Ideals with approximate unit in semicrossed products

Abstract

We characterize the ideals of the semicrossed product $C_0(X)\times _\phi {\mathbb Z}_+$, associated with suitable sequences of closed subsets of X, with left (resp. right) approximate unit. As a consequence, we obtain a complete characterization of ideals with left (resp. right) approximate unit under the assumptions that X is metrizable and the dynamical system $(X,\phi )$ contains no periodic points.

1 Introduction and notation

The semicrossed product is a nonself-adjoint operator algebra which is constructed from a dynamical system. We recall the construction of the semicrossed product we will consider in this work. Let X be a locally compact Hausdorff space, and let $\phi :X\rightarrow X$ be a continuous and proper surjection (recall that a map $\phi $ is proper if the inverse image $\phi ^{-1}(K)$ is compact for every compact $K\subseteq X$ ). The pair $(X, \phi )$ is called a dynamical system. An action of $\mathbb {Z}_+:=\mathbb N\cup \{0\}$ on $C_0(X)$ by isometric $*$ -endomorphisms $\alpha _n$ , $n\in \mathbb {Z}_+$ is obtained by defining $\alpha _n(f)=f\circ \phi ^n$ . We write the elements of the Banach space $\ell ^1({\mathbb Z}_+,C_0(X))$ as formal series $A=\sum _{n\in {\mathbb Z}_+}U^nf_n$ with the norm given by $\|A\|_1=\sum _{n\in {\mathbb Z}_+}\|f_n\|_{C_0(X)}$ . Multiplication on $\ell ^1({\mathbb Z}_+,C_0(X))$ is defined by setting

$$ \begin{align*} (U^nf)(U^mg)=U^{n+m}(\alpha^m(f)g), \end{align*} $$

and extending by linearity and continuity. With this multiplication, $\ell ^1({\mathbb Z}_+,C_0(X))$ is a Banach algebra.

The Banach algebra $\ell ^1({\mathbb Z}_+,C_0(X))$ can be faithfully represented as a (concrete) operator algebra on a Hilbert space. This is achieved by assuming a faithful action of $C_0(X)$ on a Hilbert space $\mathcal {H}_0$ . Then we can define a faithful contractive representation $\pi $ of $\ell _1({\mathbb Z}_+,C_0(X))$ on the Hilbert space $\mathcal H=\mathcal {H}_0\otimes \ell ^2({\mathbb Z}_+)$ by defining $\pi (U^nf)$ as

$$ \begin{align*} \pi(U^nf)(\xi\otimes e_k)=\alpha^k(f)\xi\otimes e_{k+n}. \end{align*} $$

The semicrossed product $C_0(X)\times _{\phi }{\mathbb Z}_+$ is the closure of the image of $\ell ^1({\mathbb Z}_+,C_0(X))$ in $\mathcal {B(H)}$ in the representation just defined. We will denote an element $\pi (U^nf)$ of $C_0(X)\times _{\phi }{\mathbb Z}_+$ by $U^nf$ to simplify the notation.

For $A=\sum _{n\in {\mathbb Z}_+}U^nf_n\in \ell ^1({\mathbb Z}_+,C_0(X))$ , we call $f_n\equiv E_n(A)$ the nth Fourier coefficient of A. The maps $E_n:\ell ^1({\mathbb Z}_+,C_0(X))\rightarrow C_0(X)$ are contractive in the (operator) norm of $C_0(X)\times _{\phi }{\mathbb Z}_+$ , and therefore they extend to contractions $E_n:C_0(X)\times _{\phi }{\mathbb Z}_+ \rightarrow C_0 (X)$ . An element A of the semicrossed product $C_0(X)\times _{\phi }{\mathbb Z}_+$ is $0$ if and only if $E_n(A)=0$ , for all $n \in {\mathbb Z}_+$ , and thus A is completely determined by its Fourier coefficients. We will denote A by the formal series $A=\sum _{n\in {\mathbb Z}_+}U^nf_n$ , where $f_n=E_n(A)$ . Note, however, that the series $\sum _{n\in {\mathbb Z}_+}U^nf_n$ does not in general converge to A [6, II.9]. The kth arithmetic mean of A is defined to be $\bar A_k=\frac {1}{k+1}\sum _{l=0}^k S_l(A)$ , where $S_l(A)=\sum _{n=0}^l U^nf_n$ . Then, the sequence $\{\bar A_k\}_{k\in \mathbb {Z}_+}$ is norm convergent to A [6, Remark, p. 524]. We refer to [3, 4, 6] for more information about the semicrossed product.

Let $\{X_n\}_{n=0}^{\infty }$ be a sequence of closed subsets of X satisfying

(*) $$ \begin{align} X_{n+1}\cup\phi(X_{n+1})\subseteq X_n, \end{align} $$

for all $n\in \mathbb N$ . Peters proved in [7] that the subspace $\mathcal I=\{A\in C_0(X)\times _{\phi }{\mathbb Z}_+:E_n(A)(X_n)=\{0\}\}$ is a closed two-sided ideal of $C_0(X)\times _{\phi }{\mathbb Z}_+$ . We will write this as $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ . We note that if $A\in \mathcal I\sim \{X_n\}_{n=0}^{\infty }$ , then $U^nE_n(A)\in \mathcal I$ for all $n\in {\mathbb Z}_+$ . Peters proved in [7] that there is a one-to-one correspondence between closed two-sided ideals $\mathcal I\subseteq C_0(X)\times _{\phi }{\mathbb Z}_+$ and sequences $\{X_n\}_{n=0}^{\infty }$ of closed subsets of X satisfying (*), under the assumptions that X is metrizable and the dynamical system $(X,\phi )$ contains no periodic points. Moreover, he characterizes the maximal and prime ideals of the semicrossed product $C_0(X)\times _{\phi }{\mathbb Z}_+$ in this case.

Donsig, Katavolos, and Manousos obtained in [4] a characterization of the Jacobson radical for the semicrossed product $C_0(X)\times _{\phi }{\mathbb Z}_+$ , where X is a locally compact metrizable space and $\phi :X\rightarrow X$ is a continuous and proper surjection. Andreolas, Anoussis, and the author characterized in [2] the ideal generated by the compact elements and in [1] the hypocompact and the scattered radical of the semicrossed product $C_0(X)\times _{\phi }{\mathbb Z}_+$ , where X is a locally compact Hausdorff space and $\phi :X\rightarrow X$ is a homeomorphism. All these ideals are of the form $\mathcal I\sim \{X_n\}_{n=0}^\infty $ for suitable families of closed subsets $\{X_n\}_{n=0}^\infty $ .

In the present paper, we characterize the closed two-sided ideals $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ of $C_0(X)\times _\phi {\mathbb Z}_+$ with left (resp. right) approximate unit. As a consequence, we obtain a complete characterization of ideals with left (resp. right) approximate unit under the additional assumptions that X is metrizable and the dynamical system $(X,\phi )$ contains no periodic points.

Recall that a left (resp. right) approximate unit of a Banach algebra $\mathcal A$ is a net $\{u_\lambda \}_{\lambda \in \Lambda }$ of elements of $\mathcal A$ such that:

1. (1) for some positive number r, $\|u_{\lambda }\|\leq r$ for all $\lambda \in \Lambda $ ,

2. (2) $\lim u_\lambda a=a$ (resp. $\lim au_\lambda =a$ ), for all $a\in \mathcal A$ , in the norm topology of $\mathcal A$ .

A net which is both a left and a right approximate unit of $\mathcal A$ is called an approximate unit of $\mathcal A$ . A left (resp. right) approximate unit $\{u_\lambda \}_{\lambda \in \Lambda }$ that satisfies $\|u_{\lambda }\|\leq 1$ for all $\lambda \in \Lambda $ is called a contractive left (resp. right) approximate unit.

We will say that an ideal $\mathcal I$ of a Banach algebra $\mathcal A$ has a left (resp. right) approximate unit if it has a left (resp. right) approximate unit as an algebra.

2 Ideals with approximate unit

In the following theorem, the ideals $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ with right approximate unit are characterized.

Theorem 2.1 Let $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ be a nonzero ideal of $C_0(X)\times _\phi {\mathbb Z}_+$ . The following are equivalent:

1. (1) $\mathcal I$ has a right approximate unit.

2. (2) $X_n=X_{n+1}$ , for all $n\in {\mathbb Z}_+$ .

Proof We start by proving that (1) $\Rightarrow $ (2). Let $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ be an ideal with right approximate unit $\{V_{\lambda }\}_{\lambda \in \Lambda }$ . We suppose that there exists $n\in {\mathbb Z}_+$ such that $X_{n+1}\subsetneq X_{n}$ . Let

$$ \begin{align*} n_0=\min\{n\in{\mathbb Z}_+:X_{n+1}\subsetneq X_{n}\}, \end{align*} $$

$x_0\in X_{n_0}\backslash X_{n_0+1}$ , and $f\in C_0(X)$ such that $f(x_0)=1$ , $f(X_{n_0+1})=\{0\}$ , and $\|f\|=1$ . Then, for $A=U^{n_0+1}f$ , we have $A\in \mathcal I$ and

$$ \begin{align*} \|AV_{\lambda}-A\| \ge \|E_{n_0+1}(AV_{\lambda}-A)\| = \|fE_0(V_{\lambda})-f\|\ge|(fE_0(V_{\lambda})-f)(x_0)|=1, \end{align*} $$

for all $\lambda \in \Lambda $ , since $x_0\in X_{n_0}$ and $E_0(V_{\lambda })(X_{n_0})=0$ , which is a contradiction. Therefore, $X_n=X_{n+1}$ for all $n\in {\mathbb Z}_+$ .

For (2) $\Rightarrow $ (1), assume that $X_n=X_{n+1}$ for all $n\in {\mathbb Z}_+$ . By (*), we get that $\phi (X_0)\subseteq X_0$ . We will show that if $\{u_{\lambda }\}_{\lambda \in \Lambda }$ is a contractive approximate unit of the ideal $C_0(X\backslash X_0)$ of $C_0(X)$ , then $\{U^0u_{\lambda }\}_{\lambda \in \Lambda }$ is a right approximate unit of $\mathcal I$ . Since $\|u_{\lambda }\|\leq 1$ , we have $\|U^0u_{\lambda }\|\leq 1$ .

Let $A\in \mathcal I$ and $\varepsilon>0$ . Then there exists $k\in {\mathbb Z}_+$ such that

$$ \begin{align*} \|A-\bar A_k\|<\frac{\varepsilon}{4}, \end{align*} $$

where $\bar A_k$ is the kth arithmetic mean of A. Since $X_n=X_0$ , $E_n(\bar A_k)\in C_0(X\backslash X_0)$ and $\{u_{\lambda }\}_{\lambda \in \Lambda }$ is an approximate unit of $C_0(X\backslash X_0)$ , there exists $\lambda _0\in \Lambda $ such that

$$ \begin{align*} \|E_l(\bar A_k)u_\lambda-E_l(\bar A_k)\|<\frac{\varepsilon}{2(k+1)}, \end{align*} $$

for all $l\leq k$ and $\lambda> \lambda _0$ . So, for $\lambda>\lambda _0$ , we get that

$$ \begin{align*} \|AU^0u_\lambda-A\| & = \|AU^0u_\lambda-\bar A_kU^0u_\lambda+\bar A_kU^0u_\lambda-\bar A_k+\bar A_k-A\| \\ & \leq \|AU^0u_\lambda-\bar A_kU^0u_\lambda\|+\|\bar A_kU^0u_\lambda-\bar A_k\|+\|A-\bar A_k\| \\ & < \|\bar A_kU^0u_\lambda-\bar A_k\|+\frac{\varepsilon}{2}\\ &\leq \sum_{l=0}^k\|E_l(\bar A_k)u_\lambda -E_l(\bar A_k)\|+\frac{\varepsilon}{2}\\ &< \varepsilon, \end{align*} $$

which concludes the proof.

In the following theorem, the ideals $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ with left approximate unit are characterized.

Theorem 2.2 Let $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ be a nonzero ideal of $C_0(X)\times _\phi {\mathbb Z}_+$ . The following are equivalent:

1. (1) $\mathcal I$ has a left approximate unit.

2. (2) $X_0\subsetneq X$ and $\phi ^{n}(X\backslash X_n)= X\backslash X_0$ , for all $n\in {\mathbb Z}_+$ .

3. (3) $\phi (X\backslash X_{1})= X\backslash X_{0}$ and $\phi (X_{n+1}\backslash X_{n+2})=X_{n}\backslash X_{n+1}$ , for all $n\in {\mathbb Z}_+$ .

Proof We start by proving that (1) $\Rightarrow $ (2). Let $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ be an ideal with left approximate unit $\{V_{\lambda }\}_{\lambda \in \Lambda }$ .

First, we prove that $X_0\subsetneq X$ . We suppose that $X_0=X$ . Then $E_0(V_\lambda )=0$ , for all ${\lambda \in \Lambda }$ , and hence for every $U^nf\in \mathcal I$ , we have

$$ \begin{align*} \|V_\lambda U^nf-U^nf\|\ge\|E_n(V_\lambda U^nf-U^nf)\|=\|E_0(V_\lambda)\circ\phi^nf-f\|=\|f\|, \end{align*} $$

for all $\lambda \in \Lambda $ , which is a contradiction. Therefore, $X_0\subsetneq X$ .

Now, we prove that $\phi ^{n}(X\backslash X_n)= X\backslash X_0$ , for all $n\in {\mathbb Z}_+$ . We suppose that there exists $n\in {\mathbb Z}_+$ such that $\phi ^{n}(X\backslash X_n)\not \subseteq X\backslash X_0$ and let

$$ \begin{align*} n_0=\min\{n\in{\mathbb Z}_+:\phi^{n}(X\backslash X_n)\not\subseteq X\backslash X_0\}. \end{align*} $$

The set $X\backslash X_{n_0}$ is nonempty, since $X_{n_0}\subseteq X_0\subsetneq X$ . Then, there exist $x_0\in X\backslash X_{n_0}$ such that $\phi ^{n_0}(x_0)\in X_{0}$ and a function $f\in C_0(X)$ such that $f(x_0)=1$ , $f(X_{n_0})=\{0\}$ , and $\|f\|=1$ . If $A=U^{n_0}f$ , by the choice of f, we have that $A\in \mathcal I$ , $\|A\|=1$ and

$$ \begin{align*} \|V_{\lambda}A-A\| & \ge \|E_{n_0}(V_{\lambda}A-A)\|\\ & = \|E_0(V_{\lambda})\circ\phi^{n_0}f-f\|\\ &\ge|(E_0(V_{\lambda})\circ\phi^{n_0}f-f)(x_0)|\\ &=1, \end{align*} $$

for all $\lambda \in \Lambda $ , since $\phi ^{n_0}(x_0)\in X_0$ and $E_0(V_{\lambda })(X_0)=\{0\}$ , which is a contradiction. Therefore, $\phi ^{n}(X\backslash X_n)\subseteq X\backslash X_0$ . Furthermore, by (*), we get that $\phi ^n(X_n)\subseteq X_0$ , for all $n\in {\mathbb Z}_+$ , and hence

$$ \begin{align*} X = \phi^n(X) =\phi^n(X_n\cup(X\backslash X_n)) = \phi^n(X_n)\cup\phi^n(X\backslash X_n) \subseteq X_0\cup \phi^n(X\backslash X_n). \end{align*} $$

Since $\phi ^{n}(X\backslash X_n)\subseteq X\backslash X_0$ and $\phi $ is surjective, $\phi ^n(X\backslash X_n)=X\backslash X_0$ , for all $n\in {\mathbb Z}_+$ .

For (2) $\Rightarrow $ (1), assume that $X_0\subsetneq X$ and $\phi ^{n}(X\backslash X_n)= X\backslash X_0$ , for all $n\in {\mathbb Z}_+$ . We will show that if $\{u_{\lambda }\}_{\lambda \in \Lambda }$ is a contractive approximate unit of the ideal $C_0(X\backslash X_0)$ of $C_0(X)$ , then $\{U^0u_{\lambda }\}_{\lambda \in \Lambda }$ is a left approximate unit of $\mathcal I$ . Since $\|u_{\lambda }\|\leq 1$ , we have $\|U^0u_{\lambda }\|\leq 1$ .

Let A be a norm-one element of $\mathcal I$ and $\varepsilon>0$ . Then there exists $k\in {\mathbb Z}_+$ such that

$$ \begin{align*} \|A-\bar A_k\|<\frac{\varepsilon}{4}, \end{align*} $$

where $\bar A_k$ is the kth arithmetic mean of A. For $l\leq k$ , let

$$ \begin{align*} D_\varepsilon(E_l(\bar A_k))=\left\{x\in X: |E_l(\bar A_k)(x)|\ge\frac{\varepsilon}{4(k+1)} \right\}. \end{align*} $$

Since $A\in \mathcal I$ , we have $E_l(\bar A_k)(X_l)=\{0\}$ and hence $D_\varepsilon (E_l(\bar A_k))\subseteq X\backslash X_l$ . Furthermore, since $\phi ^{n}(X\backslash X_n)= X\backslash X_0$ , for all $n\in {\mathbb Z}_+$ , we have that $\phi ^{l}(D_\varepsilon (E_l(\bar A_k)))\subseteq X\backslash X_0$ . Moreover, the set $D_\varepsilon (E_l(\bar A_k))$ is compact, since $E_l(\bar A_k)\in C_0(X)$ , and hence the set $\phi ^{l}(D_\varepsilon (E_l(\bar A_k)))$ is also compact. By Urysohn’s lemma for locally compact Hausdorff spaces [8, p. 39], there is a norm-one function $v_l\in C_0(X)$ such that

$$ \begin{align*} v_l(x)=\left\{\begin{array}{ll} 1, & x\in \phi^l(D_\varepsilon(E_l(\bar A_k))),\\ 0, & x\in X_0. \end{array} \right. \end{align*} $$

Then, there exists $\lambda _0\in \Lambda $ such that

$$ \begin{align*} \|u_\lambda v_l-v_l\|<\frac{\varepsilon}{2(k+1)}, \end{align*} $$

for all $l\leq k$ and $\lambda>\lambda _0$ , and hence

$$ \begin{align*} |u_\lambda(x)-1|<\frac{\varepsilon}{2(k+1)}, \end{align*} $$

for all $x\in \cup _{l=0}^k\phi ^{l}(D_\varepsilon (E_l(\bar A_k)))$ and $\lambda>\lambda _0$ . Therefore, if $x\in \cup _{l=0}^k(D_\varepsilon (E_l(\bar A_k)))$ , then $\phi ^l(x)\in \cup _{l=0}^k\phi ^{l}(D_\varepsilon (E_l(\bar A_k)))$ and hence

$$ \begin{align*} \|((u_\lambda\circ\phi^l) E_l(\bar A_k)-E_l(\bar A_k))(x)\|<\frac{\varepsilon}{2(k+1)}, \end{align*} $$

for all $l\leq k$ and $\lambda> \lambda _0$ . On the other hand, if $x\not \in \cup _{l=0}^k(D_\varepsilon (E_l(\bar A_k)))$ , then

$$ \begin{align*} |E_l(\bar A_k)(x)|<\frac{\varepsilon}{4(k+1)}, \end{align*} $$

for all $l\leq k$ , and hence

$$ \begin{align*} \|((u_\lambda\circ\phi^l) E_l(\bar A_k)-E_l(\bar A_k))(x)\|<\frac{\varepsilon}{2(k+1)}. \end{align*} $$

From what we said so far, we get that

$$ \begin{align*} \|U^0u_\lambda A-A\| & < \|U^0u_\lambda\bar A_k-\bar A_k\|+\frac{\varepsilon}{2}\\ & \leq \sum_{l=0}^k\|(u_\lambda\circ\phi^l)E_l(\bar A_k)-E_l(\bar A_k)\|+\frac{\varepsilon}{2}\\ &< \varepsilon, \end{align*} $$

for all $\lambda>\lambda _0$ .

Now, we show that (2) $\Rightarrow $ (3). We assume that $\phi ^{n}(X\backslash X_{n})= X\backslash X_0$ , for all $n\in {\mathbb Z}_+$ . Then, $\phi (X\backslash X_{n+2})\subseteq X\backslash X_{n+1}$ . Indeed, if $x\in X\backslash X_{n+2}$ and $\phi (x)\in X_{n+1}$ , then ${\phi ^{n+2}(x)\in X_{0}}$ , by (*), which is a contradiction. Furthermore, by (*), we know that $\phi (X_{n+1})\subseteq X_{n}$ and hence $\phi (X_{n+1}\backslash X_{n+2})\subseteq X_n\backslash X_{n+1}$ for all $n\in {\mathbb Z}_+$ .

To prove that $\phi (X_{n+1}\backslash X_{n+2})= X_n\backslash X_{n+1}$ for all $n\in {\mathbb Z}_+$ , we suppose that there exists $n\in {\mathbb Z}_+$ such that $\phi (X_{n+1}\backslash X_{n+2})\subsetneq X_n\backslash X_{n+1}$ . If

$$ \begin{align*} n_0 = \min\{n\in{\mathbb Z}_+:\phi(X_{n+1}\backslash X_{n+2})\subsetneq X_n\backslash X_{n+1}\}, \end{align*} $$

then

$$ \begin{align*} \phi(X_{n_0+1}) & = \phi(X_{n_0+2}\cup(X_{n_0+1}\backslash X_{n_0+2}))\\ &= \phi(X_{n_0+2})\cup\phi(X_{n_0+1}\backslash X_{n_0+2})\\ & \subseteq X_{n_0+1}\cup\phi(X_{n_0+1}\backslash X_{n_0+2})\\ &\subsetneq X_{n_0+1}\cup(X_{n_0}\backslash X_{n_0+1})\\ &= X_{n_0}, \end{align*} $$

and hence

$$ \begin{align*} \phi(X) & = \phi(X_{n_0+1}\cup(X\backslash X_{n_0+1}))\\ &= \phi(X_{n_0+1})\cup\phi(X\backslash X_{n_0+1})\\ & \subseteq \phi(X_{n_0+1})\cup (X\backslash X_{n_0})\\ &\subsetneq X, \end{align*} $$

which is a contradiction, since $\phi $ is surjective. Therefore, $\phi (X_{n+1}\backslash X_{n+2})= X_n\backslash X_{n+1}$ for all $n\in {\mathbb Z}_+$ .

Finally, we show that (3) $\Rightarrow $ (2). We assume that $\phi (X\backslash X_{1})= X\backslash X_{0}$ and $\phi (X_{n+1}\backslash X_{n+2})=X_{n}\backslash X_{n+1}$ , for all $n\in {\mathbb Z}_+$ . Then, $X_0\subsetneq X$ . Indeed, if $X_0=X$ , then $\mathcal I\equiv \{0\}$ , which is a contradiction. If $n>1$ , we have that

$$ \begin{align*} \phi(X\backslash X_n)&=\phi\left[(X\backslash X_1)\cup(X_1\backslash X_2)\cup\dots\cup(X_{n-1}\backslash X_n)\right]\\ &=\phi(X\backslash X_1)\cup\phi(X_1\backslash X_2)\cup\dots\cup\phi(X_{n-1}\backslash X_n)\\ &=(X\backslash X_0)\cup(X_0\backslash X_1)\cup\dots\cup(X_{n-2}\backslash X_{n-1})\\ &=X\backslash X_{n-1} , \end{align*} $$

and hence $\phi ^n(X\backslash X_n)=X\backslash X_{0}$ , for all $n\in {\mathbb Z}_+$ .

Remark 2.3 It follows from the proofs of Theorems 2.1 and 2.2 that if $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ is an ideal of $C_0(X)\times _\phi {\mathbb Z}_+$ with a left (resp. right) approximate unit, then it has a contractive left (resp. right) approximate unit of the form $\{U^0u_\lambda \}_{\lambda \in \Lambda }$ where $\{u_\lambda \}_{\lambda \in \Lambda }$ a contractive approximate unit of the ideal $C_0(X\backslash X_0)$ of $C_0(X)$ .

By Theorem 2.2, if $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ is an ideal of $C_0(X)\times _\phi {\mathbb Z}_+$ with a left approximate unit, then $X_{n+1}= X_n$ or $X_{n+1}\subsetneq X_n$ for all $n\in {\mathbb Z}_+$ . If $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ and $X_{n+1}= X_n$ , for all $n\in {\mathbb Z}_+$ , we will write $\mathcal I\sim \{X_0\}$ . We obtain the following characterization.

Corollary 2.4 Let $\mathcal I\sim \{X_0\}$ be a nonzero ideal of $C_0(X)\times _\phi {\mathbb Z}_+$ . The following are equivalent:

1. (1) $\mathcal I$ has a left approximate unit.

2. (2) $\phi (X_0)= X_0$ and $\phi (X\backslash X_0)= X\backslash X_0$ .

Proof By Theorem 2.2, we have $\phi (X\backslash X_0)= X\backslash X_0$ . By (*), we have $\phi (X_0)\subseteq X_0$ , and since $\phi $ is surjective, we get $\phi (X_0)= X_0$ .

In the following proposition, the ideals $\mathcal I\sim \{X_n\}_{n=1}^\infty $ of $C_0(X)\times _\phi {\mathbb Z}_+$ with a left approximate unit are characterized, when $\phi $ is a homeomorphism.

Proposition 2.5 Let $\mathcal I\sim \{X_n\}_{n=1}^\infty $ be a nonzero ideal of $C_0(X)\times _\phi {\mathbb Z}_+$ , where $\phi $ is a homeomorphism. The following are equivalent:

1. (1) $\mathcal I$ has a left approximate unit.

2. (2) There exist $S,W\subsetneq X$ such that S is closed and $\phi (S)=S$ , the sets $\phi ^{-1}(W)$ , $\phi ^{-2}(W),\dots $ are pairwise disjoint and $\phi ^k(W)\cap S=\emptyset $ , for all $k\in {\mathbb Z}$ , and

   $$ \begin{align*} X_n=S\cup(\cup_{k=n}^{\infty}\phi^{-k}(W)), \end{align*} $$
   for all $n\in {\mathbb Z}_+$ .

Proof The second condition implies the second condition of Theorem 2.2 and hence the implication (2) $\Rightarrow $ (1) is immediate. We will prove the implication (1) $\Rightarrow $ (2).

We set $S=\cap _{n=0}^\infty X_n$ . Clearly, the set S is closed and, by (*), we have $\phi (S)\subseteq S$ . We will prove that $\phi (S)=S$ . We suppose $\phi (S)\subsetneq S$ . Since $\phi $ is surjective, there exists $x\in X\backslash S$ such that $\phi (x)\in S$ . Moreover, $\phi ^n(x)\in S$ for all $n\ge 1$ . However, since $x\notin S$ , there exists $n_0$ such that $x\notin X_{n_0}$ and hence $\phi ^{n_0}(x)\in X\backslash X_0$ , by Theorem 2.2, which is a contradiction since $S\cap (X\backslash X_0)=\emptyset $ .

By Theorem 2.2, $\phi (X_{n+1}\backslash X_{n+2})= X_n\backslash X_{n+1}$ for all $n\in {\mathbb Z}_+$ and hence $\phi ^n (X_{n}\backslash X_{n+1})= X_0\backslash X_{1}$ or equivalently $ X_n\backslash X_{n+1}=\phi ^{-n} (X_{0}\backslash X_{1})$ since $\phi $ is a homeomorphism. Furthermore, the sets $\phi ^{-1}(X_0\backslash X_1),\phi ^{-2}(X_0\backslash X_1),\dots $ are pairwise disjoint.

We set $W=X_0\backslash X_1$ . Clearly, $\phi ^k(W)\cap S=\emptyset $ for all $k\in {\mathbb Z}$ , since $\phi (S)=S$ and $\phi (W)\subseteq X\backslash X_0$ . Also, $X_0=S\cup (X_0\backslash X_1)\cup (X_1\backslash X_2)\cup \cdots $ and hence

$$ \begin{align*} X_0=S\cup(\cup_{k=0}^{\infty}\phi^{-k}(W)). \end{align*} $$

Finally, for all $n\in {\mathbb Z}_+$ we have that

$$ \begin{align*} X_0=X_n\cup(\cup_{k=1}^n (X_{k-1}\backslash X_k)) = X_n\cup(\cup_{k=1}^n \phi^{-k+1}(W))= X_n\cup(\cup_{k=0}^{n-1} \phi^{-k}(W)), \end{align*} $$

and so

$$ \begin{align*} X_n=X_0\backslash(\cup_{k=0}^{n-1} \phi^{-k}(W)) =S\cup(\cup_{k=n}^{\infty} \phi^{-k}(W)).\\[-35pt] \end{align*} $$

In the following corollary, the ideals with an approximate unit are characterized.

Corollary 2.6 Let $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ be a nonzero ideal of $C_0(X)\times _\phi {\mathbb Z}_+$ . The following are equivalent:

1. (1) $\mathcal I$ has an approximate unit.

2. (2) $X_n=X_{n+1}$ , for all $n\in {\mathbb Z}_+$ , and $\phi (X\backslash X_0)= X\backslash X_0$ .

Proof (1) $\Rightarrow $ (2) is immediate from Theorem 2.1 and Corollary 2.4.

We show (2) $\Rightarrow $ (1). If $X_n=X_{n+1}$ , by (*), we have $\phi (X_0)\subseteq X_0$ . Since $\phi (X\backslash X_0)= X\backslash X_0$ and $\phi $ surjective, we have $\phi (X_0)= X_0$ . Theorem 2.1 and Corollary 2.4 conclude the proof.

Let B be a Banach space, and let C be a subspace of B. The set of linear functionals that vanish on a subspace C of B is called the annihilator of C. A subspace C of a Banach space B is an M-ideal in B if its annihilator is the kernel of a projection P on $B^*$ such that $\|y\|=\|P(y)\|+\|y-P(y)\|$ , for all y, where $B^*$ is the dual space of B.

Effros and Ruan proved that the M-ideals in a unital operator algebra are the closed two-sided ideals with an approximate unit [5, Theorem 2.2]. Therefore, we obtain the following corollary about the M-ideals of a semicrossed product.

Corollary 2.7 Let $\mathcal I\sim \{X_n\}_{n=0}^{\infty }$ be a nonzero ideal of $C(X)\times _\phi {\mathbb Z}_+$ , where X is compact. The following are equivalent:

1. (1) $\mathcal I$ is an M-ideal.

2. (2) $\mathcal I$ has an approximate unit.

3. (3) $X_n=X_{n+1}$ , for all $n\in {\mathbb Z}_+$ , and $\phi (X\backslash X_0)= X\backslash X_0$ .