Compact elements and the hypocompact radical of crossed products

Abstract

We characterize the compact elements and the hypocompact radical of a crossed product $C_0(X)\times _\phi \mathbb Z$, where X is a locally compact metrizable space and $\phi :X\rightarrow X$ is a homeomorphism, in terms of the corresponding dynamical system $(X,\phi )$.

1 Introduction and notation

Let $\mathcal A$ be a Banach algebra and $a,b\in \mathcal A$ . The map $M_{a,b}:\mathcal A\rightarrow \mathcal A$ given by $M_{a,b}(x)=axb$ is called a multiplication operator. Vala [22] proved that if $\mathcal {B(X)}$ is the algebra of all linear bounded operators on a Banach space $\mathcal X$ and $A,B\in \mathcal {B(X)}$ , then the multiplication operator $M_{A, B}:\mathcal {B(X)}\rightarrow \mathcal {B(X)}$ , is compact if and only if A and B are compact operators. It follows that $A\in \mathcal {B(X)}$ is a compact operator if and only if the multiplication operator $M_{A,A}:\mathcal {B(X)}\rightarrow \mathcal {B(X)}$ is compact. Vala [23] defines an element a of a normed algebra $\mathcal A$ to be compact if the multiplication operator $M_{a,a}:\mathcal A\rightarrow \mathcal A$ is compact. He also proves that the set of compact elements of a Banach algebra is closed.

Ylinen [24] studied compact elements for abstract C^*-algebras and showed that a is a compact element of a $C^*$ -algebra $\mathcal {A}$ if and only if there exists an isometric $*$ -representation $\pi $ of $\mathcal {A}$ on a Hilbert space $\mathcal H$ such that the operator $\pi (a)$ is compact.

Multiplication operators on algebras of operators have been studied, among others, by Akemann and Wright [1], Saksman and Tylli [16, 17], Johnson and Schechtman [9], Lindström, Saksman and Tylli [11], Mathieu and Tradacete [14] and also by Mathieu [12, 13], Timoney [20] in the more general framework of elementary operators.

Compactness properties of multiplication operators on nest algebras are studied by Andreolas and Anoussis [2]. Andreolas, Anoussis, and Magiatis obtain in [3] a characterization of the compact multiplication operators on semicrossed products.

If $\mathcal H$ is a separable Hilbert space, the set of compact elements coincides with the ideal $ \mathcal {K}(\mathcal H)$ of compact operators on $\mathcal H$ , while the Calkin algebra $\mathcal {B}(\mathcal H)/\mathcal {K}(\mathcal H)$ does not have any nonzero compact element [8, Section 5]. However, this is not a general phenomenon.

Shulman and Turovskii observe in [18, p. 298] that there exist Banach spaces $\mathcal X$ , such that the quotient $\mathcal {B}(\mathcal X)/\mathcal {K}(\mathcal X)$ contains compact elements. A stronger instance of this phenomenon appears in the algebra of bounded linear operators on the Argyros–Haydon space [5]. This space is a Banach space $\mathcal X$ with the property that every operator in $\mathcal {B}(\mathcal X)$ is a scalar multiple of the identity plus a compact operator. It follows that $\mathcal {B}(\mathcal X)/\mathcal {K}(\mathcal X)$ is finite-dimensional and hence consists of compact elements.

Shulman and Turovskii prove in [18] that if $\mathcal A$ is a Banach algebra, then there exists a closed ideal $\mathcal I$ in $\mathcal A$ such that if $\mathcal J$ is a closed ideal in $\mathcal A$ and $\mathcal A/\mathcal J$ does not have compact elements, then $\mathcal I$ is contained in $\mathcal J$ . This ideal is called the hypocompact radical of $\mathcal {A}$ and is denoted by $\mathcal {A}_{\mathrm {hc}} $ . This definition of the hypocompact radical is not the one given in [18, 21], but it is equivalent to it as we show in Proposition 3.1.

A characterization of the hypocompact radical of nest algebras is proved in [2]. A characterization of the hypocompact radical of a semicrossed product $C_0(X)\times _\phi \mathbb Z_+$ is proved in [4].

In this article, we characterize the compact elements and the hypocompact radical of a crossed product $C_0(X)\times _\phi \mathbb Z$ , where X is a locally compact metrizable space and $\phi :X\rightarrow X$ is a homeomorphism, in terms of the corresponding dynamical system $(X,\phi )$ .

Throughout this paper, X will be a locally compact metrizable space and $\phi :X\rightarrow X$ a homeomorphism. The pair $(X,\phi )$ is called dynamical system. If $C_0(X)$ is the $C^*$ -algebra of continuous functions vanishing at infinity, an action of $\mathbb Z$ on $C_0(X)$ by isometric $*$ -automorphisms $\alpha _n$ , $n\in \mathbb Z$ is obtained by defining $\alpha _n(f)=f\circ \phi ^{-n}$ .

We follow the notation of [7]. We will denote by $C_0(X)\times _\phi \mathbb Z$ the crossed product corresponding to the dynamical system $(X,\phi )$ . If $n\in \mathbb Z$ and $A\in C_0(X)\times _\phi \mathbb Z$ , we write $E_n(A)$ for the nth Fourier coefficient of A. Also, we denote an element $A\in C_0(X)\times _\phi \mathbb Z$ as a formal series

$$ \begin{align*} A=\sum_{n\in\mathbb Z}f_nU^n, \end{align*} $$

where $f_n\equiv E_n(A)$ , $n\in \mathbb Z$ . We recall that if $fU^m,gU^n\in C_0(X)\times _\phi \mathbb Z$ , the multiplication is defined by setting

$$ \begin{align*} fU^mgU^n=f(g\circ\phi^{-m})U^{m+n}, \end{align*} $$

and extending by linearity and continuity.

If $\mathcal X$ is a Banach space, we denote by $\mathcal X_1$ the unit ball of $\mathcal X$ . If $\mathcal A$ is a Banach algebra, we will denote by $\mathcal C(\mathcal A)$ the set of compact elements of $\mathcal A$ .

2 The compact elements of the crossed product

In this section, we characterize the compact elements of the crossed product $C_0(X)\times _\phi \mathbb Z$ . We will denote by $X_{\mathrm a}$ the set of accumulation points of X, by $X_{\mathrm i}$ the set of isolated points of X and by $X_{\mathrm p}$ the set of periodic points of the dynamical system $(X,\phi )$ .

Proposition 2.1 If $f\in C_0(X)$ , the following are equivalent:

1. (1) The element f is a compact element of the Banach algebra $C_0(X)$ .

2. (2) $f(X_{\mathrm a})=\{0\}$ .

Proof $1 \Rightarrow 2$ . Let $x_0\in X_{\mathrm a}$ such that $f(x_0)\neq 0$ . Then there exists an open neighborhood U of $x_0$ such that $|f(x)|>\frac {|f(x_0)|}{2}$ for all $x\in U$ . We consider a sequence $\{x_i\}_{i=1}^\infty \subseteq U$ such that $x_i\neq x_j$ for $i\neq j$ and a sequence of norm one functions $\{g_{i}\}_{i=1}^\infty \subseteq C_0(X)$ such that $g_{i}(x_i)=1$ and $g_i(x_j)=0$ for $i\neq j$ . Then, for $i\neq j$ we have

$$ \begin{align*} \|M_{f,f}(g_i)-M_{f,f}(g_j)\|&=\sup_{x\in X} |[f^2(g_i-g_j)](x)|\\ &\ge |[f^2(g_i-g_j)](x_i)|\\ &\ge \frac{|f^2(x_0)|}{4}, \end{align*} $$

and thus the sequence $\{M_{f,f}(g_i)\}_{i=1}^{\infty }$ has no convergent subsequence.

$2 \Rightarrow 1$ . Let $f\in C_0(X)$ be such that $f(X_{\mathrm a})=\{0\}$ .

For $n\in \mathbb N$ , the set

$$ \begin{align*} S_n=\left\{x\in X:|f(x)|\ge \frac{1}{n}\right\} \end{align*} $$

is compact and hence finite. We denote by $f_n\in C_0(X)$ the function defined by

$$ \begin{align*} f_n(x)=\left\{\begin{matrix} f(x), & x\in S_n, \\ 0, & x\in X\,{\backslash}\, S_n. \end{matrix}\right. \end{align*} $$

We have that $\|f-f_n\|\leq \frac {1}{n}$ . Moreover, the operators $M_{f_n, f_n}$ are finite-rank operators, and $M_{f, f}$ is the norm limit of the sequence $\{M_{f_n, f_n}\}_{n=1}^{\infty }$ .

Lemma 2.2 If $fU^0\in C_0(X)\times _\phi \mathbb Z$ is a compact element, then $f(X_{\mathrm a})=\{0\}$ .

Proof If $S=\{gU^0\in C_0(X)\times _\phi \mathbb Z:\|gU^0\|\leq 1\}$ , then

$$ \begin{align*} E_0(M_{fU^0,fU^0}(S))=M_{f,f}(C_0(X)_1). \end{align*} $$

Therefore, if the multiplication operator $M_{fU^0,fU^0}$ is compact, f is a compact element of $C_0(X),$ which implies that $f(X_{\mathrm a})=\{0\}$ , by Proposition 2.1.

Lemma 2.3 If $fU^0\in C_0(X)\times _\phi \mathbb Z$ is a compact element, then $f(X_{\mathrm p})=\{0\}$ .

Proof If $fU^0$ is a compact element, then $f(X_{\mathrm p}\cap X_{\mathrm a})=\{0\}$ , by Lemma 2.2. We assume that there exists $x_0\in X_{\mathrm p}\cap X_{\mathrm i}$ such that $f(x_0)\neq 0$ , and we prove that the multiplication operator $M_{fU^0,fU^0}$ is not compact.

Since $x_0\in X_{\mathrm p}\cap X_{\mathrm i}$ , there exists $n_0\in \mathbb Z$ , $n_0\neq 0$ , such that $\phi ^{n_0}(x_0)=x_0$ . We consider the sequence $\{\chi U^{in_0}\}_{i=1}^{\infty }$ , where $\chi $ is the characteristic function of the singleton $\{x_0\}$ . For $i,j\in \mathbb N$ , $i\neq j$ , we have

$$ \begin{align*} \|M_{fU^0,fU^0}(\chi U^{in_0})-M_{fU^0,fU^0}(\chi U^{jn_0})\| &\ge \|E_{in_0}(M_{fU^0,fU^0}(\chi U^{in_0}-\chi U^{jn_0}))\|\\ & = \|E_{in_0}(M_{fU^0,fU^0}(\chi U^{in_0}))\|\\ & = |f^2(x_0)|. \end{align*} $$

Hence, the sequence $\{M_{fU^0,fU^0}(\chi U^{in_0})\}_{i=1}^{\infty }$ has no convergent subsequence.

Lemma 2.4 Let $m,n\in \mathbb Z$ , $x,y\in X_{\mathrm i}\,{\backslash}\, X_{\mathrm p}$ and $\chi _x, \chi _y$ be the characteristic functions of the singletons $\{x\}$ and $\{y\}$ , respectively. Then, the multiplication operator $M_{\chi _xU^m,\chi _yU^n}:C_0(X)\times _\phi \mathbb Z\rightarrow C_0(X)\times _\phi \mathbb Z$ is compact.

Proof To prove that $M_{\chi _xU^m,\chi _yU^n}$ is compact, we distinguish two cases. If there is no $k\in \mathbb Z$ such that $\phi ^k(x)=y$ , then, $M_{\chi _xU^m,\chi _yU^n}\equiv 0$ . Indeed, for an element $C=\sum _l h_lU^l$ , we have

$$ \begin{align*} M_{\chi_xU^m,\chi_yU^n}(C) & = \chi_xU^m\left(\sum_l h_lU^l\right)\chi_yU^n \\ & = \sum_l \chi_xh_l\circ\phi^{-m}\chi_y\circ\phi^{-m-l}U^{m+l+n}\\ & =0, \end{align*} $$

since $\chi _x\chi _y\circ \phi ^{-m-l}=0$ .

Now, assume that there exists $k\in \mathbb Z$ such that $\phi ^k(x)=y$ . Then k is unique, since $x,y\notin X_{\mathrm p}$ . It is easy to see that

$$ \begin{align*} M_{\chi_xU^m,\chi_yU^n}((C_0(X) \times_\phi\mathbb Z)_1)=\left\{z\chi_{x}U^{n-k}:z\in\mathbb C_1\right\}, \end{align*} $$

hence $M_{\chi _xU^m,\chi _yU^n}$ is a rank-one operator.

Remark 2.5 The set of compact elements of a $C^*$ -algebra is an ideal [24]. Moreover, it is invariant under the $*$ -automorphisms of the algebra, and hence an element ${A=\sum _{n}f_nU^n }$ of the crossed product $C_0(X)\times _\phi \mathbb Z$ is a compact element, if and only if $f_nU^n$ is a compact element for all $n \in \mathbb Z$ . Note also that if a is a compact element of a Banach algebra $\mathcal A$ and $b \in \mathcal A$ , then $ab$ and $ba$ are also compact elements. We shall use these facts in the sequel.

Recall that we denote by $\mathcal C(C_0(X)\times _\phi \mathbb Z)$ the set of compact elements of ${C_0(X)\times _\phi \mathbb Z}$ .

Theorem 2.6 The set of compact elements $\mathcal C(C_0(X)\times _\phi \mathbb Z)$ of the crossed product $C_0(X)\times _\phi \mathbb Z$ is the set

$$ \begin{align*}\left\{\sum f_nU^n:f_n(X_{\mathrm a}\cup X_{\mathrm p})=\{0\}, \forall n\in\mathbb Z\right\}.\end{align*} $$

Proof If $A=\sum _{n}f_nU^n$ is a compact element, then $f_nU^n$ is a compact element for all $n\in \mathbb Z$ and hence $f_nU^0$ is a compact element for all $n\in \mathbb Z$ . Therefore, $f_n(X_{\mathrm a}\cup X_{\mathrm p})=\{0\}$ for all $n\in \mathbb Z$ , by Lemmas 2.2 and 2.3.

We will show the opposite direction. It is enough to prove that if $A=fU^0$ is such that $f(X_{\mathrm a}\cup X_{\mathrm p})=\{0\}$ , then A is a compact element. To prove this, we will prove that the element A is the norm limit of a sequence of compact elements $\{A_m\}_{m\in \mathbb N}$ of $C_0(X)\times _\phi \mathbb Z$ .

For $m\in \mathbb N$ , the set

$$ \begin{align*} S_m=\left\{x\in X: |f(x)|\ge\frac{1}{m}\right\}, \end{align*} $$

is a finite subset of $X_{\mathrm i}\,{\backslash}\, X_{\mathrm p}$ . Let $f_m\in C_0(X)$ be the function defined as follows:

$$ \begin{align*} f_m(x)=\left\{\begin{matrix} f(x), & x\in S_m, \\ 0, & x\in X\,{\backslash}\, S_m. \end{matrix}\right. \end{align*} $$

Then if $A_m=f_mU^0$ , it follows from Lemma 2.4 that $A_m$ is a compact element of $C_0(X)\times _\phi \mathbb Z$ . Moreover, $\|A-A_m\|=\|f-f_m\|\leq \frac {1}{m}$ , which concludes the proof.

Remark 2.7 If X is a locally compact metrizable space and $\phi :X\rightarrow X$ is a homeomorphism, the semicrossed product $C_0(X)\times _\phi \mathbb Z_+$ is isomorphic to a nonself-adjoint subalgebra of the crossed product $C_0(X)\times _\phi \mathbb Z$ [15, Proposition II.4]. The compact elements of the semicrossed product $C_0(X)\times _\phi \mathbb Z_+$ are characterized in [3]. Considering the semicrossed product $C_0(X)\times _\phi \mathbb Z_+$ as a subalgebra of the crossed product $C_0(X)\times _\phi \mathbb Z$ , we have that $\mathcal C(C_0\times _\phi \mathbb Z)\cap (C_0\times _\phi \mathbb Z_+)\subseteq \mathcal C(C_0\times _\phi \mathbb Z_+)$ , but in general, this inclusion is strict and more compact elements appear in the semicrossed product.

Remark 2.8 An elementary operator on a Banach algebra is a finite sum of multiplication operators. R. M. Timoney proved in [20, Theorem 3.1] that if $\mathcal A$ is a $C^*$ -algebra, then an elementary operator $\Phi :\mathcal A\rightarrow \mathcal A$ is compact if and only if there exist compact elements $a_i,b_i\in \mathcal A$ , for $i=1,\dots ,n$ , such that $\Phi =\sum _{i=1}^n M_{a_i,b_i}$ . It follows from the proof of this theorem that if $M_{a, b}$ is a compact multiplication operator on a $C^*$ -algebra $\mathcal A$ , then there exist compact elements $c, d \in \mathcal A$ such that $M_{a, b}=M_{c, d}$ . Hence, the knowledge of the compact elements implies the knowledge of the compact multiplication operators in this sense.

3 The hypocompact radical of the crossed product

Shulman and Turovskii in [18, 21] call a Banach algebra $\mathcal {A}$ hypocompact if any nonzero quotient $\mathcal {A}/\mathcal {I}$ by a closed ideal $\mathcal {I}$ contains a nonzero compact element. An ideal $\mathcal I$ of a Banach algebra $\mathcal A$ is hypocompact if it is hypocompact as an algebra. Shulman and Turovskii have proved that any Banach algebra $\mathcal {A}$ has a largest hypocompact ideal [18, Corollary 3.10]. This ideal is closed, is called the hypocompact radical of $\mathcal {A}$ , and is denoted by $\mathcal {A}_{\mathrm {hc}} $ .

If $\mathcal H$ is a separable Hilbert space, the ideal $\mathcal {K}(\mathcal H)$ of compact operators on $\mathcal H$ is the only proper ideal of $\mathcal {B}(\mathcal H)$ , while the Calkin algebra $\mathcal {B}(\mathcal H)/\mathcal {K}(\mathcal H)$ does not have any nonzero compact element [8, Section 5]. It follows that the hypocompact radical of $\mathcal {B}(\mathcal H)$ is $\mathcal {K}(\mathcal H)$ .

We already mentioned in the introduction that there are Banach spaces such that the hypocompact radical contains strictly the ideal of compact operators [18, Lemma 3.12, p. 298]. Moreover, if $\mathcal X$ is the Argyros-Haydon space, it follows from [18, Corollary 3.9] that the hypocompact radical of $\mathcal {B}(\mathcal X)$ coincides with $\mathcal {B}(\mathcal X)$ .

The hypocompact radical of a nest algebra is characterized in [2] and the hypocompact radical of a semicrossed product $C_0(X)\times _\phi \mathbb Z_+$ is characterized in [4]. In this section, we characterize the hypocompact radical of the crossed product $C_0(X)\times _\phi \mathbb Z$ .

We noted in the introduction that the hypocompact radical of a Banach algebra $\mathcal A$ is the smallest closed ideal $\mathcal I$ of $\mathcal A$ , such that $\mathcal A/\mathcal I$ does not contain compact elements. This follows from the results of [18], though we could not find the exact statement in that paper. We state it as a proposition because it is important for our view.

Proposition 3.1 Let $\mathcal A$ be a Banach algebra and $\mathcal J$ a closed ideal of $\mathcal A$ such that $\mathcal A/\mathcal J$ has no compact elements. Then, $ \mathcal J$ contains the hypocompact radical $\mathcal A_{\mathrm hc}$ of $\mathcal A$ .

Proof It follows from [18, Lemma 3.12] that $\mathcal A/\mathcal A_{\mathrm hc}$ does not have compact elements. Set $\mathcal I=\mathcal A_{\mathrm hc}$ and let $\mathcal J$ be a closed ideal of $\mathcal A$ . The hypocompact radical of $\overline {\mathcal I+\mathcal J}$ is $\mathcal I$ by [18, Lemma 3.11]. Let $\pi : \overline {\mathcal {I+J}}\rightarrow \overline {\mathcal {I+J}}/\mathcal J$ be the natural quotient map. It follows from [18, Proposition 3.8], that $\pi (\mathcal I)$ is $\{0\}$ or contains compact elements of $\overline {\mathcal {I+J}}/\mathcal J$ . If $\overline {\mathcal {I+J}}/\mathcal J$ contains compact elements, it follows from [18, Lemma 3.5] that $\mathcal A/\mathcal J$ contains compact elements, which is contrary to our assumption. Hence, $\pi (\mathcal I)=\{0\}$ and $\mathcal I \subseteq \mathcal J$ .

Let $X_1=X_{\mathrm a}\cup \overline {X_{\mathrm p}}$ and $\phi _1=\phi |_{X_1}$ be the restriction of $\phi $ to $X_1$ . We thus obtain a dynamical system $(X_1,\phi _1)$ . Define by transfinite recursion a family $(X_\gamma ,\phi _\gamma )$ of dynamical systems. If $(X_\gamma ,\phi _\gamma )$ is defined, then $X_{\gamma +1}$ is the union of the set of accumulation points of $X_{\gamma }$ and of $\overline {X_{\mathrm p}}$ and $\phi _{\gamma +1}=\phi |_{X_{\gamma +1}}$ , the restriction of $\phi $ to $X_{\gamma +1}$ . If $\gamma $ is a limit ordinal and the system $(X_\beta ,\phi _\beta )$ have been defined for all $\beta <\gamma $ , set $X_{\gamma }=\cap _{\beta <\gamma } X_{\beta }$ and $\phi _{\gamma }=\phi |_{X_{\gamma }}$ , the restriction of $\phi $ to $X_{\gamma }$ . This process must stop at some ordinal $\gamma _0$ since the cardinality of the family cannot exceed the cardinality of the power set of X. The set $X_{\gamma }$ is a closed subset of X for all $\gamma \leq \gamma _0$ .

A subset Y of a topological space is said to be dense in itself, if it contains no isolated points. If Y is closed and dense in itself, it is called a perfect set. A set Y is called scattered, if it does not contain dense in themselves subsets. It is well known that every space is the disjoint union of a perfect set and a scattered one, and this decomposition is unique [10, Theorem 3, p. 79]. If X is a locally compact metrizable space, we denote by $X_{\mathrm {pp}}$ the perfect set and by $X_{\mathrm s}$ the scattered set in this decomposition. We thus have $X = X_{\mathrm {pp}}\cup X_{\mathrm s}$ .

Lemma 3.2 $X_{\gamma _0}=\overline {X_{\mathrm p}}\cup X_{\mathrm {pp}}$ .

Proof Clearly $X_{\mathrm {pp}}\subseteq X_{\beta }$ for all $\beta <\gamma _0$ , and hence $\overline {X_{\mathrm p}}\cup X_{\mathrm {pp}}\subseteq X_{\gamma _0}$ .

We prove that $X_{\gamma _0} \subseteq \overline {X_{\mathrm p}}\cup X_{\mathrm {pp}}$ . Since $\overline {X_{\mathrm p}}\subseteq X_{\gamma _0}$ , it is enough to prove that ${X_{\gamma _0}\,{\backslash}\, \overline {X_{\mathrm p}}\subseteq X_{\mathrm {pp}}}$ .

Let $x\in X_{\gamma _0}\,{\backslash}\, \overline {X_{\mathrm p}}$ . If x is an isolated point of $X_{\gamma _0}\,{\backslash}\, \overline {X_{\mathrm p}}$ , then x is an isolated point of $X_{\gamma _0}$ , which is a contradiction since $X_{\gamma _0}= X_{\gamma _0+1}$ . Therefore, the set $X_{\gamma _0}\,{\backslash}\, \overline {X_{\mathrm p}}$ is dense in itself and hence $X_{\gamma _0}\,{\backslash}\, \overline {X_{\mathrm p}}\subseteq X_{\mathrm {pp}}$ .

If $\gamma \leq \gamma _0$ , we will denote by $\mathcal I_\gamma $ the ideal

$$ \begin{align*} \{A\in C_0(X)\times_\phi\mathbb Z: E_n(A)(X_\gamma)=\{0\},\forall n\in\mathbb Z\}. \end{align*} $$

The proof of the following lemma is straightforward and is omitted.

Lemma 3.3 If $\gamma $ is a limit ordinal, then $\mathcal I_{\mathrm {\gamma }}=\overline {\cup _{\beta <\gamma }\mathcal I_\beta }$ .

Theorem 3.4

$$ \begin{align*} (C_0(X)\times_\phi\mathbb Z)_{\mathrm{hc}}=\mathcal I_{\gamma_0}=\left\{A=\sum_{n}f_nU^n:f_n(\overline{X_{\mathrm p}}\cup X_{\mathrm{pp}})=\{0\}\right\}. \end{align*} $$

Proof 1st step: First we shall prove that $\mathcal I_{\gamma _0}\subseteq (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}$ . Assume the contrary.

It follows from Theorem 2.6 that $\mathcal I_1=\mathcal C(C_0(X)\times _\phi \mathbb Z)$ . The hypocompact radical contains the ideal of compact elements [6, Lemma 8.2], and hence ${\mathcal I_1\subseteq (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}}$ .

Let $\beta $ be the least ordinal $\beta \leq \gamma _0$ such that $\mathcal I_\beta $ is not contained in $(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}$ . We show that $\beta $ is a successor. If not, since $\mathcal I_{\gamma }\subseteq (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}$ for all $\gamma <\beta $ , we obtain from Lemma 3.3 that $\mathcal I_\beta =\overline {\cup _{\gamma <\beta }\mathcal I_\gamma }\subseteq (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}$ , which is absurd.

We are going to prove that $\mathcal I_{\beta }$ is a hypocompact algebra. Consider the algebra $\mathcal I_{\beta }/\mathcal I_{\beta -1}$ . It suffices to show that $\mathcal I_{\beta }/\mathcal I_{\beta -1}$ is a hypocompact algebra since the class of hypocompact algebras is closed under extensions and the ideal $\mathcal I_{\beta -1}$ is hypocompact, [18, Corollary 3.9].

We show that the algebra $\mathcal I_{\beta }/\mathcal I_{\beta -1}$ is generated by the compact elements it contains and hence is a hypocompact algebra by [6, Lemma 8.2].

Let $A \in \mathcal I_{\beta }$ . It follows from the condition defining $\mathcal I_{\beta }$ , that $E_n(A)U^n \in \mathcal I_{\beta }$ , for all $n\in \mathbb Z$ . Hence, it suffices to show that the image of $E_n(A)U^n$ under the natural map $\pi : \mathcal I_{\beta }\rightarrow \mathcal I_{\beta }/\mathcal I_{\beta -1}$ is contained in the ideal generated by the compact elements of $\mathcal I_{\beta }/\mathcal I_{\beta -1}$ . It suffices to see this for an element of $\mathcal I_{\beta }$ of the form $fU^n$ with f compactly supported.

Let $fU^n\in \mathcal I_{\beta }$ with f compactly supported and

$$ \begin{align*} S(f)=\{x\in X_{\beta-1}:f(x)\neq 0\}. \end{align*} $$

The set $S(f)$ is finite since f is compactly supported and vanishes $X_{\beta }$ . It follows that the multiplication operator $M_{\pi (fU^n),\pi (fU^n)}$ is finite rank and hence it is compact.

2nd step: Now we prove that $(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\subseteq \left \{A=\sum _{n}f_nU^n:f_n(\overline {X_{\mathrm {p}}})=\{0\}\right \}$ . Let $\mathcal P=\left \{A=\sum _{n}f_nU^n:f_n(\overline {X_{\mathrm {p}}})=\{0\}\right \}$ and $\mathcal P_{\mathrm h}=(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\cap \mathcal P$ . We suppose that $(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\not \subseteq \mathcal P$ , and we will prove that the quotient algebra $(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}/ \mathcal P_{\mathrm h}$ contains no nonzero compact elements.

Let $\pi : (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\rightarrow (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}/ \mathcal P_{\mathrm h}$ be the quotient map and ${A\in (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\,{\backslash}\, \mathcal P_{\mathrm h}}$ . We will prove that the multiplication operator $M_{\pi (A),\pi (A)}:(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}/ \mathcal P_{\mathrm h}\rightarrow (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}/ \mathcal P_{\mathrm h}$ is not compact.

It is enough to consider the case $A=fU^0$ . Then, there exists $x_0\in X_{\mathrm p}$ such that $f(x_{0})\neq 0$ . We denote $k_0=\min \{k\in \mathbb N:\phi ^{k}(x_0)=x_0\}$ .

We will prove that the sequence $\{M_{\pi (A),\pi (A)}(\pi (fU^{ik_0}))\}_{i=1}^\infty $ has no convergent subsequence.

We estimate the quantity

$$ \begin{align*} \left\|M_{\pi(A),\pi(A)}(\pi(fU^{ik_0}))-M_{\pi(A),\pi(A)}(\pi(fU^{jk_0}))\right\|, \end{align*} $$

for $i,j\in \mathbb {N}$ with $i\neq j$ . We have

$$ \begin{align*} \left\|M_{\pi(A),\pi(A)}(\pi(fU^{ik_0}))-M_{\pi(A),\pi(A)}(\pi(fU^{jk_0}))\right\| &\\ = \inf_{B\in \mathcal P_{\mathrm h}}\left\|AfU^{ik_0}A-AfU^{jk_0}A+B\right\| &\\ = \inf_{B\in \mathcal P_{\mathrm h}}\left\|f^2(f\circ\phi^{-ik_0})U^{ik_0}-f^2(f\circ\phi^{-jk_0})U^{jk_0}+B\right\| & \\ \ge \inf_{B\in \mathcal P_{\mathrm h}}\|E_{ik_0}\left(f^2(f\circ\phi^{-ik_0})U^{ik_0}-f^2(f\circ\phi^{-jk_0})U^{jk_0}+B\right)\| & \\ = \inf_{B\in \mathcal P_{\mathrm h}}\|E_{ik_0}\left(f^2(f\circ\phi^{-ik_0})U^{ik_0}+B\right)\| & \\ \ge \inf_{B\in \mathcal P_{\mathrm h}}\left|E_{ik_0}\left(f^2(f\circ\phi^{-ik_0})U^{ik_0}+B\right)(x_0)\right| = |f^3(x_0)|, \end{align*} $$

and the proof is complete.

3rd step: Now we prove $(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\subseteq \left \{A=\sum _{n}f_nU^n:f_n(X_{\mathrm {pp}})=\{0\}\right \}$ . Let $\mathcal R=\left \{A=\sum _{n}f_nU^n:f_n(X_{\mathrm {pp}})=\{0\}\right \}$ and $\mathcal R_{\mathrm h}=(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\cap \mathcal R$ . We suppose that $(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\not \subseteq \mathcal R$ , and we will prove that the quotient algebra $(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}/ \mathcal R_{\mathrm h}$ contains no nonzero compact elements.

Let $\pi : (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\rightarrow (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}/ \mathcal R_{\mathrm h}$ be the quotient map and ${A\in (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\,{\backslash}\, \mathcal R_{\mathrm h}}$ . We shall prove that the multiplication operator $M_{\pi (A),\pi (A)}:(C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}/ \mathcal R_{\mathrm h}\rightarrow (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}/ \mathcal R_{\mathrm h}$ is not compact.

It is enough to consider the case $A=fU^0$ . Then, there exists $x_0\in X_{\mathrm {pp}}$ such that $f(x_{0})\neq 0$ . Let $S_0$ be an open neighborhood of $x_0$ such that

$$ \begin{align*} |f(x)|>\frac{f(x_0)}{2}, \end{align*} $$

for all $x\in S_0$ . By the second step, the set $S_0$ contains no periodic points.

Since $x_0\in S_0\cap X_{\mathrm {pp}}$ there exist a sequence of points $\{x_i\}_{i=1}^\infty \subseteq S_0\cap X_{\mathrm {pp}}$ , a sequence of open subsets $\{W_i\}_{i=1}^\infty \subseteq S_0$ with $x_i\in W_i$ and $W_i\cap W_j$ , for $i\neq j$ and a sequence of norm one functions $\{h_i\}_{i=1}^\infty \subseteq C_0(X)$ with $h_i(x_i)=1$ and $h_i(X\,{\backslash}\, W_i)=\{0\}$ , for all $i\in \mathbb N$ . Let $g_i=fh_i\in C_0(X)$ . It follows that $g_iU^0\in (C_0(X)\times _\phi \mathbb Z)_{\mathrm {hc}}\,{\backslash}\, \mathcal R_{\mathrm h}$ .

We will prove that the sequence $\{M_{\pi (A),\pi (A)}(\pi (g_iU^{0}))\}_{i=1}^\infty $ has no convergent subsequence.

We estimate the quantity

$$ \begin{align*} \left\|M_{\pi(A),\pi(A)}\left(\pi(g_iU^{0})\right)-M_{\pi(A),\pi(A)}\left(\pi(g_jU^{0})\right)\right\|, \end{align*} $$

for $i,j\in \mathbb {N}$ with $i< j$ . We have

$$ \begin{align*} \left\|M_{\pi(A),\pi(A)}\left(\pi(g_iU^{0})\right)-M_{\pi(A),\pi(A)}\left(\pi(g_jU^{0})\right)\right\| & \\= \inf_{B\in \mathcal R_{\mathrm h}}\left\|M_{A,A}\left(g_iU^{0}\right)-M_{A,A}\left(g_jU^{0}\right)+B\right\| & \\= \inf_{B\in \mathcal R_{\mathrm h}}\left\|f^3h_iU^{0}-f^3h_jU^{0}+B\right\| & \\ \ge \inf_{B\in \mathcal R_{\mathrm h}}\|E_{0}\left(f^3h_iU^{0}-f^3h_jU^{0}+B\right)\| & \\ \ge \inf_{B\in \mathcal R_{\mathrm h}}\left|E_{0}\left(f^3h_iU^{0}-f^3h_jU^{0}+B\right)(x_i)\right|& \\= |f^3(x_i)|>|f^3(x_0)|, \end{align*} $$

and the proof is complete.

Remark 3.5 The hypocompact radical of the semicrossed product $C_0(X)\times _\phi \mathbb Z_+$ is determined in [4].

Remark 3.6 Shulman and Turovskii call a Banach algebra scattered if the spectrum of every element $a\in \mathcal A$ is finite or countable [19, 21]. They show that a Banach algebra $\mathcal A$ has a largest scattered ideal denoted by $\mathcal A_{\mathrm s}$ [19, Theorem 8.10]. This ideal is closed and is called the scattered radical of $\mathcal A$ . It follows from Theorem 3.4 and [19, Theorem 8.22] that

$$ \begin{align*} (C_0(X)\times_\phi\mathbb Z)_{\mathrm{hc}}=(C_0(X)\times_\phi\mathbb Z)_{\mathrm{s}}. \end{align*} $$