^\times )^{\mathrm {op}}$ , note that $1-v_{(r,p)}v_{(r,p)}^*=1-e_{(r,p) \times (R_+ \rtimes R^\times )^{\mathrm {op}}}=0$ because $\{(r,p)\times (R_+ \rtimes R^\times )^{\mathrm {op}} \}$ is a foundation set. So each $v_{(r,p)}$ is a unitary.

For $r \in R_+$ , define $U_r:=v_{(r,1)}$ . For any $r,s \in R_+$ ,

$$ \begin{align*} U_r U_s=v_{(r,1)}v_{(s,1)}=v_{(s,1)(r,1)}=v_{(r+s,1)}=v_{(r,1)(s,1)}=v_{(s,1)}v_{(r,1)}=U_sU_r, \end{align*} $$

so $j_A:C^*(R_+) \to \mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}}),u_r \mapsto v_{(r,1)}$ is a homomorphism by the universal property of $C^*(R_+)$ . For $p \in R^\times $ , define $j_P(p):=v_{(0,p)}^*$ . For any $p,q \in R^\times $ ,

$$ \begin{align*} j_P(p)j_P(q)=v_{(0,p)}^*v_{(0,q)}^*=(v_{(0,q)}v_{(0,p)})^*=(v_{(0,p)(0,q)})^*=v_{(0,pq)}^*=j_P(pq), \end{align*} $$

so $j_P:R^{\times } \to \mathrm {Isom}(\mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}}))$ is a semigroup homomorphism. For any $p \in R^\times $ , $r \in R_+$ , we compute

$$ \begin{align*} j_P(p)j_A(u_r)j_P(p)^*=v_{(0,p)}^*v_{(r,1)}v_{(0,p)}=v_{(0,p)}^*v_{(pr,p)}=v_{(pr,1)}=j_A(u_{pr})=j_A(\alpha_p(u_r)). \end{align*} $$

By the universal property of $C^*(R_+) \rtimes R^\times $ , there exists a unique homomorphism $\Phi :C^*(R_+) \rtimes R^\times \to \mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}})$ such that $\Phi \circ i_A=j_A$ and $\Phi \circ i_P=j_P$ . Since $v_{(r,p)}=v_{(0,p)}v_{(r,1)}$ for any $(r,p) \in (R_+ \rtimes R^\times )^{\mathrm {op}}$ , we see that $\Phi $ is surjective. So, $C^*(R_+) \rtimes R^\times $ is an extension of $\mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}})$ .

Now, we assume that R is a GCD domain. By [23, Proposition 2.23], $R^\times $ is right LCM. For $(r_1,p_1),(r_2,p_2) \in (R_+ \rtimes R^\times )^{\mathrm {op}}$ , suppose that $p_1R^\times \cap p_2R^\times =pR^\times $ for some $p \in R^\times $ . We claim that

$$ \begin{align*}(r_1,p_1)\times (R_+ \rtimes R^\times)^{\mathrm{op}}\cap(r_2,p_2) \times (R_+ \rtimes R^\times)^{\mathrm{op}}=(0,p) \times (R_+ \rtimes R^\times)^{\mathrm{op}}.\end{align*} $$

Indeed, for any $(s_1,q_1),(s_2,q_2) \in (R_+ \rtimes R^\times )^{\mathrm {op}}$ , if $(r_1,p_1)\times (s_1,q_1)=(r_2,p_2) \times (s_2,q_2)$ , then $(r_1,p_1)\times (s_1,q_1)=(r_2,p_2) \times (s_2,q_2)=(0,p) \times (s_1+q_1r_1,{q_1p_1}/{p})$ . Conversely, for any $(s,q) \in (R_+ \rtimes R^\times )^{\mathrm {op}}$ ,

$$ \begin{align*}(0,p) \times (s,q)=(r_1,p_1)\times \bigg(s-\frac{pqr_1}{p_1},\frac{pq}{p_1}\bigg)=(r_2,p_2) \times \bigg(s-\frac{pqr_2}{p_2},\frac{pq}{p_2}\bigg).\end{align*} $$

This proves the claim. Hence, $(R_+ \rtimes R^\times )^{\mathrm {op}}$ is right LCM as well.

Since $(R_+ \rtimes R^\times )^{\mathrm {op}}$ is right LCM, it follows from [24, Lemma 3.4] that ${\mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}})}$ is the universal unital $C^*$ -algebra generated by a family of unitaries $\{v_{(r,p)}:(r,p) \in (R_+ \rtimes R^\times )^{\mathrm {op}}\}$ satisfying the conditions:

1. (1) $v_{(r_1,p_1)}v_{(r_2,p_2)}=v_{(r_1,p_1) \times (r_2,p_2)}$ ;

2. (2) $v_{(r_1,p_1)}^*v_{(r_2,p_2)}=v_{(s_1,q_1)}v_{(s_2,q_2)}^*$ , whenever $(r_1,p_1)\times (s_1,q_1)=(r_2,p_2) \times (s_2,q_2)$ and $(r_1,p_1)\times (R_+ \rtimes R^\times )^{\mathrm {op}}\cap (r_2,p_2)\times (R_+ \rtimes R^\times )^{\mathrm {op}}=(r_1,p_1)\times (s_1,q_1)\times (R_+ \rtimes R^\times )^{\mathrm {op}}$ .

For $(r,p) \in (R_+ \rtimes R^\times )^{\mathrm {op}}$ , define $V_{(r,p)}:=i_P(p)^*i_A(u_r)$ . Finally, we check that $\{V_{(r,p)}\}$ satisfies the above two conditions. For any $(r_1,p_1) ,(r_2,p_2) \in (R_+ \rtimes R^\times )^{\mathrm {op}}$ ,

$$ \begin{align*} V_{(r_1,p_1)}V_{(r_2,p_2)}&=i_P(p_1)^*i_A(u_{r_1})i_P(p_2)^*i_A(u_{r_2}) =i_P(p_1)^*i_P(p_2)^*i_A(\alpha_{p_2}(u_{r_1}))i_A(u_{r_2}) \\ &=(i_P(p_2)i_P(p_1))^*i_A(u_{p_2r_1})i_A(u_{r_2}) =i_P(p_1p_2)^*i_A(u_{r_2+p_2r_1}) \\ &=V_{(r_2+p_2r_1,p_1p_2)} =V_{(r_1,p_1) \times (r_2,p_2)}. \end{align*} $$

For $(r_1,p_1),(r_2,p_2) \in (R_+ \rtimes R^\times )^{\mathrm {op}}$ , suppose that $p_1R^\times \cap p_2R^\times =pR^\times $ for some $p \in R^\times $ . By the above claim, $(r_1,p_1)\times (R_+ \rtimes R^\times )^{\mathrm {op}}\cap (r_2,p_2)\times (R_+ \rtimes R^\times )^{\mathrm {op}}=(0,p)\times (R_+ \rtimes R^\times )^{\mathrm {op}}$ . It is not hard to see that $(r_1,p_1) \times (-{pr_1}/{p_1},{p}/{p_1})=(r_2,p_2) \times (-{pr_2}/{p_2},{p}/{p_2})=(0,p)$ . So,

$$ \begin{align*} V_{(r_1,p_1)}^*V_{(r_2,p_2)}&=i_A(u_{-r_1})i_P(p_1)i_P(p_2)^*i_A(u_{r_2}) =i_A(u_{-r_1})i_P\bigg(\frac{p}{p_1}\bigg)^*i_P\bigg(\frac{p}{p_2})i_A(u_{r_2}\bigg) \\ &=i_P\bigg(\frac{p}{p_1}\bigg)^*i_A(u_{-{pr_1}/{p_1}})i_A(u_{{pr_2}/{p_2}})i_P\bigg(\frac{p}{p_2}\bigg) \\&=V_{(-{pr_1}/{p_1},{p}/{p_1})}V_{(-{pr_2}/{p_2},{p}/{p_2})}^*. \end{align*} $$

By the universal property of $\mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}})$ , there exists a homomorphism ${\Psi :\mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}}) \to C^*(R_+) \rtimes R^\times }$ such that $\Psi (v_{(r,p)})=i_P(p)^*i_A(u_r)$ . Since

$$ \begin{align*} \Phi \circ \Psi(v_{(r,p)})=\Phi(i_P(p)^*i_A(u_r))=j_P(p)^*j_A(u_r)=v_{(0,p)}v_{(r,1)}=v_{(r,p)}, \end{align*} $$

$$ \begin{align*} \Psi\circ \Phi(i_A(u_r))=\Psi(j_A(u_r))=\Psi(v_{(r,1)})=i_A(u_r), \end{align*} $$

$$ \begin{align*} \Psi\circ \Phi(i_P(p))=\Psi(j_P(p))=\Psi(v_{(0,p)})^*=i_P(p), \end{align*} $$

we conclude that $C^*(R_+) \rtimes R^\times \cong \mathcal {Q}((R_+ \rtimes R^\times )^{\mathrm {op}})$ .