The primitive ideal space of the partial-isometric crossed product of a system by a single automorphism

TL;DR

This paper characterizes the primitive ideal space of the partial-isometric crossed product of a $C^*$-algebra by a single automorphism, using its realization as a full corner of a classical crossed product.

Contribution

It provides a detailed description of the primitive ideal space for the partial-isometric crossed product via realization as a full corner, extending understanding of such structures.

Findings

Primitive ideal space described explicitly

Realization as a full corner used in analysis

Connections to classical crossed product theory

Abstract

Let $(A,\alpha)$ be a system consisting of a $C^*$-algebra $A$ and an automorphism $\alpha$ of $A$. We describe the primitive ideal space of the partial-isometric crossed product $A\times_{\alpha}^{\textrm{piso}}\mathbb{N}$ of the system by using its realization as a full corner of a classical crossed product and applying some results of Williams and Echterhoff.

Equations

$$\pi ({\alpha }_n(a))=V_n\pi (a)V_n^{\ast }\text{and}{V}_n^{\ast }{V}_n\pi (a)=\pi (a)V_n^{\ast }{V}_n$$

$$\mathcal{H}:=\overline{\mathrm{span}}\{\xi \in {\ell }^2(\mathbb{N},H):\xi (n)\in \overline{\pi }({\overline{\alpha }}_n(1))H\text{for all}n\},$$

$$\textrm{ker}\ \big{(}\operatorname{Ind}_{\Gamma_{x}}^{\Gamma}(\varepsilon_{x}\times\gamma|_{\Gamma_{x}})\big{)}=\textrm{ker}\ \big{(}\operatorname{Ind}_{\Gamma_{y}}^{\Gamma}(\varepsilon_{y}\times\mu|_{\Gamma_{y}})\big{)}.$$

$$\overline{\Gamma \cdot x}=\overline{\Gamma \cdot y}\text{and}\gamma {|}_{{\Gamma }_x}=\mu {|}_{{\Gamma }_x}.$$

$$\Phi(x,\gamma):=\ker\ \big{(}\operatorname{Ind}_{\Gamma_{x}}^{\Gamma}(\varepsilon_{x}\times\gamma|_{\Gamma_{x}})\big{)}$$

$$(\tilde{\pi }(a)\xi )(s)=\pi ({\alpha }_s^{-1}(a))\xi (s)\text{and}(U_t\xi )(s)=\xi (t^{-1}s)$$

$$\mathcal{O}(\mathrm{Prim}A)\to \mathrm{Prim}(A{\times }_{\alpha }\Gamma )$$

$$\mathcal{O}(P)\mapsto \mathrm{Ind}P=\ker (\mathrm{Ind}\pi )$$

$$\mathcal{K}({\ell }^2(\mathbb{N}))=\overline{\mathrm{span}}\{T_n(1-T{T}^{\ast })T_m^{\ast }:n,m\in \mathbb{N}\}.$$

$$0⟶(\mathcal{K}({\ell }^2(\mathbb{N}))\otimes A)\stackrel{\mu }{⟶}A{\times }_{\alpha }^{\mathrm{piso}}\mathbb{N}\stackrel{q}{⟶}A{\times }_{\alpha }\mathbb{Z}⟶0,$$

$$\mathcal{H}:=\overline{\mathrm{span}}\{(\Pi \times V{)}_P(v_n^{\ast }{j}_A(a)(1-v^{\ast }v)v_m)(\xi ):a\in A,n,m\in \mathbb{N}\}$$

$$\displaystyle\begin{array}[]{rcl}e_{n}\otimes\big{(}\pi(a)\xi(m)\big{)}&=&(V_{n}^{*}\Pi(a)(1-V^{*}V)V_{m})(\xi)\\ &=&(\Pi\times V)_{P}(v_{n}^{*}j_{A}(a)(1-v^{*}v)v_{m})(\xi),\end{array}$$

$$(\Pi \times V{)}_P(j_A(a)(1-v^{\ast }v))(e_0\otimes h)=\Pi (a)(1-V^{\ast }V)(e_0\otimes h)=e_0\otimes \pi (a)h,$$

$$\mathcal{U}:=\{\mathcal{I}\in \mathrm{Prim}(A{\times }_{\alpha }^{\mathrm{piso}}\mathbb{N}):\mathcal{K}({\ell }^2(\mathbb{N}))\otimes A\simeq \ker q\not\subset \mathcal{I}\}$$

$$\displaystyle\begin{array}[]{rcl}{\mathcal{I}}_{P}\cap({\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A)&=&\ker((\Pi\times V)_{P}|_{{\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A})\\ &=&\ker(\operatorname{id}\otimes\pi)={\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes P.\end{array}$$

$$X:=\{-1\}\cup \{-1+1/(1-n):n\in \mathbb{Z},n<0\}\cup \{1-1/(1+n):n\in \mathbb{Z},n\ge 0\}\cup \{1\}$$

$$f\in B_{\mathbb{Z}}\mapsto \tilde{f}\in C(\{-\infty \}\cup \mathbb{Z}\cup \{\infty \}),$$

$$\tilde{f}(r):=\{\begin{array}{ll}{\lim }_{n\to \infty }f(n)& \text{if}\text{r=∞,}\\ f(r)& \text{if}\text{r∈Z,}\text{and}\\ 0& \text{if}\text{r=−∞,}\end{array}$$

$$I:=\{\tilde{f}\in C(\{-\infty \}\cup \mathbb{Z}\cup \{\infty \}):\tilde{f}(-\infty )=0\}.$$

$$\{\pi \in C(\{-\infty \}\cup \mathbb{Z}\cup \{\infty \}{)}^{\wedge }:\pi {|}_I\ne 0\}=\{{\tilde{\epsilon }}_r:r\in (\mathbb{Z}\cup \{\infty \})\}$$

$$\mathcal{B}:=\{\{n\}:n\in \mathbb{Z}\}\cup \{J_n:n\in \mathbb{Z}\}$$

$$\begin{diagram}$$

$$n\cdot (m,[\pi ])=(m+n,[\pi ])\text{and}n\cdot (\infty ,[\pi ])=(n+\infty ,n\cdot [\pi ])=(\infty ,[\pi \circ {\alpha }_n])$$

$$n\cdot (m,P)=(m+n,P)\text{and}n\cdot (\infty ,P)=(\infty ,{\alpha }_n^{-1}(P))$$

$$n\cdot (m,\varphi )=(m+n,\varphi )\text{and}n\cdot (\infty ,\varphi )=(\infty ,\varphi \circ {\alpha }_n)$$

$$\displaystyle\begin{array}[]{rcl}((m,\phi),z)\sim((n,\psi),w)&\Longleftrightarrow&\overline{{\mathbb{Z}}\cdot(m,\phi)}=\overline{{\mathbb{Z}}\cdot(n,\psi)}\\ &\Longleftrightarrow&\overline{{\mathbb{Z}}\times\{\phi\}}=\overline{{\mathbb{Z}}\times\{\psi\}}\\ &\Longleftrightarrow&\overline{{\mathbb{Z}}}\times\overline{\{\phi\}}=\overline{{\mathbb{Z}}}\times\overline{\{\psi\}}\\ &\Longleftrightarrow&(\mathbb{Z}\cup\{\infty\})\times\overline{\{\phi\}}=(\mathbb{Z}\cup\{\infty\})\times\overline{\{\psi\}}\\ &\Longleftrightarrow&(\mathbb{Z}\cup\{\infty\})\times\{\phi\}=(\mathbb{Z}\cup\{\infty\})\times\{\psi\}.\end{array}$$

$$\overline{\mathbb{Z}\cdot (\infty ,\psi )}=\overline{\{\infty \}\times \mathbb{Z}\cdot \psi }=\overline{\{\infty \}}\times \overline{\mathbb{Z}\cdot \psi }=\{\infty \}\times \overline{\mathbb{Z}\cdot \psi }.$$

$$\displaystyle\begin{array}[]{rcl}((\infty,\phi),z)\sim((\infty,\psi),w)&\Longleftrightarrow&\overline{{\mathbb{Z}}\cdot(\infty,\phi)}=\overline{{\mathbb{Z}}\cdot(\infty,\psi)}\ \textrm{and}\ \gamma_{z}|_{{\mathbb{Z}}_{\phi}}=\gamma_{w}|_{{\mathbb{Z}}_{\phi}}\\ &\Longleftrightarrow&\{\infty\}\times\overline{{\mathbb{Z}}\cdot\phi}=\{\infty\}\times\overline{{\mathbb{Z}}\cdot\psi}\ \textrm{and}\ \gamma_{z}|_{{\mathbb{Z}}_{\phi}}=\gamma_{w}|_{{\mathbb{Z}}_{\phi}}.\end{array}$$

$$((\infty ,\varphi ),z)\sim ((\infty ,\psi ),w)⟺\overline{\mathbb{Z}\cdot \varphi }=\overline{\mathbb{Z}\cdot \psi }\text{and}{\gamma }_z{|}_{{\mathbb{Z}}_{\varphi }}={\gamma }_w{|}_{{\mathbb{Z}}_{\varphi }},$$

$$\Omega (A)\bigsqcup (\Omega (A)\times \mathbb{T}/\sim ).$$

Full text

1. Introduction

Lindiarni and Raeburn in [8] introduced the partial-isometric crossed product of a dynamical system $(A,\Gamma^{+},\alpha)$ in which $\Gamma^{+}$ is the positive cone of a totally ordered abelian group $\Gamma$ and $\alpha$ is an action of $\Gamma^{+}$ by endomorphisms of A. Note that since the $C^{*}$-algebra $A$ is not necessarily unital, we require that each endomorphism $\alpha_{s}$ extends to a strictly continuous endomorphism $\overline{\alpha}_{s}$ of the multiplier algebra $M(A)$. This for an endomorphism $\alpha$ of $A$ happens if and only if there exists an approximate identity $(a_{\lambda})$ in $A$ and a projection $p\in M(A)$ such that $\alpha(a_{\lambda})$ converges strictly to $p$ in $M(A)$. We stress that if $\alpha$ is extendible, then we may not have $\overline{\alpha}(1_{M(A)})=1_{M(A)}$. A covariant representation of the system $(A,\Gamma^{+},\alpha)$ is defined for which the endomorphisms $\alpha_{s}$ are implemented by partial isometries, and the associated partial-isometric crossed product $A\times_{\alpha}^{\operatorname{piso}}\Gamma^{+}$ of the system is a $C^{*}$-algebra generated by a universal covariant representation such that there is a bijection between covariant representations of the system and nondegenerate representations of $A\times_{\alpha}^{\operatorname{piso}}\Gamma^{+}$. This generalizes the covariant isometric representation theory that uses isometries to represent the semigroup of endomorphisms in a covariant representation of the system (see [3]). The authors of [8], in particular, studied the structure of the partial-isometric crossed product of the distinguished system $(B_{\Gamma^{+}},\Gamma^{+},\tau)$, where the action $\tau$ of $\Gamma^{+}$ on the subalgebra $B_{\Gamma^{+}}$ of $\ell^{\infty}(\Gamma^{+})$ is given by the right translation. Later, in [4], the authors showed that $A\times_{\alpha}^{\operatorname{piso}}\Gamma^{+}$ is a full corner in a subalgebra of the $C^{*}$-algebra ${\mathcal{L}}(\ell^{2}(\Gamma^{+})\otimes A)$ of adjointable operators on the Hilbert $A$-module $\ell^{2}(\Gamma^{+})\otimes A\simeq\ell^{2}(\Gamma^{+},A)$. This realization led them to identify the kernel of the natural homomorphism $q:A\times_{\alpha}^{\operatorname{piso}}\Gamma^{+}\rightarrow A\times_{\alpha}^{\operatorname{iso}}\Gamma^{+}$ as a full corner of the compact operators ${\mathcal{K}}(\ell^{2}({\mathbb{N}})\otimes A)$, when $\Gamma^{+}$ is ${\mathbb{N}}:={\mathbb{Z}}^{+}$. So as an application, they recovered the Pimsner-Voiculescu exact sequence in [10]. Then in their subsequent work [5], they proved that for an extendible $\alpha$-invariant ideal $I$ of $A$ (see the definition in [1]), the partial-isometric crossed product $I\times_{\alpha}^{\operatorname{piso}}\Gamma^{+}$ sits naturally as an ideal in $A\times_{\alpha}^{\operatorname{piso}}\Gamma^{+}$ such that $(A\times_{\alpha}^{\operatorname{piso}}\Gamma^{+})/(I\times_{\alpha}^{\operatorname{piso}}\Gamma^{+})\simeq A/I\times_{\tilde{\alpha}}^{\operatorname{piso}}\Gamma^{+}$. This is actually a generalization of [2, Theorem 2.2]. They then combined these results to show that the large commutative diagram of [8, Theorem 5.6] associated to the system $(B_{\Gamma^{+}},\Gamma^{+},\tau)$ is valid for any totally ordered abelian group, not only for subgroups of ${\mathbb{R}}$. In particular, they use this large commutative diagram for $\Gamma^{+}={\mathbb{N}}$ to describe the ideal structure of the algebra $B_{{\mathbb{N}}}\times_{\tau}^{\operatorname{piso}}{\mathbb{N}}$ explicitly.

Now here we consider a system $(A,\alpha)$ consisting of a $C^{*}$-algebra $A$ and an automorphism $\alpha$ of $A$. So we actually have an action of the positive cone ${\mathbb{N}}={\mathbb{Z}}^{+}$ of integers ${\mathbb{Z}}$ by automorphisms of $A$. In the present work, we want to study $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$, the primitive ideal space of the partial-isometric crossed product $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ of the system. Since $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is in fact a full corner of the classical crossed product $(B_{{\mathbb{Z}}}\otimes A)\times{\mathbb{Z}}$ (see [4, §5]), $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ is homeomorphic to $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times{\mathbb{Z}})$. Therefore it is enough to describe $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times{\mathbb{Z}})$. To do this, we apply the results on describing the primitive ideal space (ideal structure) of the classical crossed products from [12, 6]. So we consider the following two conditions:

• (1)

when $A$ is separable and abelian;

• (2)

when $A$ is separable and ${\mathbb{Z}}$ acts on $\operatorname{Prim}A$ freely (see §2).

For the first condition, by applying a theorem of Williams, $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times{\mathbb{Z}})$ is homeomorphic to a quotient space of $\Omega(B_{{\mathbb{Z}}})\times\Omega(A)\times{\mathbb{T}}$, where $\Omega(B_{{\mathbb{Z}}})$ and $\Omega(A)$ are the spectrums of the $C^{*}$-algebras $B_{{\mathbb{Z}}}$ and $A$ respectively (recall that the dual $\hat{{\mathbb{Z}}}$ is identified with ${\mathbb{T}}$ via the map $z\mapsto(\gamma_{z}:n\mapsto z^{n}$)). By computing $\Omega(B_{{\mathbb{Z}}})$, we parameterize the quotient space as a disjoint union, and then we precisely identify the open sets. For the second condition, we apply a result of Echterhoff which shows that $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times{\mathbb{Z}})$ is homeomorphic to the quasi-orbit space of $\operatorname{Prim}(B_{{\mathbb{Z}}}\otimes A)=\operatorname{Prim}B_{{\mathbb{Z}}}\times\operatorname{Prim}A$ (see in §2 that this is a quotient space of $\operatorname{Prim}(B_{{\mathbb{Z}}}\otimes A)$). Again by a similar argument to the first condition, we describe the quotient space and its topology precisely.

We begin with a preliminary section in which we recall the theory of the partial-isometric crossed products, and some discussions on the primitive ideal space of the classical crossed products briefly. In section 3, for a system $(A,\alpha)$ consisting of a $C^{*}$-algebra $A$ and an automorphism $\alpha$ of $A$, we apply the works of Williams and Echterhoff to describe $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ using the realization of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ as a full corner of the classical crossed product $(B_{{\mathbb{Z}}}\otimes A)\times{\mathbb{Z}}$. As some examples, we compute the primitive ideal space of $C({\mathbb{T}})\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ where the action $\alpha$ is given by rotation through the angle $2\pi\theta$ with $\theta$ rational and irrational. Moreover the description of the primitive ideal space of the Pimsner-Voiculescu Toeplitz algebra associated to the system $(A,\alpha)$ is completely obtained, as it is isomorphic to $A\times_{\alpha^{-1}}^{\operatorname{piso}}{\mathbb{N}}$. We also discuss necessary and sufficient conditions under which $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is GCR (postliminal or type I). Finally in the last section, we discuss the primitivity and simplicity of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$.

2. Preliminaries

2.1. The partial-isometric crossed product

A partial-isometric representation of ${\mathbb{N}}$ on a Hilbert space $H$ is a map $V:{\mathbb{N}}\rightarrow B(H)$ such that each $V_{n}:=V(n)$ is a partial isometry, and $V_{n+m}=V_{n}V_{m}$ for all $n,m\in{\mathbb{N}}$.

A covariant partial-isometric representation of $(A,\alpha)$ on a Hilbert space $H$ is a pair $(\pi,V)$ consisting of a nondegenerate representation $\pi:A\rightarrow B(H)$ and a partial-isometric representation $V:{\mathbb{N}}\rightarrow B(H)$ such that

[TABLE]

for all $a\in A$ and $n\in{\mathbb{N}}$.

Note that every system $(A,\alpha)$ admits a nontrivial covariant partial-isometric representation [8, Example 4.6]: let $\pi$ be a nondegenerate representation of $A$ on $H$. Define $\Pi:A\rightarrow B(\ell^{2}({\mathbb{N}},H))$ by $(\Pi(a)\xi)(n)=\pi(\alpha_{n}(a))\xi(n)$. If

[TABLE]

then the representation $\Pi$ is nondegenerate on ${\mathcal{H}}$. Now for every $m\in{\mathbb{N}}$, define $V_{m}$ on ${\mathcal{H}}$ by $(V_{m}\xi)(n)=\xi(n+m)$. Then the pair $(\Pi|_{{\mathcal{H}}},V)$ is a partial-isometric covariant representation of $(A,\alpha)$ on ${\mathcal{H}}$. One can see that if we take $\pi$ faithful, then $\Pi$ will be faithful as well, and ${\mathcal{H}}=\ell^{2}({\mathbb{N}},H)$ whenever $\overline{\alpha}(1)=1$ (e.g. when $\alpha$ is an automorphism).

Definition 2.1*.*

A partial-isometric crossed product of $(A,\alpha)$ is a triple $(B,j_{A},j_{{\mathbb{N}}})$ consisting of a $C^{*}$-algebra $B$, a nondegenerate homomorphism $i_{A}:A\rightarrow B$, and a partial-isometric representation $i_{{\mathbb{N}}}:{\mathbb{N}}\rightarrow M(B)$ such that:

• (i)

the pair $(j_{A},j_{{\mathbb{N}}})$ is a covariant representation of $(A,\alpha)$ in $B$;

• (ii)

for every covariant partial-isometric representation $(\pi,V)$ of $(A,\alpha)$ on a Hilbert space $H$, there exists a nondegenerate representation $\pi\times V:B\rightarrow B(H)$ such that $(\pi\times V)\circ i_{A}=\pi$ and $(\overline{\pi\times V})\circ i_{{\mathbb{N}}}=V$; and

• (iii)

the $C^{*}$-algebra $B$ is spanned by $\{i_{{\mathbb{N}}}(n)^{*}i_{A}(a)i_{{\mathbb{N}}}(m):n,m\in{\mathbb{N}},a\in A\}$.

By [8, Proposition 4.7], the partial-isometric crossed product of $(A,\alpha)$ always exists, and it is unique up to isomorphism. Thus we write the partial-isometric crossed product $B$ as $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$.

We recall that by [8, Theorem 4.8], a covariant representation $(\pi,V)$ of $(A,\alpha)$ on $H$ induces a faithful representation $\pi\times V$ of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ if and only if $\pi$ is faithful on the range of $(1-V_{n}^{*}V_{n})$ for every $n>0$ (one can actually see that it is enough to verify that $\pi$ is faithful on the range of $(1-V^{*}V)$, where $V:=V_{1}$).

2.2. The primitive ideal space of crossed products associated to second countable locally compact transformation groups

Let $\Gamma$ be a discrete group which acts on a topological space $X$. For every $x\in X$, the set $\Gamma\cdot x:=\{s\cdot x:s\in\Gamma\}$ is called the $\Gamma$-orbit of $x$. The set $\Gamma_{x}:=\{s\in\Gamma:s\cdot x=x\}$, which is a subgroup of $\Gamma$, is called the stability group of $x$. We say the $\Gamma$-action is free or $\Gamma$ acts on $X$ freely if $\Gamma_{x}=\{e\}$ for all $x\in X$. Consider a relation $\sim$ on $X$ such that for $x,y\in X$, $x\sim y$ if and only if $\overline{\Gamma\cdot x}=\overline{\Gamma\cdot y}$. One can see that this is an equivalence relation on $X$. The set of all equivalence classes equipped with the quotient topology is denoted by ${\mathcal{O}}(X)$ and called the quasi-orbit space, which is always a $T_{0}$-topological space. The equivalence class of each $x\in X$ is denoted by ${\mathcal{O}}(x)$ and called the quasi-orbit of $x$.

Now let $\Gamma$ be an abelian countable discrete group which acts on a second countable locally compact Hausdorff space $X$. So $(\Gamma,X)$ is a second countable locally compact transformation group with $\Gamma$ abelian. Then the associated dynamical system $(C_{0}(X),\Gamma,\tau)$ is separable with $\Gamma$ abelian, and so the primitive ideals of $C_{0}(X)\times_{\tau}\Gamma$ are known (see [12, Theorem 8.21]). Furthermore, the topology of $\operatorname{Prim}(C_{0}(X)\times_{\tau}\Gamma)$ has been beautifully described [12, Theorem 8.39]. So here we want to recall the discussion on $\operatorname{Prim}(C_{0}(X)\times_{\tau}\Gamma)$ in brief. See more in [12] that this is indeed a huge and deep discussion.

Let $N$ be a subgroup of $\Gamma$. If we restrict the action $\tau$ to $N$, then we obtain a dynamical system $(C_{0}(X),N,\tau|_{N})$ with the associated crossed product $C_{0}(X)\times_{\tau|_{N}}N$. Suppose that $\textsf{X}_{N}^{\Gamma}$ is the Green’s $((C_{0}(X)\otimes C_{0}(\Gamma/N))\times_{\tau\otimes\operatorname{lt}}\Gamma)-(C_{0}(X)\times_{\tau|_{N}}N)$-imprimitivity bimodule whose structure can be found in [12, Theorem 4.22]. If $(\pi,V)$ is a covariant representation of $(C_{0}(X),N,\tau|_{N})$, then $\operatorname{Ind}_{N}^{\Gamma}(\pi\times V)$ denotes the representation of $C_{0}(X)\times_{\tau}\Gamma$ induced from the representation $\pi\times V$ of $C_{0}(X)\times_{\tau|_{N}}N$ via $\textsf{X}_{N}^{\Gamma}$. Now for $x\in X$, let $\varepsilon_{x}:C_{0}(X)\rightarrow\mathbb{C}\simeq B(\mathbb{C})$ be the evaluation map at $x$ and $w$ a character of $\Gamma_{x}$. Then the pair $(\varepsilon_{x},w)$ is a covariant representation of $(C_{0}(X),\Gamma_{x},\tau|_{\Gamma_{x}})$ such that the associated representation $\varepsilon_{x}\times w$ of $C_{0}(X)\times\Gamma_{x}$ is irreducible, and hence by [12, Proposition 8.27], $\operatorname{Ind}_{\Gamma_{x}}^{\Gamma}(\varepsilon_{x}\times w)$ is an irreducible representation of $C_{0}(X)\times_{\tau}\Gamma$. So \textrm{ker}\ \big{(}\operatorname{Ind}_{\Gamma_{x}}^{\Gamma}(\varepsilon_{x}\times w)\big{)} is a primitive ideal of $C_{0}(X)\times_{\tau}\Gamma$. Note if a primitive ideal is obtained in this way, then we say it is induced from a stability group. In fact by [12, Theorem 8.21], all primitive ideals of $C_{0}(X)\times_{\tau}\Gamma$ are induced from stability groups. Moreover since for every $w\in\widehat{\Gamma_{x}}$ there is a $\gamma\in\widehat{\Gamma}$ such that $w=\gamma|_{\Gamma_{x}}$, every primitive ideal of $C_{0}(X)\times_{\tau}\Gamma$ is actually given by the kernel of an induced irreducible representation $\operatorname{Ind}_{\Gamma_{x}}^{\Gamma}(\varepsilon_{x}\times\gamma|_{\Gamma_{x}})$ correspondent to a pair $(x,\gamma)$ in $X\times\widehat{\Gamma}$. To see the description of the topology of $\operatorname{Prim}(C_{0}(X)\times_{\tau}\Gamma)$, first note that if $(x,\gamma)$ and $(y,\mu)$ belong to $X\times\widehat{\Gamma}$ such that $\overline{\Gamma\cdot x}=\overline{\Gamma\cdot y}$ (which implies that $\Gamma_{x}=\Gamma_{y}$) and $\gamma|_{\Gamma_{x}}=\mu|_{\Gamma_{x}}$, then by [12, Lemma 8.34],

[TABLE]

So define a relation on $X\times\widehat{\Gamma}$ such that $(x,\gamma)\sim(y,\mu)$ if

[TABLE]

One can see that $\sim$ is an equivalence relation on $X\times\widehat{\Gamma}$. Now consider the quotient space $X\times\widehat{\Gamma}/\sim$ equipped with the quotient topology. Then we have:

Theorem 2.2**.**

[12*, Theorem 8.39]**

Let $(\Gamma,X)$ be a second countable locally compact transformation group with $\Gamma$ abelian. Then the map $\Phi:X\times\widehat{\Gamma}\rightarrow\operatorname{Prim}(C_{0}(X)\times_{\tau}\Gamma)$ defined by*

[TABLE]

is a continuous and open surjection, and factors through a homeomorphism of $X\times\widehat{\Gamma}/\sim$ onto $\operatorname{Prim}(C_{0}(X)\times_{\tau}\Gamma)$.

Remark 2.3*.*

In the theorem above, note that $\operatorname{Prim}(C_{0}(X)\times_{\tau}\Gamma)$ is then a second countable space. This is because as it is mentioned in [12, Remark 8.40], the quotient map $\textsf{q}:X\times\widehat{\Gamma}\rightarrow X\times\widehat{\Gamma}/\sim$ is open. Moreover, $X$ and $\widehat{\Gamma}$ both are second countable.

Theorem 2.2 can be applied to see that the primitive ideal space of the rational rotation algebra is homeomorphic to $\mathbb{T}^{2}$. We skip it here and refer readers to [12, Example 8.45] for more details.

2.3. The primitive ideal space of crossed products by free actions

Let $(A,\Gamma,\alpha)$ be a classical dynamical system with $\Gamma$ discrete. Then the system gives an action of $\Gamma$ on the spectrum $\hat{A}$ of $A$ by $s\cdot[\pi]:=[\pi\circ\alpha_{s}^{-1}]$ for every $s\in\Gamma$ and $[\pi]\in\hat{A}$ (see [12, Lemma 2.8] and [11, Lemma 7.1]). This also induces an action of $\Gamma$ on $\operatorname{Prim}A$ such that $s\cdot P:=\alpha_{s}(P)$ for each $s\in\Gamma$ and $P\in\operatorname{Prim}A$.

Recall that if $\pi$ is a (nondegenerate) representation of $A$ on $H$ with $\ker\pi=J$, then $\operatorname{Ind}\pi$ denotes the induced representation $\tilde{\pi}\times U$ of $A\times_{\alpha}\Gamma$ on $\ell^{2}(\Gamma,H)$ associated to the covariant pair $(\tilde{\pi},U)$ of $(A,\Gamma,\alpha)$ defined by

[TABLE]

for all every $a\in A$, $\xi\in\ell^{2}(\Gamma,H)$, and $s,t\in\Gamma$. Note that by $\operatorname{Ind}J$, we mean $\ker(\operatorname{Ind}\pi)$.

Now let $(A,\Gamma,\alpha)$ be a classical dynamical system in which $A$ is separable and $\Gamma$ is an abelian discrete countable group. If $\Gamma$ acts on $\operatorname{Prim}A$ freely, then each primitive ideal $\ker\pi=P$ of $A$ induces a primitive ideal of $A\times_{\alpha}\Gamma$, namely $\operatorname{Ind}P=\ker(\operatorname{Ind}\pi)$, and the description of $\operatorname{Prim}(A\times_{\alpha}\Gamma)$ is completely available:

Theorem 2.4**.**

[6*, Corollary 10.16]**

Suppose in the system $(A,\Gamma,\alpha)$ that $A$ is separable and $\Gamma$ is an amenable discrete countable group. If $\Gamma$ acts on $\operatorname{Prim}A$ freely, then the map*

[TABLE]

[TABLE]

is a homeomorphism, where $\pi$ is an irreducible representation of $A$ with $\ker\pi=P$. In particular, $A\times_{\alpha}\Gamma$ is simple if and only if every $\Gamma$-orbit is dense in $\operatorname{Prim}A$.

We can apply the above Theorem to see that the irrational rotation algebras are simple. Readers can refer to [6, Example 10.18] or [12, Example 8.46] for more details.

3. The Primitive Ideal Space of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ by automorphic action

First recall that if $T$ is the isometry in $B(\ell^{2}({\mathbb{N}}))$ such that $T(e_{n})=e_{n+1}$ on the usual orthonormal basis $\{e_{n}\}_{n=0}^{\infty}$ of $\ell^{2}(\mathbb{N})$, then we have

[TABLE]

Now consider a system $(A,\alpha)$ consisting of a $C^{*}$-algebra $A$ and an automorphism $\alpha$ of $A$. Let the triples $(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}},j_{A},v)$ and $(A\times_{\alpha}{\mathbb{Z}},i_{A},u)$ be the partial-isometric crossed product and the classical crossed product of the system respectively. Here our goal is to describe the primitive ideal space of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ and its topology completely. See in [4] that the kernel of the natural homomorphism $q:(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}},j_{A},v)\rightarrow(A\times_{\alpha}{\mathbb{Z}},i_{A},u)$ given by $q(v_{n}^{*}j_{A}(a)v_{m})=u_{n}^{*}i_{A}(a)u_{m}$, is isomorphic to the algebra of compact operators ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$. Therefore we have a short exact sequence

[TABLE]

where $\mu(T_{n}(1-TT^{*})T_{m}^{*}\otimes a)=v_{n}^{*}j_{A}(a)(1-v^{*}v)v_{m}$ for all $a\in A$ and $n,m\in{\mathbb{N}}$. So $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ as a set, is given by the sets $\operatorname{Prim}({\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A)$ and \operatorname{Prim}\big{(}A\times_{\alpha}{\mathbb{Z}}). With no condition on the system, we do not have much information about \operatorname{Prim}\big{(}A\times_{\alpha}{\mathbb{Z}}) in general. However, by [4, Proposition 2.5], we do know that $\ker q\simeq{\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$ is an essential ideal of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$. Therefore $\operatorname{Prim}({\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A)$ which is homeomorphic to $\operatorname{Prim}A$, sits in $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ as an open dense subset. We will identify this open dense subset, namely the primitive ideals $\{{\mathcal{I}}_{P}:P\in\operatorname{Prim}A\}$ of $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ coming from $\operatorname{Prim}A$, shortly. Moreover see in [4, §5] that $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is a full corner of the classical crossed product $(B_{{\mathbb{Z}}}\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}}$, where $B_{\mathbb{Z}}:=\overline{\operatorname{span}}\{1_{n}:n\in\mathbb{Z}\}\subset\ell^{\infty}(\mathbb{Z})$, and the action $\beta$ of $\mathbb{Z}$ on $B_{\mathbb{Z}}$ is given by translation such that $\beta_{m}(1_{n})=1_{n+m}$ for all $m,n\in\mathbb{Z}$. Thus $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ is homeomorphic to $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}})$, and hence it suffices to describe $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}})$ and its topology. To do this, we will consider two conditions on the system that make us able to apply a theorem of Williams and a result by Echterhoff. We will also identify those primitive ideals of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ coming from \operatorname{Prim}\big{(}A\times_{\alpha}{\mathbb{Z}}), which form a closed subset of $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$. But first, let us identify the primitive ideals ${\mathcal{I}}_{P}$.

Proposition 3.1**.**

Let $\pi:A\rightarrow B(H)$ be a nonzero irreducible representation of $A$ with $P:=\ker\pi$. If the pair $(\Pi,V)$ is defined as in [8, Example 4.6] (see §2), then the associated representation of $(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}},j_{A},v)$, which we denote by $(\Pi\times V)_{P}$, is irreducible on $\ell^{2}({\mathbb{N}},H)$, and does not vanish on $\ker q\simeq{\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$.

Proof.

To see that $(\Pi\times V)_{P}$ is irreducible, we show that every $\xi\in\ell^{2}({\mathbb{N}},H)\backslash\{0\}$ is a cyclic vector for $(\Pi\times V)_{P}$, that is $\ell^{2}({\mathbb{N}},H)=\overline{\operatorname{span}}\{(\Pi\times V)_{P}(x)(\xi):x\in(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})\}$. We show that

[TABLE]

equals $\ell^{2}({\mathbb{N}},H)$ which is enough. By viewing $\ell^{2}(\mathbb{N},H)$ as the Hilbert space $\ell^{2}(\mathbb{N})\otimes H$, it suffices to see that each $e_{n}\otimes h$ belongs to ${\mathcal{H}}$, where $\{e_{n}\}_{n=0}^{\infty}$ is the usual orthonormal basis of $\ell^{2}(\mathbb{N})$ and $h\in H$. Since $\xi\neq 0$ in $\ell^{2}(\mathbb{N},H)$, there is $m\in\mathbb{N}$ such that $\xi(m)\neq 0$ in $H$. But $\xi(m)$ is a cyclic vector for the representation $\pi:A\rightarrow B(H)$ as $\pi$ is irreducible. Thus we have $\overline{\textrm{span}}\{\pi(a)(\xi(m)):a\in A\}=H$, and hence \textrm{span}\{e_{n}\otimes\big{(}\pi(a)\xi(m)\big{)}:n\in{\mathbb{N}},a\in A\} is dense in $\ell^{2}(\mathbb{N})\otimes H\simeq\ell^{2}(\mathbb{N},H)$. So we only have to show that ${\mathcal{H}}$ contains each element e_{n}\otimes\big{(}\pi(a)\xi(m)\big{)}. Calculation shows that

[TABLE]

and therefore $e_{n}\otimes(\pi(a)\xi(m))\in{\mathcal{H}}$ for every $a\in A$ and $n\in{\mathbb{N}}$. So we have ${\mathcal{H}}=\ell^{2}(\mathbb{N},H)$.

To show that $(\Pi\times V)_{P}$ does not vanish on ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$, first note that since $\pi$ is nonzero, $\pi(a)h\neq 0$ for some $a\in A$ and $h\in H$. Now if we take $(1-TT^{*})\otimes a\in{\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$, then $(\Pi\times V)_{P}(\mu((1-TT^{*})\otimes a))=(\Pi\times V)_{P}(j(a)(1-v^{*}v))\neq 0$. This is because for $(e_{0}\otimes h)\in\ell^{2}(\mathbb{N},H)$, we have

[TABLE]

which is not zero in $\ell^{2}({\mathbb{N}},H)$ as $\pi(a)h\neq 0$. ∎

Remark 3.2*.*

The primitive ideals ${\mathcal{I}}_{P}$ are actually the kernels of the irreducible representations $(\Pi\times V)_{P}$ which form the open dense subset

[TABLE]

of \operatorname{Prim}\big{(}A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}\big{)} homeomorphic to $\operatorname{Prim}({\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A)$. Now $\operatorname{Prim}({\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A)$ itself is homeomorphic to $\operatorname{Prim}A$ via the (Rieffel) homeomorphism $P\mapsto{\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes P$. But ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes P$ is the kernel of the irreducible representation $(\operatorname{id}\otimes\pi)$ of ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$, which indeed equals the restriction $(\Pi\times V)_{P}|_{{\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A}$. Therefore we have

[TABLE]

Consequently the map $P\mapsto{\mathcal{I}}_{P}$ is a homeomorphism of $\operatorname{Prim}A$ onto the open dense subset ${\mathcal{U}}$ of $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$.

Now we want to describe the topology of $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}})\simeq\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ and identify the primitive ideals of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ coming from $A\times_{\alpha}{\mathbb{Z}}$ under the following two conditions:

• (1)

when $A$ is separable and abelian, by applying a theorem of Williams, namely Theorem 2.2;

• (2)

when $A$ is separable and ${\mathbb{Z}}$ acts on $\operatorname{Prim}A$ freely, by applying Theorem 2.4.

3.1. The topology of $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}})$ when $A$ is separable and abelian

Suppose that $A$ is separable and abelian. Then $(B_{{\mathbb{Z}}}\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}}$ is isomorphic to the crossed product $C_{0}(\Omega(B_{{\mathbb{Z}}}\otimes A))\times_{\tau}{\mathbb{Z}}$ associated to the second countable locally compact transformation group $({\mathbb{Z}},\Omega(B_{{\mathbb{Z}}}\otimes A))$. Therefore by Theorem 2.2, $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}})$ is homeomorphic to $\Omega(B_{{\mathbb{Z}}}\otimes A)\times{\mathbb{T}}/\sim$. But we want to describe $\Omega(B_{{\mathbb{Z}}}\otimes A)\times{\mathbb{T}}/\sim$ precisely. To do this, we need to analyze $\Omega(B_{{\mathbb{Z}}}\otimes A)$, and since $\Omega(B_{{\mathbb{Z}}}\otimes A)\simeq\Omega(B_{{\mathbb{Z}}})\times\Omega(A)$ (see [11, Theorem B.37] or [11, Theorem B.45]), we have to compute $\Omega(B_{{\mathbb{Z}}})$ first.

Lemma 3.3**.**

Let $\{-\infty\}\cup\mathbb{Z}\cup\{\infty\}$ be the two-point compactification of $\mathbb{Z}$. Then $\Omega(B_{{\mathbb{Z}}})$ is homeomorphic to the open dense subset $\mathbb{Z}\cup\{\infty\}$.

Proof.

First note that $B_{\mathbb{Z}}$ exactly consists of those functions $f:\mathbb{Z}\rightarrow\mathbb{C}$ such that $\lim_{n\rightarrow-\infty}f(n)=0$ and $\lim_{n\rightarrow\infty}f(n)$ exists. Thus the complex homomorphisms (irreducible representations) of $B_{\mathbb{Z}}$ are given by the evaluation maps $\{\varepsilon_{n}:n\in\mathbb{Z}\}$, and the map $\varepsilon_{\infty}:B_{\mathbb{Z}}\rightarrow\mathbb{C}$ defined by $\varepsilon_{\infty}(f):=\lim_{n\rightarrow\infty}f(n)$ for all $f\in B_{\mathbb{Z}}$. So we have $\Omega(B_{\mathbb{Z}})=\{\varepsilon_{n}:n\in\mathbb{Z}\}\cup\{\varepsilon_{\infty}\}$. Note that the kernel of $\varepsilon_{\infty}$ is the ideal $C_{0}(\mathbb{Z})=\overline{\operatorname{span}}\{1_{n}-1_{m}:n<m\in\mathbb{Z}\}$ of $B_{\mathbb{Z}}$. Now let $\{-\infty\}\cup\mathbb{Z}\cup\{\infty\}$ be the two-point compactification of $\mathbb{Z}$ which is homeomorphic to the subspace

[TABLE]

of $\mathbb{R}$. Then the map

[TABLE]

where

[TABLE]

embeds $B_{\mathbb{Z}}$ in $C(\{-\infty\}\cup\mathbb{Z}\cup\{\infty\})$ as the maximal ideal

[TABLE]

Thus it follows that $\Omega(B_{\mathbb{Z}})$ is homeomorphic to $\hat{I}$, and $\hat{I}$ itelf is homeomorphic to the open subset

[TABLE]

of $C(\{-\infty\}\cup\mathbb{Z}\cup\{\infty\})^{\wedge}$ in which each $\tilde{\varepsilon}_{r}$ is an evaluation map. So by the homeomorphism between $C(\{-\infty\}\cup\mathbb{Z}\cup\{\infty\})^{\wedge}$ and $\{-\infty\}\cup\mathbb{Z}\cup\{\infty\}$, the open subset $\{\tilde{\varepsilon}_{r}:r\in(\mathbb{Z}\cup\{\infty\})\}$ is homeomorphic to the open (dense) subset $\mathbb{Z}\cup\{\infty\}$ of $\{-\infty\}\cup\mathbb{Z}\cup\{\infty\}$ equipped with the relative topology. Therefore $\Omega(B_{\mathbb{Z}})$ is in fact homeomorphic to $\mathbb{Z}\cup\{\infty\}$. One can see that $\mathbb{Z}\cup\{\infty\}$ is indeed a second countable locally compact Hausdorff space with

[TABLE]

as a countable basis for its topology, where $J_{n}:=\{n,n+1,n+2,...\}\cup\{\infty\}$ for every $n\in\mathbb{Z}$. ∎

Remark 3.4*.*

Before we continue, we need to mention that, if $A$ is a separable $C^{*}$-algebra (not necessarily abelian), then by [11, Theorem B.45] and using Lemma 3.3, $(C_{0}({{\mathbb{Z}}})\otimes A)^{\widehat{}}$ and $(B_{{\mathbb{Z}}}\otimes A)^{\widehat{}}$ are homeomorphic to ${\mathbb{Z}}\times\hat{A}$ and $(\mathbb{Z}\cup\{\infty\})\times\hat{A}$ respectively. Also $\operatorname{Prim}(C_{0}({{\mathbb{Z}}})\otimes A)$ and $\operatorname{Prim}(B_{{\mathbb{Z}}}\otimes A)$ are homeomorphic to ${\mathbb{Z}}\times\operatorname{Prim}A$ and $(\mathbb{Z}\cup\{\infty\})\times\operatorname{Prim}A$ respectively (note that these homeomorphisms are ${\mathbb{Z}}$-equivariant for the action of ${\mathbb{Z}}$). Since $C_{0}({{\mathbb{Z}}})\otimes A$ is an (essential) ideal of $B_{{\mathbb{Z}}}\otimes A$, we have the following commutative diagram:

[TABLE]

where $\Theta$ and $\tilde{\Theta}$ are the canonical continuous, open surjections, and $\iota$ an $\tilde{\iota}$ are the canonical embedding maps. Now to see how ${\mathbb{Z}}$ acts on $(\mathbb{Z}\cup\{\infty\})\times\hat{A}$ (and accordingly on $(\mathbb{Z}\cup\{\infty\})\times\operatorname{Prim}A)$, note that since the crossed products $(C_{0}({{\mathbb{Z}}})\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}}$ and $(C_{0}({{\mathbb{Z}}})\otimes A)\times_{\beta\otimes\operatorname{id}}{\mathbb{Z}}$ are isomorphic (see [12, Lemma 7.4]), we have

[TABLE]

for all $n,m\in{\mathbb{Z}}$ and $[\pi]\in\hat{A}$. Accordingly

[TABLE]

for all $n,m\in{\mathbb{Z}}$ and $P\in\operatorname{Prim}A$.

So when $A$ is separable and abelian, using Lemma 3.3, $\Omega(B_{{\mathbb{Z}}}\otimes A)=(\mathbb{Z}\cup\{\infty\})\times\Omega(A)$. Now to describe $((\mathbb{Z}\cup\{\infty\})\times\Omega(A))\times{\mathbb{T}}/\sim$, note that by Remark 3.4, ${\mathbb{Z}}$ acts on $(\mathbb{Z}\cup\{\infty\})\times\Omega(A)$ as follows:

[TABLE]

for all $n,m\in{\mathbb{Z}}$ and $\phi\in\Omega(A)$. Therefore, the stability group of each $(m,\phi)$ is $\{0\}$, and the stability group of each $(\infty,\phi)$ equals the stability group ${\mathbb{Z}}_{\phi}$ of $\phi$. Accordingly, the $\mathbb{Z}$-orbit of each $(m,\phi)$ is ${\mathbb{Z}}\times\{\phi\}$, and the $\mathbb{Z}$-orbit of $(\infty,\phi)$ is $\{\infty\}\times{\mathbb{Z}}\cdot\phi$, where ${\mathbb{Z}}\cdot\phi$ is the $\mathbb{Z}$-orbit of $\phi$. So for the pairs (or triples) $((m,\phi),z)$ and $((n,\psi),w)$ of $(\mathbb{Z}\times\Omega(A))\times\mathbb{T}$, we have

[TABLE]

The last equivalence follows from the fact that $\Omega(A)$ is Hausdorff. Therefore $((m,\phi),z)$ and $((n,\psi),w)$ are in the same equivalence class in $((\mathbb{Z}\cup\{\infty\})\times\Omega(A))\times{\mathbb{T}}/\sim$ if and only if $\phi=\psi$, while $((m,\phi),z)\nsim((\infty,\psi),w)$ for every $\psi\in\Omega(A)$ and $w\in\mathbb{T}$, because

[TABLE]

Thus if $\phi\in\Omega(A)$, then all pairs $((m,\phi),z)$ for every $m\in{\mathbb{Z}}$ and $z\in{\mathbb{T}}$ are in the same equivalence class, which can be parameterized by $\phi\in\Omega(A)$. On the other hand, for the pairs $((\infty,\phi),z)$ and $((\infty,\psi),w)$, we have

[TABLE]

Therefore

[TABLE]

which means if and only if the pairs $(\phi,z)$ and $(\psi,w)$ are in the same equivalence class in the quotient space $\Omega(A)\times{\mathbb{T}}/\sim$ homeomorphic to \operatorname{Prim}\big{(}A\times_{\alpha}{\mathbb{Z}}). Therefore $((\infty,\phi),z)\sim((\infty,\psi),w)$ in $((\mathbb{Z}\cup\{\infty\})\times\Omega(A))\times{\mathbb{T}}/\sim$ precisely when $(\phi,z)\sim(\psi,w)$ in $\Omega(A)\times{\mathbb{T}}/\sim$, and hence the class of each $((\infty,\phi),z)$ in $((\mathbb{Z}\cup\{\infty\})\times\Omega(A))\times{\mathbb{T}}/\sim$ can be parameterized by the class of $(\phi,z)$ in $\Omega(A)\times{\mathbb{T}}/\sim$. So we can identify $((\mathbb{Z}\cup\{\infty\})\times\Omega(A))\times{\mathbb{T}}/\sim$ with the disjoint union

[TABLE]

Now we have:

Theorem 3.5**.**

Let $(A,\alpha)$ be a system consisting of a separable abelian $C^{*}$-algebra $A$ and an automorphism $\alpha$ of $A$. Then $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ is homeomorphic to $\Omega(A)\sqcup(\Omega(A)\times{\mathbb{T}}/\sim)$, equipped with the (quotient) topology in which the open sets are of the form

[TABLE]

[TABLE]

Proof.

Since the quotient map $\textsf{q}:((\mathbb{Z}\cup\{\infty\})\times\Omega(A))\times{\mathbb{T}}\rightarrow\Omega(A)\sqcup(\Omega(A)\times{\mathbb{T}}/\sim)$ is open, as well as $\tilde{\textsf{q}}:\Omega(A)\times{\mathbb{T}}\rightarrow\Omega(A)\times{\mathbb{T}}/\sim$, for every $n\in{\mathbb{Z}}$, every open subset $O$ of $\Omega(A)$, and every open subset $V$ of ${\mathbb{T}}$, the forward image of open subsets $\{n\}\times O\times V$ and $J_{n}\times O\times V$ by q, forms a basis for the topology of $\Omega(A)\sqcup(\Omega(A)\times{\mathbb{T}}/\sim)$, which is

[TABLE]

[TABLE]

As the open subsets $\tilde{\textsf{q}}(O\times V)$ also form a basis for the quotient topology of $\Omega(A)\times{\mathbb{T}}/\sim$, we can see that each open subset of $\Omega(A)\sqcup(\Omega(A)\times{\mathbb{T}}/\sim)$ is either an open subset $U$ of $\Omega(A)$ or of the form $U\cup W$ for some nonempty open subset $U$ in $\Omega(A)$ and some open subset $W$ in $\Omega(A)\times{\mathbb{T}}/\sim$. ∎

Remark 3.6*.*

Under the condition of Theorem 3.5, the primitive ideals of $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ coming from $\operatorname{Prim}(A\times_{\alpha}{\mathbb{Z}})$, which form the closed subset

[TABLE]

are the kernels of the irreducible representations $(\operatorname{Ind}_{{\mathbb{Z}}_{\phi}}^{{\mathbb{Z}}}(\phi\times\gamma_{z}|_{{\mathbb{Z}}_{\phi}}))\circ q$ corresponding to the equivalence classes of the pairs $(\phi,z)$ in $\Omega(A)\times{\mathbb{T}}/\sim$ (again by using Theorem 2.2). Therefore if ${\mathcal{J}}_{[(\phi,z)]}$ denotes $\ker(\operatorname{Ind}_{{\mathbb{Z}}_{\phi}}^{{\mathbb{Z}}}(\phi\times\gamma_{z}|_{{\mathbb{Z}}_{\phi}})\circ q)$, then $\mathcal{F}=\{{\mathcal{J}}_{[(\phi,z)]}:\phi\in\Omega(A),z\in{\mathbb{T}}\}$, and the map $[(\phi,z)]\mapsto{\mathcal{J}}_{[(\phi,z)]}$ is homeomorphism of $\operatorname{Prim}(A\times_{\alpha}{\mathbb{Z}})\simeq\Omega(A)\times{\mathbb{T}}/\sim$ onto $\mathcal{F}$.

Proposition 3.7**.**

Let $(A,\alpha)$ be a system consisting of a separable abelian $C^{*}$-algebra $A$ and an automorphism $\alpha$ of $A$. Then $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is GCR if and only if ${\mathbb{Z}}\backslash\Omega(A)$ is a $T_{0}$ space.

Proof.

By [9, Theorem 5.6.2], $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is GCR if and only if ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A\simeq\ker q$ and $A\times_{\alpha}{\mathbb{Z}}\simeq C_{0}(\Omega(A))\times_{\tau}{\mathbb{Z}}$ are GCR. But since $A$ is abelian, ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$ is automatically CCR, and hence it is GCR. Therefore $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is GCR precisely when $A\times_{\alpha}{\mathbb{Z}}$ is GCR. By [12, Theorem 8.43], $A\times_{\alpha}{\mathbb{Z}}$ is GCR if and only if ${\mathbb{Z}}\backslash\Omega(A)$ is $T_{0}$. ∎

Proposition 3.8**.**

Let $(A,\alpha)$ be a system consisting of a separable abelian $C^{*}$-algebra $A$ and an automorphism $\alpha$ of $A$. Then $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is not CCR.

Proof.

Note that $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is CCR if and only if $(B_{{\mathbb{Z}}}\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}}\simeq C_{0}(\Omega(B_{{\mathbb{Z}}}\otimes A))\times_{\tau}{\mathbb{Z}}$ is CCR, because they are Morita equivalent (see [12, Proposition I.43]). Since for the $\mathbb{Z}$-orbit of a pair $(m,\phi)$, we have

[TABLE]

it follows that $\mathbb{Z}$-orbit of $(m,\phi)$ is not closed in $\Omega(B_{{\mathbb{Z}}}\otimes A)=(\mathbb{Z}\cup\{\infty\})\times\Omega(A)$. Therefore by [12, Theorem 8.44], $C_{0}(\Omega(B_{{\mathbb{Z}}}\otimes A))\times_{\tau}{\mathbb{Z}}$ is not CCR, and hence $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is not CCR. ∎

Example 3.9*.*

(Pimsner-Voiculescu Toeplitz algebra)

Suppose ${\mathcal{T}}(A,\alpha)$ is the Pimsner-Voiculescu Toeplitz algebra associated to the system $(A,\alpha)$ (see [10]). It was shown in [4, §5] that ${\mathcal{T}}(A,\alpha)$ is isomorphic to the partial-isometric crossed product $A\times_{\alpha^{-1}}^{\operatorname{piso}}{\mathbb{N}}$ associated to the system $(A,\alpha^{-1})$. Therefore when $A$ is abelian and separable, the description of $\operatorname{Prim}({\mathcal{T}}(A,\alpha))$ follows completely from Theorem 3.5. In particular, for the trivial system $({\mathbb{C}},\operatorname{id})$, ${\mathcal{T}}({\mathbb{C}},\operatorname{id})$ is the Toeplitz algebra ${\mathcal{T}}({\mathbb{Z}})$ of integers isomorphic to ${\mathbb{C}}\times_{\operatorname{id}}^{\operatorname{piso}}{\mathbb{N}}$. So again by Theorem 3.5, $\operatorname{Prim}({\mathcal{T}}({\mathbb{Z}}))$ corresponds to the disjoint union $\{0\}\sqcup{\mathbb{T}}$ in which every (nonempty) open set is of the form $\{0\}\cup W$ for some open subset $W$ of $\mathbb{T}$. This description is known which coincides with the description of $\operatorname{Prim}({\mathcal{T}}({\mathbb{Z}}))$ obtained from the well-known short exact sequence $0\rightarrow{\mathcal{K}}(\ell^{2}({\mathbb{N}}))\rightarrow{\mathcal{T}}({\mathbb{Z}})\rightarrow C({\mathbb{T}})\rightarrow 0$.

Example 3.10*.*

Consider the system $(C({\mathbb{T}}),\alpha)$ in which the action $\alpha$ is given by rotation through the angle $2\pi\theta$ with $\theta$ rational. By using the discussion in [12, Example 8.46], $\operatorname{Prim}(C({\mathbb{T}})\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ can be identified with the disjoint union

[TABLE]

in which by Theorem 3.5, each open set is given by

[TABLE]

[TABLE]

Moreover the orbit space ${\mathbb{Z}}\backslash\ {\mathbb{T}}$ is homeomorphic to ${\mathbb{T}}$, which is obviously $T_{0}$ (in fact Hausdorff). So it follows by Proposition 3.7 that $C({\mathbb{T}})\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is GCR.

3.2. The topology of $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}})$ when $A$ is separable and ${\mathbb{Z}}$ acts on $\operatorname{Prim}A$ freely

Consider a system $(A,\alpha)$ in which $A$ is separable, and ${\mathbb{Z}}$ acts on $\operatorname{Prim}A$ freely. It follows that ${\mathbb{Z}}$ acts on $\operatorname{Prim}(B_{{\mathbb{Z}}}\otimes A)$ freely too. This is because, firstly, by [11, Theorem B.45], $\operatorname{Prim}(B_{{\mathbb{Z}}}\otimes A)$ is homeomorphic to $\operatorname{Prim}B_{{\mathbb{Z}}}\times\operatorname{Prim}A$, and hence it is homeomorphic to $(\mathbb{Z}\cup\{\infty\})\times\operatorname{Prim}A$. Then ${\mathbb{Z}}$ acts on $(\mathbb{Z}\cup\{\infty\})\times\operatorname{Prim}A$ such that

[TABLE]

for all $n,m\in{\mathbb{Z}}$ and $P\in\operatorname{Prim}A$. Therefore the stability group of each $(\infty,P)$ equals the stability group ${\mathbb{Z}}_{P}$ of $P$, which is $\{0\}$ as ${\mathbb{Z}}$ acts on $\operatorname{Prim}A$ freely, and stability group of each $(m,P)$ is clearly $\{0\}$. So in the separable system $(B_{{\mathbb{Z}}}\otimes A,{\mathbb{Z}},\beta\otimes\alpha^{-1})$ (with ${\mathbb{Z}}$ abelian), ${\mathbb{Z}}$ acts on $\operatorname{Prim}(B_{{\mathbb{Z}}}\otimes A)\simeq(\mathbb{Z}\cup\{\infty\})\times\operatorname{Prim}A$ freely. Therefore by Theorem 2.4, $\operatorname{Prim}((B_{{\mathbb{Z}}}\otimes A)\times_{\beta\otimes\alpha^{-1}}{\mathbb{Z}})$ is homeomorphic to the quasi-orbit space ${\mathcal{O}}(\operatorname{Prim}(B_{{\mathbb{Z}}}\otimes A))={\mathcal{O}}((\mathbb{Z}\cup\{\infty\})\times\operatorname{Prim}A)$, which describes $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ as well. We want to describe the quotient topology of ${\mathcal{O}}((\mathbb{Z}\cup\{\infty\})\times\operatorname{Prim}A)$ precisely, and identify the primitive ideals of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ coming from $\operatorname{Prim}(A\times_{\alpha}{\mathbb{Z}})$. We have

[TABLE]

Therefore ${\mathcal{O}}(m,P)={\mathcal{O}}(n,Q)$ if and only if $\overline{\{P\}}=\overline{\{Q\}}$, and this happens precisely when $P=Q$ by the definition of the hull-kernel (Jacobson) topology on $\operatorname{Prim}A$ (that is why the primitive ideal space of any $C^{*}$-algebra is always $T_{0}$ [9, Theorem 5.4.7]). So all pairs $(m,P)$ for every $m\in{\mathbb{Z}}$ have the same quasi-orbit which can be parameterized by $P\in\operatorname{Prim}A$, and since

[TABLE]

${\mathcal{O}}(m,P)\neq{\mathcal{O}}(\infty,Q)$ for all $m\in{\mathbb{Z}}$ and $P,Q\in\operatorname{Prim}A$. Moreover

[TABLE]

Thus ${\mathcal{O}}(\infty,P)={\mathcal{O}}(\infty,Q)$ if and only if $\overline{{\mathbb{Z}}\cdot P}=\overline{{\mathbb{Z}}\cdot Q}$, which means if and only if $P$ and $Q$ have the same quasi-orbit (${\mathcal{O}}(P)={\mathcal{O}}(Q)$) in ${\mathcal{O}}(\operatorname{Prim}A)\simeq\operatorname{Prim}(A\times_{\alpha}{\mathbb{Z}})$. So each quasi-orbit ${\mathcal{O}}(\infty,P)$ can be parameterized by the quasi-orbit ${\mathcal{O}}(P)$ in ${\mathcal{O}}(\operatorname{Prim}A)$, and we can therefore identify ${\mathcal{O}}((\mathbb{Z}\cup\{\infty\})\times\operatorname{Prim}A)$ by the disjoint union

[TABLE]

Then we have:

Theorem 3.11**.**

Let $(A,\alpha)$ be a system consisting of a separable $C^{*}$-algebra $A$ and an automorphism $\alpha$ of $A$. Suppose that ${\mathbb{Z}}$ acts on $\operatorname{Prim}A$ freely. Then $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ is homeomorphic to $\operatorname{Prim}A\sqcup{\mathcal{O}}(\operatorname{Prim}A)$, equipped with the (quotient) topology in which the open sets are of the form

[TABLE]

[TABLE]

Proof.

Note that since by [12, Lemma 6.12], the quasi-orbit map $\textsf{q}:\operatorname{Prim}(B_{{\mathbb{Z}}}\otimes A)\rightarrow{\mathcal{O}}(\operatorname{Prim}(B_{{\mathbb{Z}}}\otimes A))$ is continuous and open, the proof follows from a similar argument to the proof of Theorem 3.5. So we skip it here. ∎

Remark 3.12*.*

Under the condition of Theorem 3.11, we want to identify the primitive ideals of $\operatorname{Prim}(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ coming from $\operatorname{Prim}(A\times_{\alpha}{\mathbb{Z}})$, which form the closed subset

[TABLE]

homeomorphic to $\operatorname{Prim}(A\times_{\alpha}{\mathbb{Z}})\simeq{\mathcal{O}}(\operatorname{Prim}A)$ (see Theorem 2.4). These ideals are actually the kernels of the irreducible representations $(\operatorname{Ind}\pi)\circ q=(\tilde{\pi}\times U)\circ q$ of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$, where $\pi$ is an irreducible representation of $A$ with $\ker\pi=P$. But since the pair $(\tilde{\pi},U)$ is clearly a covariant partial-isometric representation of $(A,\alpha)$, one can see that in fact, $(\operatorname{Ind}\pi)\circ q=\tilde{\pi}\times^{\operatorname{piso}}U$, where $\tilde{\pi}\times^{\operatorname{piso}}U$ is the associated representation of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ correspondent to the pair $(\tilde{\pi},U)$. Thus each element of $\mathcal{F}$ is of the form $\ker(\tilde{\pi}\times^{\operatorname{piso}}U)$ correspondent to the quasi-orbit ${\mathcal{O}}(P)$, and therefore we denote $\ker(\tilde{\pi}\times^{\operatorname{piso}}U)$ by ${\mathcal{J}}_{{\mathcal{O}}(P)}$. So the map ${\mathcal{O}}(P)\rightarrow{\mathcal{J}}_{{\mathcal{O}}(P)}$ is a homeomorphism of ${\mathcal{O}}(\operatorname{Prim}A)$ onto the closed subspace $\mathcal{F}$ of \operatorname{Prim}\big{(}A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}\big{)}.

For the following remark, we need to recall that the primitive ideal space of any $C^{*}$-algebra $A$ is locally compact [7, Corollary 3.3.8]. A locally compact space $X$ (not necessarily Hausdorff) is called almost Hausdorff if each locally compact subspace $U$ contains a relatively open nonempty Hausdorff subset (see [12, Definition 6.1.]). If a $C^{*}$-algebra is GCR, then it is almost Hausdorrff (see the discussion on pages 171 and 172 of [12]). Finally if $A$ is separable, then by applying [11, Theorem A.38] and [11, Proposition A.46], it follows that $\operatorname{Prim}A$ is second countable.

Remark 3.13*.*

It follows from [13] that if $(A,{\mathbb{Z}},\alpha)$ is a separable system in which ${\mathbb{Z}}$ acts on $\hat{A}$ freely, then $A\times_{\alpha}{\mathbb{Z}}$ is GCR if and only if $A$ is GCR and every ${\mathbb{Z}}$-orbit in $\hat{A}$ is discrete. But every ${\mathbb{Z}}$-orbit in $\hat{A}$ is discrete if and only if for each $[\pi]\in\hat{A}$, the map ${\mathbb{Z}}\rightarrow{\mathbb{Z}}\cdot[\pi]$ defined by $n\mapsto n\cdot[\pi]=[\pi\circ\alpha_{n}^{-1}]$ is a homeomorphism, and this statement itself, by [12, Theorem 6.2 (Mackey-Glimm Dichotomy)], is equivalent to saying that the orbit space ${\mathbb{Z}}\backslash\hat{A}$ is $T_{0}$. Therefore we can rephrase the statement of [13] to say that if $(A,{\mathbb{Z}},\alpha)$ is a separable system in which ${\mathbb{Z}}$ acts on $\hat{A}$ freely, then $A\times_{\alpha}{\mathbb{Z}}$ is GCR if and only if $A$ is GCR and the orbit space ${\mathbb{Z}}\backslash\hat{A}$ is $T_{0}$.

Proposition 3.14**.**

Let $(A,\alpha)$ be a system consisting of a separable $C^{*}$-algebra $A$ and an automorphism $\alpha$ of $A$. Suppose that ${\mathbb{Z}}$ acts on $\hat{A}$ freely. Then $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is GCR if and only if $A$ is GCR and the orbit space ${\mathbb{Z}}\backslash\hat{A}$ is $T_{0}$.

Proof.

The proof follows from a similar argument to the proof of Proposition 3.7 and Remark 3.13. ∎

Example 3.15*.*

Consider the system $(C({\mathbb{T}}),\alpha)$ in which the action $\alpha$ is given by rotation through the angle $2\pi\theta$ with $\theta$ irrational. Then ${\mathbb{Z}}$ acts on $\operatorname{Prim}(C({\mathbb{T}}))=C({\mathbb{T}})^{\widehat{}}={\mathbb{T}}$ freely (see [12, Example 8.45] or [6, Example 10.18]). Therefore by Theorem 3.11, $\operatorname{Prim}(C({\mathbb{T}})\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ can be identified with the disjoint union ${\mathbb{T}}\sqcup{\mathcal{O}}({\mathbb{T}})$. But the quasi-orbit space ${\mathcal{O}}({\mathbb{T}})$ contains only one point as each ${\mathbb{Z}}$-orbit is dense in ${\mathbb{T}}$ (see [12, Lemma 3.29]). Let us parameterize this only point by [math] (note that ${\mathcal{O}}({\mathbb{T}})$ is homeomorphic to the primitive ideal space of the irrational rotation algebra $A_{\theta}:=C({\mathbb{T}})\times_{\alpha}{\mathbb{Z}}$ which is simple). So $\operatorname{Prim}(C({\mathbb{T}})\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})$ is actually identified with

[TABLE]

where each open set is given by

[TABLE]

Here we would like to mention that [math] in ${\mathbb{T}}\sqcup\{0\}$ corresponds to the primitive ideal ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes C({\mathbb{T}})$ of $C({\mathbb{T}})\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$. Finally, although $C({\mathbb{T}})$ is GCR (in fact CCR), the orbit space ${\mathbb{Z}}\backslash{\mathbb{T}}$ is not $T_{0}$ as each ${\mathbb{Z}}$-orbit is dense in ${\mathbb{T}}$. So it follows by Proposition 3.14 that $C({\mathbb{T}})\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is not GCR.

Remark 3.16*.*

Recall that since the Pimsner-Voiculescu Toeplitz algebra ${\mathcal{T}}(A,\alpha)$ is isomorphic to $A\times_{\alpha^{-1}}^{\operatorname{piso}}{\mathbb{N}}$ (see Example 3.9), if $A$ is separable and ${\mathbb{Z}}$ acts on $\operatorname{Prim}A$ freely, then the description of $\operatorname{Prim}({\mathcal{T}}(A,\alpha))$ is obtained completely from Theorem 3.11.

4. Primitivity and Simplicity of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$

In this section, we want to discuss the primitivity and simplicity of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$. Recall that a $C^{*}$-algebra is called primitive if it has a faithful nonzero irreducible representation, and it is called simple if it has no nontrivial ideal.

Theorem 4.1**.**

Let $(A,\alpha)$ be a system consisting of a $C^{*}$-algebra $A$ and an automorphism $\alpha$ of $A$. Then $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is primitive if and only $A$ is primitive.

Proof.

If $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is primitive, it has a faithful nonzero irreducible representation $\rho:A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}\rightarrow B({\mathcal{H}})$. Then since the restriction of $\rho$ to the ideal ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A\simeq\ker q$ is nonzero, it gives an irreducible representation of ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$ which is clearly faithful. So it follows that ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$ is primitive, and therefore $A$ must be primitive as well.

Conversely, if $A$ is primitive, then it has a faithful nonzero irreducible representation $\pi$ on some Hilbert space $H$ ($P=\ker\pi=\{0\}$). We show that the associated irreducible representation $(\Pi\times V)_{P}$ of $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ on $\ell^{2}({\mathbb{N}},H)$ is faithful. By [8, Theorem 4.8], it is enough to see that if $\Pi(a)(1-V^{*}V)=0$, then $a=0$. If $\Pi(a)(1-V^{*}V)=0$, then

[TABLE]

It follows that $\pi(a)h=0$ for all $h\in H$, and therefore $\pi(a)=0$. Since $\pi$ is faithful, we must have $a=0$. This completes the proof. ∎

Remark 4.2*.*

Note that Theorem 4.1 simply means that in the homeomorphism $P\mapsto{\mathcal{I}}_{P}$ mentioned in Remark 3.2, $P$ is the zero ideal if and only if ${\mathcal{I}}_{P}$ is the zero ideal. This is because if $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is primitive, then its zero ideal as one of its primitive ideals is of the form ${\mathcal{I}}_{P}$ (coming from $\operatorname{Prim}A$), as ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A\neq 0$.

Finally it is not difficult to see that $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is not simple. This is because as we see, it contains ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$ as a nonzero ideal. Moreover if ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A=A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$, then $A\times_{\alpha}{\mathbb{Z}}\simeq(A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}})/({\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$) must be the zero algebra. So it follows that $A=0$, which is a contradiction as we have $A\neq 0$. Therefore $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ contains ${\mathcal{K}}(\ell^{2}({\mathbb{N}}))\otimes A$ as a proper nonzero ideal, and hence we have proved the following:

Theorem 4.3**.**

Let $(A,\alpha)$ be a system consisting of a $C^{*}$-algebra $A$ and an automorphism $\alpha$ of $A$. Then $A\times_{\alpha}^{\operatorname{piso}}{\mathbb{N}}$ is not simple.

Acknowledgements

This research is supported by Rachadapisek Sompote Fund for Postdoctoral Fellowship, Chulalongkorn University.