2 Preliminaries

We will denote the set of all nonnegative real numbers by ${\mathbb R}^+$ . If A is a complex Banach algebra with unit $1$ , then the set of all invertible elements will be indicated by $A^{-1}$ and elements of the form $\lambda 1$ in A will be denoted by $\lambda $ . If a is an element of A, then the spectrum $\{\lambda \in \mathbb C: a-\lambda \not \in A^{-1}\}$ of a in A will be denoted by $\sigma (a)$ (or by $\sigma (a, A)$ , if necessary, to avoid confusion). We will use the symbols $\partial \sigma (a)$ and $\eta \sigma (a)$ for the boundary and the connected hull, respectively, of $\sigma (a)$ , and $r(a)$ will indicate the spectral radius of a.

The boundary spectrum of a in A was introduced in [Reference Mouton4] as

$$ \begin{align*}S_{\partial}(a):= \{\lambda \in \mathbb C: a-\lambda \in \partial_A (A\backslash A^{-1})\},\end{align*} $$

and the exponential spectrum of a in A was introduced in [Reference Harte2] as

$$ \begin{align*}\varepsilon(a):=\{\lambda \in \mathbb C: a-\lambda \not \in \mathrm{Exp}(A)\},\end{align*} $$

where $\partial _A (A\backslash A^{-1})$ denotes the norm boundary of $A\backslash A^{-1}$ in A and $\mathrm {Exp}(A)$ the set $\{e^{a_1}e^{a_2}\cdots e^{a_k}: k \in \mathbb N,\ a_1, \ldots , a_k \in A\}$ . Note that $\mathrm { Exp}(A)$ equals the component $\mathrm {Comp}_A(1, A^{-1})$ of $1$ in $A^{-1}$ (see, for instance, [Reference Aupetit1, Theorem 3.3.7]).

Both the boundary spectrum and the exponential spectrum of a are nonempty compact subsets of the complex plane, for every $a \in A$ , and

$$ \begin{align*} \partial \sigma(a) \subseteq S_{\partial}(a) \subseteq \sigma(a) \subseteq \varepsilon(a) \subseteq \eta \sigma(a). \end{align*} $$

The boundary spectrum plays a role in spectral continuity if A is an ordered Banach algebra (see [Reference Mouton6]) and the exponential spectrum is of importance in the study of Fredholm theory (see [Reference Harte2]).

Let $K(\mathbb C)$ denote the metric space of all compact subsets of $\mathbb C$ with the Hausdorff metric (see [Reference Aupetit1, page 48] and also [Reference Mouton6, page 1778]). A mapping $\omega :A \rightarrow K(\mathbb C)$ is said to be a Mobius spectrum [Reference Harte and Wickstead3] on A if $\omega (f(a)) = f(\omega (a))$ for all $a \in A$ , for all functions f of the form $f(\lambda )=\alpha \lambda + \beta $ ( $\alpha , \beta \in \mathbb C$ ) and for $f(\lambda )={1}/{\lambda }$ , such that f is well defined on $\omega (a) \cup \sigma (a)$ . Note that $\sigma $ , $\varepsilon $ , $\partial \sigma $ , $\eta \sigma $ [Reference Harte and Wickstead3] and $S_{\partial }$ [Reference Mouton5] are Mobius spectra on any Banach algebra.

Let A be a complex Banach algebra with unit 1 and B a closed subalgebra of A such that $1 \in B$ . If $K \subseteq A$ is arbitrary, then (see [Reference Mouton and Harte8, Definition 3.2]) the restricted closure of K in A relative to B is given by

$$ \begin{align*}\mathrm{cl}^B(K)=\{a\in A:\text{for all } U\in\mathrm{Nbd}_B(0), (a-U)\cap K\neq\emptyset\},\end{align*} $$

where $\mathrm {Nbd}_B(0)$ indicates the set of all neighbourhoods of 0 in B in the relative (norm) topology. The restricted closure does, in fact, define a topology (see [Reference Mouton and Harte8, Theorem 3.5]), called the restricted topology or the B-topology. It is also shown in [Reference Mouton and Harte8] that, in the case where $K \subseteq B$ , the restricted closure of K in A coincides with the relative closure of K in B, and if $B=A$ , then the B-topology coincides with the norm topology. The restricted boundary and interior of K in A will be denoted by $\partial ^B(K)$ and $\mathrm {Int}^B(K)$ , respectively, and the norm closure, boundary and interior of K in A by $\mathrm {cl}_A(K)$ , $\partial _A(K)$ (as above with $K=A\backslash A^{-1}$ ) and $\mathrm {Int}_A(K)$ , respectively. We recall the following from [Reference Mouton and Harte8] (where more general versions were given). If $K\subseteq A$ , then

(2.1) $$ \begin{align} \mathrm{Int}^B(K)=\{a\in A:\text{there exists } U\in\mathrm{Nbd}_B(0), a-U\subseteq K\}, \end{align} $$

(2.2) $$ \begin{align} {\partial}^B(K) = \{a \in A: \text{for all } U \in \mathrm{Nbd}_B(0), (a-U) \cap K \neq \emptyset \text{ and } (a-U) \cap A\backslash K \neq \emptyset\} \end{align} $$

and, with the set of all neighbourhoods of a in A in the B-topology denoted by $\mathrm {Nbd}^B(a)$ ,

(2.3) $$ \begin{align} \mathrm{Nbd}^B(a)=\{U\subseteq A:a-U \in\mathrm{Nbd}_B(0)\} \end{align} $$

for all $a \in A$ .

In [Reference Mouton and Harte8], we then defined

$$ \begin{align*} {\eta}^B(A\backslash A^{-1}) = A\backslash \mathrm{Comp}^B(1, A^{-1}), \end{align*} $$

where $\mathrm {Comp}^B(1, A^{-1})$ is the component of $A^{-1}$ containing $1$ in the B-topology.

Finally, the restricted boundary and the restricted connected hull of an element $a \in A$ were defined in [Reference Mouton and Harte8] as

(2.4) $$ \begin{align} \partial_{B}(a):=\partial_{B}(a,A) = \{\lambda \in \mathbb C:a-\lambda \in \partial^{B}(A\backslash A^{-1})\} \end{align} $$

and

$$ \begin{align*} \eta_{B}(a):= \eta_{B}(a,A) = \{\lambda \in \mathbb C:a-\lambda \in \eta^{B}(A\backslash A^{-1})\}, \end{align*} $$

respectively. The following results regarding these sets have been obtained in [Reference Mouton and Harte8], where we recall that $\sigma (a):=\sigma (a,A)$ .

Theorem 2.1 [Reference Mouton and Harte8, Corollary 6.4].

For any $a \in A$ ,

$$ \begin{align*}\partial_{\mathbb C}(a) \subseteq \partial_{B}(a) \subseteq \partial_{A}(a),\end{align*} $$

with $\partial _{\mathbb C}(a) = \partial \sigma (a)$ and $\partial _{A}(a)= S_{\partial }(a)$ .

Theorem 2.2 [Reference Mouton and Harte8, Corollary 6.9].

For any $a \in A$ ,

$$ \begin{align*}\eta_A(a) \subseteq \eta_B(a) \subseteq \eta_{\mathbb C}(a),\end{align*} $$

with $\eta _A(a)=\varepsilon (a)$ . In addition, if $a \notin \mathbb C$ , then $\eta _{\mathbb C}(a) = \mathbb C$ , while if $a \in \mathbb C$ , then ${\eta _{\mathbb C}(a) = \eta \sigma (a)}$ .

In conclusion, we recall that an ordered Banach algebra (OBA) is a complex unital Banach algebra which contains an algebra cone, that is, a subset C with the properties that C contains the unit and is closed under addition, nonnegative real scalar multiplication and multiplication (see [Reference Raubenheimer and Rode11]). An algebra cone C induces a partial order on A: $a \leq b$ if and only if $b-a \in C$ , and C is normal if there exists a constant $\alpha> 0$ such that $||a|| \leq \alpha ||b||$ whenever $0 \leq a \leq b$ .

Throughout the rest of this paper, A will be a complex Banach algebra with unit 1 and B a closed subalgebra of A such that $1 \in B$ .

If $a \in B$ , then we denote the open ball in B with centre a and radius $\varepsilon $ by $B_B(a, \varepsilon )$ . We note that, if $a \in A$ , then

(2.5) $$ \begin{align} a-B_B(0,\varepsilon) = B_A(a, \varepsilon) \cap (a-B), \end{align} $$

and so, if $a \in B$ , then

(2.6) $$ \begin{align} a-B_B(0,\varepsilon) = B_B(a, \varepsilon). \end{align} $$

3 Mapping and regularity-type properties

Similarly to the definition of the spectrum of an element a, we can equivalently write $a-\lambda $ or $\lambda - a$ in the definition of $\partial _B(a)$ (see (2.4)). To see this, we note that if $U \subseteq A$ , then clearly $U \in \mathrm {Nbd}_B(0)$ if and only if $-U \in \mathrm {Nbd}_B(0)$ (and hence U is open in A in the B-topology if and only if $-U$ is open in A in the B-topology). Therefore, by (2.3), $U \in \mathrm { Nbd}^B(-a)$ if and only if $-a-U \in \mathrm {Nbd}_B(0)$ if and only if $a-(-U) \in \mathrm {Nbd}_B(0)$ if and only if $-U \in \mathrm {Nbd}^B(a)$ .

Since $A^{-1}= -A^{-1}$ , it now follows that $\partial _B(a)$ can, indeed, equivalently be written as

$$ \begin{align*}\{\lambda \in \mathbb C: \lambda -a \in \partial^B(A\backslash A^{-1})\}\end{align*} $$

for any $a \in A$ . (This also follows from [Reference Mouton and Harte8, Lemma 6.12].)

In [Reference Mouton and Harte9], a number of mapping properties of $\eta _B$ were established. In particular, it was shown that $\eta _B$ is a Mobius spectrum on B [Reference Mouton and Harte9, Proposition 3.1]. Now, if $a \in B$ , then since $\partial _B(a,B)=S_{\partial }(a,B)$ and $S_{\partial }$ is a Mobius spectrum on any Banach algebra, an argument similar to that for $\eta _B$ shows that $\partial _B$ is a Mobius spectrum on B.

In [Reference Mouton and Harte8], the following mapping property of $\partial _B$ was presented.

In Propositions 3.4 and 3.5, as well as in Theorem 3.10, we provide a few more simple mapping properties of $\partial _B$ , which extend results from [Reference Mouton4, Reference Mouton5]. We need the following lemma.

Note that Lemma 3.3 implies that $0 \in \partial ^{B}(A\backslash A^{-1})$ .

For $a \in B$ , most of the next result follows immediately from Theorem 3.2. We also provide a direct short proof which establishes the property for all $a \in A$ .

Note that the second part of Proposition 3.5 also follows from Theorem 3.2.

We now establish a theorem which will lead to two useful corollaries.

Proof. (1) If $a_j = 0$ for at least one $j = 1, \ldots , n$ , then the assumption implies that $0 \in \mathrm {Int}^A(A\backslash A^{-1})$ . By applying [Reference Mouton and Harte8, Corollary 3.12], we then obtain a contradiction from the remark following Lemma 3.3. Therefore, our assumption implies that $a_j \neq 0$ for all $j = 1, \ldots , n$ .

If $a_1a_2 \cdots a_n \in \mathrm {Int}^A(A\backslash A^{-1})$ , say $B_A(a_1a_2\cdots a_n, {\varepsilon }_0) \subseteq A\backslash A^{-1}$ , then it follows from [Reference Mouton5, Corollary 3.3] that there exist ${\varepsilon }_j> 0$ ( $j = 1, \ldots , n$ ) such that

$$ \begin{align*}B_A(a_1, {\varepsilon}_1)B_A(a_2, {\varepsilon}_2) \cdots B_A(a_n, {\varepsilon}_n) \subseteq A\backslash A^{-1},\end{align*} $$

and so it follows from (2.5) that

(3.1) $$ \begin{align} (a_1-B_B(0, {\varepsilon}_1))(a_2-B_B(0, {\varepsilon}_2)) \cdots (a_n - B_B(0, {\varepsilon}_n)) \subseteq A\backslash A^{-1}. \end{align} $$

Now suppose $a_j \in \partial ^B(A\backslash A^{-1})$ for all $j = 1, \ldots , n$ . Then, for each $j = 1, \ldots , n$ , there exists $c_j \in (a_j-B_B(0, {\varepsilon }_j)) \cap A^{-1}$ , by (2.2). However, then $c_1c_2\cdots c_n \in A\backslash A^{-1} \cap A^{-1}$ , by (3.1). This contradiction yields the result.

(2) As in part (1), we have $a_j \neq 0$ for all $j=1, \ldots , n$ . If $a_1, \ldots , a_n \in B$ and $a_1a_2 \cdots a_n \in \mathrm {Int}^B(A\backslash A^{-1})$ , then, by (2.2) and (2.6), there exists ${\varepsilon }_0> 0$ such that

$$ \begin{align*}B_B(a_1a_2 \cdots a_n, {\varepsilon}_0) \subseteq A\backslash A^{-1}.\end{align*} $$

As before, by [Reference Mouton5, Corollary 3.3], there exists ${\varepsilon }_j> 0$ ( $j = 1, \ldots , n$ ) such that

$$ \begin{align*}B_B(a_1, {\varepsilon}_1)B_B(a_2, {\varepsilon}_2) \cdots B_B(a_n, {\varepsilon}_n) \subseteq A\backslash A^{-1}.\end{align*} $$

Now suppose that $a_j \in \partial ^B(A\backslash A^{-1})$ for all $j =1, \ldots , n$ . Then, for each $j = 1, \ldots , n$ , there exists $c_j \in B_B(a_j, {\varepsilon }_j) \cap A^{-1}$ and, as before, we obtain a contradiction.

Theorem 3.6 yields the next two corollaries which can be used to establish certain mapping and regularity-type properties of $\partial _B$ (see Theorem 3.10 and Proposition 3.11).

Using Corollary 3.8(2) and building on [Reference Mouton5, Example 3.7], we can now show that there exists a complex Banach algebra A such that A has a (proper) closed subalgebra B containing the unit and an element $a \in B$ such that $\partial \sigma (a) \subsetneq \partial _B(a)$ .

We write $K^2=f(K)$ , where $f(\lambda )={\lambda }^2$ and $K \subseteq \mathbb C$ .

Now let $R_B= A^{-1} \cup \mathrm {Int}^B(A\backslash A^{-1})$ . Then $R_B= A\backslash \partial ^B(A\backslash A^{-1})$ , by [Reference Mouton and Harte8, Corollary 3.12], and we have

$$ \begin{align*}\partial_B(a) = \{\lambda \in \mathbb C: a-\lambda \notin R_B\} = \{\lambda \in \mathbb C: a-\lambda \notin A^{-1} \cup \mathrm{Int}^B(A\backslash A^{-1})\}.\end{align*} $$

Note that $R_A$ coincides with the set $R_{\partial }$ in [Reference Mouton5].

Next, we will discuss a few regularity-type properties of $R_B$ . We refer to [Reference Müller10], where the definitions and properties of upper and lower semiregularities are presented. In particular, by [Reference Müller10, Remark 3, page 212], a nonempty subset R of A satisfying

1. (P1) if $a, b \in A$ such that $ab=ba$ and $ab \in R$ , then $a, b \in R$

is a lower semiregularity. Also, it clearly follows from the definition and [Reference Müller10, Lemma 2(iii), page 212], that a lower semiregularity satisfies the following property:

1. (P2) if $a \in A$ and $b \in A^{-1}$ such that $ab=ba$ , then $ab \in R$ if and only if $a \in R$ .

We first note that although $R_B$ is in general neither a lower semiregularity (see [Reference Raubenheimer and Swartz12, Example 2.1(1)]), nor an upper semiregularity (see [Reference Raubenheimer and Swartz12, Example 3.2]) on B, $R_B$ does satisfy the following properties, of which part (2) is a weaker version of property (P1).

Parts (1) and (2) in the above result follow easily from Corollaries 3.8(2) and 3.7(2), respectively.

The set $R_B$ also satisfies the following property, which is in some sense weaker, and in another sense stronger, than property (P2).

We note that since $\mathrm {Exp}(A) \subseteq A^{-1} \subseteq R_B$ , [Reference Mouton5, Theorem 4.4] and its obvious analogue give the following characterisation of the (Jacobson) radical:

(3.2) $$ \begin{align} \mathrm{Rad}(A) = \{a \in A: R_Ba \subseteq \mathrm{QN}(A)\} = \{a \in A: aR_B \subseteq \mathrm{QN}(A)\} \end{align} $$

for any closed subalgebra B of A such that $1 \in B$ .

Finally, we exhibit a variation of Harte’s theorem [Reference Harte2, Theorem 3] involving the restricted boundary.

Theorem 3.13. Let $A_1$ and $A_2$ be Banach algebras with the same unit 1, B a closed subalgebra of both $A_1$ and $A_2$ such that $1 \in B$ , and $T:A_1 \rightarrow A_2$ a homomorphism which is either bounded with closed range or satisfies the strong Riesz property. Then, for all $a \in A_1$ ,

$$ \begin{align*}\bigcap\limits_{Tb=0}\partial_B(a+b,A_1) \subseteq \eta \partial_B(Ta,A_2).\end{align*} $$

Theorem 3.13 follows directly from [Reference Mouton7, Theorem 5.4], which generalises [Reference Harte2, Theorem 3], via the use of abstract mappings. (Here, T has the strong Riesz property if $\sigma (a) \subseteq \eta \sigma (Ta) \cup \text {iso\ } \sigma (a)$ for all $a \in A_1$ , with $\text {iso\ } \sigma (a)$ denoting the set of all isolated points of $\sigma (a)$ .)

4 Continuity properties

For a set K in $\mathbb C$ , let $|K|$ indicate the set $\{|\lambda |: \lambda \in K\}$ in this section.

We first note that $\partial _B(a)$ is a compact set in $\mathbb C$ (see the remark following Corollary 4.2), and then show that the function $a \mapsto \partial _{B}(a)$ from A into $K(\mathbb C)$ is upper semicontinuous on B in the norm topology (see Theorem 4.5). We start with the following lemma.

Since $\partial ^B(A\backslash A^{-1})$ is closed in A in the B-topology, we have the following corollary.

Since $\partial \sigma (a) \subseteq \partial _B(a) \subseteq \sigma (a)$ , it follows that $\partial _B(a)$ is a nonempty, compact set in $\mathbb C$ , for all $a \in A$ .

To establish the upper semicontinuity of certain maps, we start with the following two lemmas (recalling that $\sigma (x):=\sigma (x,A)$ for any $x \in A$ (or B)).

Finally, we develop a spectral continuity result in OBAs (Theorem 4.8) using $\partial _B$ . We need some preliminary theory and results.

For any $a \in A$ , let

$$ \begin{align*}|\partial|_B(a) := \{\lambda \in \mathbb C: a-|\lambda|\in \partial^B(A\backslash A^{-1})\}.\end{align*} $$

By Lemma 3.1, $|\partial |_B(a)=\{\lambda \in \mathbb C: |\lambda |-a \in \partial ^B(A\backslash A^{-1})\}$ , $|\partial |_A(a)$ equals the set $T(a)$ of [Reference Mouton6] and

$$ \begin{align*}|\partial|_B(a) = \{\lambda \in \mathbb C: |\lambda| \in \partial_B(a)\} = \{\lambda \in \mathbb C: |\lambda| \in \partial_B(a) \cap {\mathbb R}^+\},\end{align*} $$

so that $||\partial |_B(a)| \subseteq \partial _B(a)$ . Clearly, $|\partial |_B(a)$ is a compact subset of $\mathbb C$ .

We now prove an important property of the map $a \mapsto |\partial |_B(a)$ .

By applying Theorem 2.3, we have the following result in OBAs.

We now arrive at our spectral continuity result for OBAs, which extends [Reference Mouton6, Theorem 4.6].