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Linking the boundary and exponential spectra via the restricted topology
J. Math. Anal. Appl. 454 (2017) 730–745
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Linking the boundary and exponential spectra via the restricted topology ✩ Sonja Mouton a,∗ , Robin Harte b a
Department of Mathematical Sciences, Private Bag X1, Stellenbosch University, Matieland 7602, South Africa b School of Mathematics, Trinity College, Dublin, Ireland
a r t i c l e
i n f o
Article history: Received 6 December 2016 Available online 11 May 2017 Submitted by M. Mathieu
a b s t r a c t We build a chain, based on subalgebras, connecting the boundary spectrum/exponential spectrum duality with the duality between the usual boundary and the connected hull. © 2017 Elsevier Inc. All rights reserved.
Keywords: Boundary spectrum Exponential spectrum Boundary Hull
1. Introduction If a is an element of a complex Banach algebra A with unit 1, then the boundary spectrum S∂ (a) = {λ ∈ C : a − λ1 ∈ ∂A−1 } of a (see [4]) is a compact set in C that lies between the usual topological boundary of the spectrum and the spectrum σ(a) itself, i.e. ∂σ(a) ⊆ S∂ (a) ⊆ σ(a). It therefore seems natural to view ∂σ(a), on the one hand, and S∂ (a), on the other hand, as the “thin” and “fat” boundaries of σ(a), respectively. In this paper we show that, using closed subalgebras B, it is possible to define a topology on A (different from the norm-topology) in such a way that a whole range of “boundaries” can be obtained, with B = C giving the “thin” boundary and B = A the “fat” boundary of σ(a) — see Corollary 6.4.
✩
The first author was supported by the National Research Foundation (NRF) of South Africa.
* Corresponding author. E-mail addresses: [email protected] (S. Mouton), [email protected] (R. Harte). http://dx.doi.org/10.1016/j.jmaa.2017.05.020 0022-247X/© 2017 Elsevier Inc. All rights reserved.
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From [3] we recall a certain duality between the boundary ∂σ(a) and the connected hull ησ(a) of the spectrum of a. Bearing in mind the fact that the exponential spectrum ε(a) of a lies between σ(a) and ησ(a), i.e. σ(a) ⊆ ε(a) ⊆ ησ(a), we also investigate the “connected hulls” accompanying these new “boundaries”, and show that B = A yields ε(a), making ε(a) the “little” connected hull of σ(a). In addition it is shown that in trivial cases, the “big” connected hull of σ(a) is ησ(a), while otherwise it is actually the whole of C (see Corollary 6.9). In the following section we provide all the relevant notation and terminology. In Section 3, given an additive topological group A and a subgroup B of A, we define a new topology on A, called the B-topology or the restricted topology, via the closure operation and show that, for elements and subsets of A lying inside B, topological concepts in the B-topology coincide with those in the relative topology (see Theorem 3.4). We also obtain a number of basic properties relating to the B-topology and provide a number of examples. In Section 4 we investigate the correspondence between subsets Hω of A and certain mappings ω from A into 2B relying on some special element e of B. (The motivation for this is the relationship between the set of all invertible elements of a Banach algebra A and the function that maps an element a ∈ A onto its spectrum, where B = C and e = 1.) Section 5 is devoted to finding the relationships between the restricted boundaries and between the restricted connected hulls of ω and A\Hω , with the main results contained in Theorems 5.2 and (its partial analogue) 5.13. Finally, in Section 6, we return to Banach algebras. Using the concepts of the restricted boundary and the restricted connected hull of the set of all non-invertible elements of a Banach algebra A, we define a range of “boundaries” and “connected hulls” contained in C (see (6.1) and (6.6)) and thereby arrive at our main results Theorems 6.2 and 6.7. In conclusion, we show that certain known results about the boundary spectrum can be generalised by using our new “boundary” concept (see, in particular, Theorem 6.15). 2. Preliminaries Let X be a topological space and let t be an element of X. Then we denote the set of all neighbourhoods of t in X by NbdX (t). In addition, if K is a subset of X, then the (topological) closure, interior and boundary of K in X will be denoted by clX (K), IntX (K) and ∂X (K), respectively. / K, then the connected hull In some sense dual to the topological boundary is the connected hull. If t ∈ ηt (K) relative to t of K is defined by X\ηt (K) = CompX (t, X\K), where CompX (t, H) ⊆ X is the (connected) component of t in H. When in particular X is a normed linear space and K is a bounded set in X, we shall also write ηK := η∞ K for the complement of the unique unbounded component of X\K. If instead X is a topological ring with identity e, and e ∈ / S ⊆ A, we shall prefer ηS := ηe S. If, in turn, 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 λ1 in A will be denoted by λ. If a is an element of A, then the spectrum {λ ∈ C : a − λ ∈ / A−1 } of a in A will be denoted by σ(a) (or by σA (a), if necessary to avoid confusion). Referring to the notation given above, we will use the symbols ∂σ(a) := ∂C σ(a) and (as already mentioned) ησ(a) := η∞ σ(a) for the boundary and the connected hull, respectively, of σ(a). It is well known that if B is a closed subalgebra of A containing 1, then σA (a) ⊆ σB (a) (see, for instance, [1], Theorem 3.2.13).
and ∂σB (a) ⊆ ∂σA (a)
(2.1)
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The boundary spectrum of a in A (which plays an interesting role regarding spectral continuity if A is an ordered Banach algebra — see [6]) was introduced in [4] as S∂ (a) := {λ ∈ C : a − λ ∈ ∂A (A\A−1 )}. It is easy to see ([4], Proposition 2.1) that ∂σ(a) ⊆ S∂ (a) ⊆ σ(a)
(2.2)
and that S∂ (a) is a closed set. Therefore the boundary spectrum of a is a non-empty compact subset of the complex plane, for every a ∈ A. Due to the first inclusion in (2.2) the boundary spectrum is sometimes called the “fat boundary”. Let Exp A denote the set {ea1 . . . eak : k ∈ N, a1 , . . . , ak ∈ A}. Then Exp A equals the component CompA (1, A−1 ) of 1 in A−1 (see, for instance, [1], Theorem 3.3.7). The exponential spectrum of a in A was introduced in [2] as ε(a) := {λ ∈ C : a − λ ∈ / Exp A}. The exponential spectrum of a is another non-empty compact subset of the complex plane, for every a ∈ A, and it satisfies σ(a) ⊆ ε(a) ⊆ ησ(a)
(2.3)
(see [2], Theorem 1). 3. The restricted closure and the B-topology If Y ⊆ X is a subset of a topological space then the relative topology of Y induced by the topology of X is familiar: in terms of the closure operation we set, for arbitrary K ⊆ Y , clY (K) = Y ∩ clX (K).
(3.1)
We now extend this idea to more general subsets K ⊆ X, when X and Y ⊆ X are topological abelian groups, by making the following Definition 3.2. Let A be an additive topological group and let B be a subgroup of A. If K ⊆ A is arbitrary then its restricted closure in A relative to B is given by clB (K) = {a ∈ A : ∀ U ∈ NbdB (0) : (a − U ) ∩ K = ∅}. By (separate) continuity of addition, NbdA (a) = a − NbdA (0), for all a ∈ A, and therefore clA (K) = clA (K),
(3.3)
i.e., if B = A then the restricted closure of K in A coincides with the topological closure of K in A. Moreover, if K ⊆ B, then the restricted closure of K in A relative to B coincides with the relative closure (3.1) of K in B: Theorem 3.4. Let B be a subgroup of a topological group A and let K ⊆ B. Then clB (K) = clB (K).
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Proof. Since clB (K) = B ∩clA (K), by (3.3) it suffices to prove that clB (K) = B ∩clA (K). So let a ∈ clB (K) and let U ∈ NbdA (0). Then V = B ∩ U ∈ NbdB (0), so that (a − V ) ∩ K = ∅, say v ∈ V and y = a − v ∈ K. Since (a − V ) ∩ K ⊆ (a − U ) ∩ K, it follows that (a − U ) ∩ K = ∅ and so a ∈ clA (K). Also, since K ⊆ B, we have that y ∈ B and since v ∈ B, it follows that a = y + v ∈ B. Conversely, let a ∈ B ∩ clA (K) and let U ∈ NbdB (0). Then U = B ∩ V with V ∈ NbdA (0). Hence (a − V ) ∩ K = ∅, say v ∈ V and y = a − v ∈ K. Since K ⊆ B, we have that y ∈ B and since a ∈ B, it follows that v = a − y ∈ B. Therefore v ∈ U , so that (a − U ) ∩ K = ∅ and so a ∈ clB (K). • In order to verify that the restricted closure does in fact define a topology, we check the Kuratowski conditions: Theorem 3.5. If B is a subgroup of a topological group A and if K, H ⊆ A then 1. 2. 3. 4. 5.
clB (∅) = ∅, K ⊆ H =⇒ clB (K) ⊆ clB (H), K ⊆ clB (K), clB clB (K) ⊆ clB (K), clB (K ∪ H) ⊆ clB (K) ∪ clB (H).
Proof. Each of (1) and (2) is clear. Towards (3), if a ∈ K and U ∈ NbdB (0) then a ∈ (a − U ) ∩ K. Towards (4), suppose a ∈ A is in the left hand side and U ∈ NbdB (0) is arbitrary. Then there are bU ∈ U and then cU ∈ U for which a − bU ∈ clB (K) , a − bU − cU ∈ K. By joint continuity of addition there is also V ∈ NbdB (0) for which V + V ⊆ U . Hence b U = bV + cV ∈ U and a − b U ∈ (a − U ) ∩ K. Finally, for (5), suppose that a ∈ A is in the left hand side and not in the closure of H. For arbitrary U ∈ NbdB (0) we have (a − U ) ∩ (K ∪ H) = ∅. There is however at least one V ∈ NbdB (0) for which (a − V ) ∩ H = ∅.
(3.6)
In particular, intersecting U and V , (a − (U ∩ V )) ∩ (K ∪ H) = ∅; but (3.6) gives (a − (U ∩ V )) ∩ H = ∅. It now follows that (a − U ) ∩ K ⊇ (a − (U ∩ V )) ∩ K = (a − (U ∩ V )) ∩ (K ∪ H) = ∅.
•
We will refer to the topology induced on an additive topological group A by the restricted closure in A relative to a subgroup B as the restricted topology or the B-topology. In order to compile a number of basic results associated with the restricted topology, let A denote a topological group and let B denote a subgroup of A. Knowing that the restricted closure defines a topology, our first proposition follows immediately from the facts that IntB (K) = A\clB (A\K) and ∂ B (K) = clB (K) ∩ clB (A\K) = clB (K)\IntB (K):
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Proposition 3.7. If K ⊆ A then IntB (K) = {a ∈ A : ∃ U ∈ NbdB (0) , a − U ⊆ K} and ∂ B (K) = {a ∈ A : ∀U ∈ NbdB (0) : (a − U ) ∩ K = ∅ and (a − U ) ∩ A\K = ∅}. Neighbourhoods in the B-topology can be described as follows: Proposition 3.8. If a ∈ A then NbdB (a) = {U ⊆ A : a − U ∈ NbdB (0)}. Proof. If U ∈ NbdB (a), then there exists a subset V of U , open in the B-topology, containing a. By Proposition 3.7 there exists a W ∈ NbdB (0) such that a − W ⊆ V . It follows that W ⊆ a − V ⊆ a − U , so that a − U ∈ NbdB (0). Conversely, suppose that a − U ∈ NbdB (0). Then a − U = B ∩ V for some V ∈ NbdA (0), so that U = a −(B ∩V ), with V containing an open set V1 in A which contains 0. It follows that a ∈ a −(B ∩V1 ) ⊆ U . If b ∈ B ∩ V1 , then B ∩ V1 ∈ NbdB (b), so that (B ∩ V1 ) − b ∈ NbdB (0). Hence, if b ∈ B ∩ V1 , then U = (B ∩ V1 ) − b ∈ NbdB (0) with a − b − U ⊆ a − (B ∩ V1 ), so that a − (B ∩ V1 ) is open in the B-topology by Proposition 3.7. It follows that U ∈ NbdB (a). • If C denotes another subgroup of A, we have the following properties: Lemma 3.9. If C ⊆ B ⊆ A and K ⊆ A, then clC (K) ⊆ clB (K). Proof. Let a ∈ clC (K) and V ∈ NbdB (0). Then V = W ∩ B for some W ∈ NbdA (0), so that U = W ∩ C ∈ NbdC (0). Hence (a − U ) ∩ K = ∅. Since (a − U ) ∩ K ⊆ (a − V ) ∩ K, it follows that (a − V ) ∩ K = ∅, so that a ∈ clB (K). • Corollary 3.10. If C ⊆ B ⊆ A, then the C-topology of A is stronger than the B-topology of A. Hence the B-topology of A is stronger than the (original) topology of A. Corollary 3.11. Let C ⊆ B ⊆ A and G ⊆ A. If G is connected in the C-topology, then G is connected in the B-topology. Corollary 3.12. Let C ⊆ B ⊆ A and K ⊆ A. Then: 1. IntB (K) ⊆ IntC (K) ⊆ clC (K) ⊆ clB (K) 2. ∂ C (K) ⊆ ∂ B (K) Proof. 1. If a ∈ IntB (K), then there exists a U ∈ NbdB (0), i.e. U = W ∩ B for some W ∈ NbdA (0), such that a − U ⊆ K. Let V = W ∩ C. Then V ∈ NbdC (0) and a − V ⊆ a − U ⊆ K, so that a ∈ IntC (K). The result now follows from Lemma 3.9. 2. Follows from (1). •
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For the purposes of illustrating some of these concepts and providing examples, we will consider the following Banach spaces and subsets: the space CC [0, 1] of all continuous complex valued functions on [0, 1] with the supremum norm, the subset Cx of CC [0, 1] of all complex valued homogeneous polynomials on [0, 1] of degree 1, the space CR [0, 1] of all continuous real valued functions on [0, 1] with the supremum norm and the subset PR [0, 1] of CR [0, 1] of all real valued polynomials on [0, 1]. Example 3.13. If A = CC [0, 1], B = C and K = Cx, then clB (K) = K = clA (K) and IntA (K) = ∅ = IntB (K). Proof. If f ∈ clA (K), then there exists a sequence (pn ) in K, say pn (x) = αn x (x ∈ [0, 1]) such that supx∈[0,1] |αn x − f (x)| = ||pn − f || → 0. Therefore αn → α where α = f (1), so that αn x → αx, but since αn x → f (x), it follows that f (x) = αx, for all x ∈ [0, 1], i.e. f ∈ K. We have shown that clA (K) = K, and therefore, by Corollary 3.12 (1), clB (K) = K. Suppose that f ∈ IntB (K). We may assume that f ∈ K, say f (x) = α0 x, for all x ∈ [0, 1]. Then, by Proposition 3.7, there exists 0 > 0 such that to every λ ∈ C with modulus less than 0 there corresponds αλ ∈ C such that (α0 − αλ )x = λ for all x ∈ [0, 1]. But the case x = 0 forces λ to be zero, which yields IntB (K) = ∅. By Corollary 3.12 (1), IntA (K) = ∅ as well. • Example 3.14. If A = CR [0, 1], B = R and K = PR [0, 1], then clB (K) = K, clA (K) = A, IntA (K) = ∅ and IntB (K) = K. Proof. By the Weierstrass Approximation Theorem clA (K) = A, so suppose that f ∈ clB (K). Then given > 0, by Definition 3.2 there exist λ ∈ (− , ) and a polynomial p ∈ PR [0, 1] such that f − λ = p, i.e. f (x) = λ + p(x) for all x ∈ [0, 1], so that f ∈ PR [0, 1] = K. Therefore clB (K) = K. Suppose that p ∈ IntA (K). Then, since p is a continuous real-valued function on [0, 1], p can be approximated by continuous piecewise linear functions. Hence there is no -neighbourhood of p consisting only of polynomials, which contradicts p ∈ IntA (K). Therefore IntA (K) = ∅. Now let p ∈ K, say p(x) = α0 + α1 x + · · · + αn xn , take any 0 > 0 and let U = (− 0 , 0 ). For λ ∈ U , define qλ (x) := (α0 − λ) + α1 x + · · · + αn xn . Then qλ ∈ PR [0, 1] and p(x) − λ = qλ (x). We have shown that p − U ⊆ PR [0, 1], and so p ∈ IntB (K), by Proposition 3.7. Therefore IntB (K) = K. • It follows from the example above that the set of all real-valued polynomials on [0, 1] is both open and closed in the R-topology and therefore the set of continuous real-valued functions on [0, 1] is not connected in the R-topology. u the space of all 2 × 2complex upper triangular matrices, B = C, K = Example 3.15. Let A = M2 (C), w z w z : z, w ∈ C . Then : z ∈ C, w ∈ Q + iQ and S = 0 w 0 w
clB (K) = S = clA (K) and IntA (K) = ∅ = IntB (K).
w z Proof. It is easy to check that cl (K) = S, so let ∈ S and let > 0. Take w ∈ Q + iQ such that 0 w w z λ 0 w z |w − w | < and let λ = w − w . Then ∈ K, so that − = 0 w 0 w 0 λ A
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w 0
z w
− B(0, )
∩ K = ∅
(where B(0, ) indicates the open ball with centre 0 and radius in C (considered as a subset of A)), i.e. w z ∈ clB (K). We have shown that S ⊆ clB (K), and since clA (K) = S, it follows from Corol0 w lary 3.12 (1) that clB (K) = S. The density of the set R\Q + iR\Q in C forces the interiors to be empty.
•
Finally, we show that if A is a locally connected topological group and B is a connected subgroup of A, then A is locally connected in the B-topology (Proposition 3.21). We need some preliminary results: Lemma 3.16. If V is a connected subset of A which is contained in B, then V is connected in the B-topology (i.e., in the relative topology). Proof. Suppose that V ⊆ B and V is not connected in the relative topology, say V = U1 ∪ U2 with U1 and U2 non-empty sets such that U1 ∩ clB (U2 ) = ∅ = U2 ∩ clB (U1 ). Then, since U1 ⊆ V ⊆ B, we have ∅ = U1 ∩ clB (U2 ) = U1 ∩ B ∩ clA (U2 ) = U1 ∩ clA (U2 ) and similarly U2 ∩ clA (U1 ) = ∅. Hence V is not connected in A.
•
Lemma 3.17. If a ∈ A and G ⊆ A, then clB (a − G) = a − clB (G), and hence if G is closed in A in the B-topology, so is a − G. Proof. It suffices to prove that clB (a − G) ⊆ a − clB (G) for all a ∈ A and all G ⊆ A. So let c ∈ clB (a − G) and let W ∈ NbdB (0). Then W = −W ∈ NbdB (0), so that (c + W ) ∩ (a − G) = ∅, say x ∈ W , y ∈ G are such that c + x = a − y. It follows that if b = a − c, then b − x = a − c − x = y ∈ (b − W ) ∩ G = ∅, so that b ∈ clB (G), and so c = a − b ∈ a − clB (G). • Corollary 3.18. If a ∈ A and H ⊆ A is open in A in the B-topology, then so is a − H. Corollary 3.19. If a ∈ A and G ⊆ A is connected in the B-topology, then so is a − G. Proof. Suppose that a −G = H1 ∪H2 with H1 and H2 non-empty sets with H1 ∩clB (H2 ) = ∅ = H2 ∩clB (H1 ). Then G = (a − H1 ) ∪ (a − H2 ) with a − H1 and a − H2 non-empty sets, and it follows from Lemma 3.17 that (a − H1 ) ∩ clB (a − H2 ) = ∅ = (a − H2 ) ∩ clB (a − H1 ). • Corollary 3.20. Suppose that B is connected in A. If V is a connected neighbourhood of 0 in A, then U = a − (B ∩ V ) is a connected neighbourhood of a in the B-topology. Proof. If V is a connected neighbourhood of 0 in A, then since B ∩ V ⊆ B, it follows from Lemma 3.16 that B ∩ V is connected in the relative topology. Then it follows from Corollary 3.19 that U = a − (B ∩ V ) is connected in the B-topology, and U is a neighbourhood of a in the B-topology, by Proposition 3.8. • Using the previous observations, we can now prove: Proposition 3.21. Suppose that A is locally connected and B is connected in A. Then A is locally connected in the B-topology.
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Proof. Let B0 be a neighbourhood base of 0 consisting of open connected sets in A and let a ∈ A. Then by Corollaries 3.18 and 3.20 BaB = {a − (B ∩ V ) : V ∈ B0 } is a neighbourhood base of a consisting of open connected sets in the B-topology. Indeed, if U ∈ NbdB (a), then a − U ∈ NbdB (0), by Proposition 3.8, and hence a − U = B ∩ W for some W ∈ NbdA (0). It follows that there exists a V ∈ B0 such that V ⊆ W , and so a − (B ∩ V ) ⊆ U . • 4. The ω-spectrum Let B denote a subgroup of a topological group A. In the presence of a distinguished element e ∈ B there is a correspondence ω↔G between subsets of A and certain mappings from A into subsets of B, where we denote elements of the form be in B by b: Definition 4.1. If G ⊆ A and if ω : A → 2B satisfies ∀ (a, b) ∈ A × B : ω(a − b) = ω(a) − b ⊆ B,
(4.2)
set Hω = {a ∈ A : 0 ∈ / ω(a)} ⊆ A and B
G (a) = {b ∈ B : a − b ∈ / G} ⊆ B.
Evidently B ω(·) = G (·) ⇐⇒ G = Hω .
We note that if x ∈ B, then x∈ / ω(a) if and only if a − x ∈ Hω .
(4.3)
The most important situation is when A is a ring, e ∈ A is an identity and B is a subring, in particular a field. For a good spectral theory for ω the subset G = Hω should be a regularity [8]. When A is a ring, in particular a linear algebra over a field B, the most fundamental “spectrum” is derived from the invertible group: σ = A−1 ; Hσ = A−1 . In the remainder of this section, let B be a subgroup of a topological group A and let ω : A → 2B be a mapping satisfying (4.2). We now establish the following basic properties relating to the map ω and the set Hω . These properties will be used in the next section. Lemma 4.4. If ω(a) is closed in A in the B-topology for all a ∈ A, then Hω is open in A in the B-topology.
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Proof. Suppose that ω(a) is closed in A in the B-topology for all a ∈ A. If a ∈ Hω , then 0 ∈ / ω(a), so that there exists a W ∈ NbdB (0) such that W ⊆ A\ω(a). It follows from (4.3) that a − W ⊆ Hω and hence Hω is open in A in the B-topology. • If G ⊆ X ⊆ A, then we will denote the closure of G in X in the B-topology by clB (G, X); so clB (G, X) = cl (G) ∩ X. B
Lemma 4.5. If G ⊆ B\ω(a), then a − clB (G, B\ω(a)) = clB (a − G, Hω ∩ (a − B)). Proof. Let b ∈ clB (G, B\ω(a)). Then b ∈ B\ω(a), so that a − b ∈ Hω , by (4.3). Let W ∈ NbdB (0). Then W = −W ∈ NbdB (0), so that (b + W ) ∩ G = ∅, say x ∈ W , y ∈ G are such that b + x = y. It follows that a − b − x = a − y ∈ (a − b − W ) ∩ (a − G) = ∅. Hence a − b ∈ clB (a − G, Hω ∩ (a − B)). Now let c ∈ clB (a − G, Hω ∩ (a − B)). Then c = a − b with b ∈ B and c ∈ Hω , so that b ∈ B\ω(a), by (4.3). If W ∈ NbdB (0), then, as before, (c + W ) ∩ (a − G) = ∅, say x ∈ W , y ∈ G are such that c + x = a − y. It follows that b − x = a − c − x = y ∈ (b − W ) ∩ G = ∅. Hence b ∈ clB (G, B\ω(a)), so that c ∈ a − clB (G, B\ω(a)). • Corollary 4.6. If G ⊆ B\ω(a) is closed in B\ω(a) in the B-topology, then a − G is closed in Hω ∩ (a − B) in the B-topology. Lemma 4.7. Suppose that B is closed in A. Then Hω ∩ (a − B) is closed in Hω in the B-topology. Proof. If B is closed in A, then B is closed in A in the B-topology, since the latter is a stronger topology. Then the result follows from Lemma 3.17. • 5. The restricted boundary and connected hull The closure in B of the “spectrum” ω(a) determines the restricted closure of the “non singulars” Hω ⊆ A: Theorem 5.1. Let B be a subgroup of a topological group A. If a ∈ A, e, b ∈ B and ω : A → 2B satisfies (4.2), then b∈ / clB ω(a) ⊆ B ⇐⇒ a − b ∈ intB Hω ⊆ A. Proof. Simply observe that for arbitrary U ∈ NbdB (b) ω(a) ∩ U = ∅ ⇐⇒ a − U ⊆ Hω
•
Theorem 5.1 relates the (restricted) topological boundaries of ω and Hω : Theorem 5.2. Let B be a subgroup of a topological group A. If a ∈ A, e ∈ B and ω : A → 2B satisfies (4.2), then ∂ B ω(a) = {b ∈ B : a − b ∈ ∂ B (A\Hω ) ⊆ A}. Proof. We first note that if b ∈ B, then by Theorem 5.1, b ∈ clB ω(a) if and only if a − b ∈ A\IntB Hω = clB (A\Hω ) and, similarly, b ∈ clB (B\ω(a)) if and only if a − b ∈ A\IntB (A\Hω ) = clB (Hω ). The result then follows by observing that
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∂ B ω(a) = clB ω(a) ∩ clB (B\ω(a)) and ∂ B (A\Hω ) = clB (A\Hω ) ∩ clB (Hω ).
•
Having discussed the boundaries of ω and Hω , we now turn to the (somewhat dual) concept of the connected hull. Similarly to the situation in general topological spaces, if A is a topological group, B is a subgroup of A and t ∈ / K ⊆ A, then the restricted connected hull ηtB (K) relative to t of K is given by A\ηtB (K) = CompB (t, A\K), and if K ⊆ B, then B\ηtB (K) = CompB (t, B\K). In particular, if A is a normed algebra, e = 1 and B is a closed subalgebra of A with 1 ∈ B, then if, for some a ∈ A, ω(a) is bounded (where ω : A → 2B satisfies (4.2)), then η B ω(a) = B\CompB (∞, B\ω(a)),
(5.3)
where CompB (∞, B\ω(a)) indicates the unique unbounded component of B\ω(a) relative to the B-topology, and if 1 ∈ Hω , then η B (A\Hω ) = A\CompB (1, Hω ),
(5.4)
where CompB (1, Hω ) is the component of Hω containing 1 in the B-topology. With A a topological group and B and C subgroups of A, we first note, directly from Corollary 3.11, the following: / K. Then Corollary 5.5. Let C ⊆ B ⊆ A and K ⊆ A such that t ∈ CompC (t, A\K) ⊆ CompB (t, A\K). Corollary 5.6. Let C ⊆ B ⊆ A and K ⊆ A such that t ∈ / K. Then ηtB (K) ⊆ ηtC (K). In order to obtain a partial analogue of Theorem 5.2 for the connected hull (Theorem 5.13), we need a number of preliminary results relating to components in the B-topology. In the following results, let B be a subgroup of a topological group A and let ω : A → 2B be a mapping satisfying (4.2). We first note the following: Proposition 5.7. Suppose that A is locally connected, B is closed and connected in A and ω(a) is closed in B for all a ∈ A. If G is a component in the B-topology of B\ω(a), then a − G is open in A in the B-topology. Proof. Since ω(a) is closed in B, we have that B\ω(a) is open in A in the B-topology (note that B is open in A in the B-topology). It follows from Proposition 3.21 that G is open in A in the B-topology. Hence the result follows from Corollary 3.18. •
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Directly from Corollary 4.6 and Lemma 4.7 we have: Corollary 5.8. Suppose that B is closed in A and b ∈ B\ω(a). If G = CompB (b, B\ω(a)), then a − G is closed in Hω in the B-topology. The previous corollary, together with results from Sections 3 and 4, yields: Proposition 5.9. Suppose that A is locally connected, B is closed and connected in A, ω(a) is closed in B for all a ∈ A and b ∈ B\ω(a). If G = CompB (b, B\ω(a)), then a − G is closed in CompB (a − b, Hω ) in the B-topology. Proof. If B is closed in A and ω(a) is closed in B for all a ∈ A, then it follows from Lemma 4.4 and Proposition 3.21 that CompB (a − b, Hω ) is open in Hω in the B-topology. By Corollary 3.19 and (4.3) we have a − G ⊆ CompB (a − b, Hω ) ⊆ Hω . The result then follows from Corollary 5.8. • Now Propositions 5.7 and 5.9 imply: Proposition 5.10. Suppose that A is locally connected, B is closed and connected in A, ω(a) is closed in B for all a ∈ A and b ∈ B\ω(a). 1. If G = CompB (b, B\ω(a)), then a − G = CompB (a − b, Hω ). 2. If G = CompB (a − b, Hω ), then a − G = CompB (b, B\ω(a)). Proof. 1. By (4.3) a − b ∈ a − G ⊆ Hω , and a − G is connected in the B-topology, by Corollary 3.19. Therefore ∅ = a − G ⊆ CompB (a − b, Hω ) ⊆ A. By Proposition 5.7 a − G is open in A, and hence in CompB (a − b, Hω ), in the B-topology, and by Proposition 5.9 a − G is closed in CompB (a − b, Hω ) in the B-topology. Hence the result follows from the connectedness in the B-topology of CompB (a − b, Hω ). 2. Follows from (1). • The following proposition provides a sufficient condition for a function f : [0, 1] → A to be continuous in the B-topology: Proposition 5.11. Suppose that f : [0, 1] → A is continuous in (the usual topology of) A and f ([0, 1]) ⊆ B. Then f is continuous in the B-topology. Proof. Let K ⊆ [0, 1] and let α ∈ cl[0,1] (K). If U ∈ NbdB (0), then U = B ∩ V for some V ∈ NbdA (0), so that f (α) − V ∈ NbdA (f (α)). Since f is continuous in A, we have f (cl[0,1] (K)) ⊆ clA (f (K)) and hence f (α) ∈ clA (f (K)), so that (f (α) − V ) ∩ f (K) = ∅, say b ∈ V and μ ∈ K are such that f (α) − b = f (μ). Then f (α) − f (μ) ∈ V , so that it follows from the assumption that f (α) − f (μ) ∈ U . Hence f (μ) ∈ f (α) − U , so that (f (α) − U ) ∩ f (K) = ∅. Hence f (α) ∈ clB (f (K)). We have proven that f (cl[0,1] (K)) ⊆ clB (f (K)) for all K ⊆ [0, 1], and so f is continuous in the B-topology.
•
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Proposition 5.11 guarantees the unboundedness of certain components in the B-topology: Proposition 5.12. Let A be a complex normed algebra with unit 1 and B a closed subalgebra of A such that 1 ∈ B. Let ω : A → 2B be a mapping satisfying (4.2) with the property that ω(λ) = λω(1) for all λ ∈ C. If 1 ∈ Hω , then C\{0} ⊆ CompB (1, Hω ) and, therefore, CompB (1, Hω ) is unbounded. Proof. If λ1 , λ2 ∈ C\{0}, then there exists a continuous function g : [0, 1] → C such that g(0) = λ1 , g(1) = λ2 and g([0, 1]) ⊆ C\{0}. The function f : [0, 1] → A, defined by f (μ) = g(μ) (:= g(μ)1), is then continuous in (the usual topology of) A and f ([0, 1]) ⊆ C\{0} ⊆ B, so that f is continuous in the B-topology, by Proposition 5.11. Also, f (0) = λ1 and f (1) = λ2 , so that C\{0} is pathwise connected, hence connected, in the B-topology. Using (4.3), we have that 0 ∈ / ω(1), since 1 ∈ Hω , and therefore C\{0} ⊆ Hω , since ω(λ) = λω(1). This yields the result. • Now, using Propositions 5.10 and 5.12, we obtain the following result, which is partially analogous to Theorem 5.2: Theorem 5.13. Let A be a complex normed algebra with unit 1 and B a closed subalgebra of A such that 1 ∈ B. Let ω : A → 2B be a mapping satisfying (4.2), such that ω(a) is bounded and closed in B for all a ∈ A and with the property that ω(λ) = λω(1) for all λ ∈ C. If 1 ∈ Hω then η B ω(a) ⊆ {b ∈ B : a − b ∈ η B (A \ Hω ) ⊆ A}.
(5.14)
Proof. Suppose that b ∈ B and a − b ∈ CompB (1, Hω ). Then b ∈ / ω(a). Now suppose that b is in a bounded component G (in the B-topology) of B\ω(a). Let H = a − G. Then H is bounded, and H = CompB (a − b, Hω ), by Proposition 5.10 (1). But since a − b ∈ CompB (1, Hω ), it follows that H = CompB (1, Hω ), so that CompB (1, Hω ) is bounded, which contradicts Proposition 5.12. Therefore b ∈ CompB (∞, B\ω(a)) (the unique unbounded component of B\ω(a) in the B-topology). The result now follows from (5.3) and (5.4). • It will be shown in the next section that we do not in general have equality in (5.14). 6. The restricted topology applied to Banach algebras Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that e := 1 ∈ B. For any a ∈ A, we define ∂B (a) := {λ ∈ C : a − λ ∈ ∂ B (A\A−1 )}.
(6.1)
For example, if B = A then we get the “fat boundary”, while if B = C then it gives back the “thin boundary”, of the spectrum of a — i.e., the boundary spectrum S∂ (a) and the usual boundary ∂σ(a) of the spectrum of a, respectively: Theorem 6.2. Let A be a complex Banach algebra with unit 1. Then 1. ∂C (a) = ∂σ(a) and 2. ∂A (a) = S∂ (a), for all a ∈ A.
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Proof. 1. If B = C in Definition 4.1, then ω = σ satisfies (4.2) and Hω = A−1 . Therefore it follows from Theorem 5.2 and (6.1) that ∂C (a) = ∂ C σ(a). By Proposition 3.7 ∂ C σ(a) = ∂σ(a), so that the result follows. 2. This follows directly from (6.1) by recalling the definition of S∂ (a). • Lemma 6.3. Let A be a complex Banach algebra with unit 1 and let B and C be closed subalgebras of A such that 1 ∈ C ⊆ B ⊆ A. If a ∈ A, then ∂C (a) ⊆ ∂B (a) ⊆ σ(a), for all a ∈ A. Proof. It follows from Corollary 3.12 (2) that ∂ C (A\A−1 ) ⊆ ∂ B (A\A−1 ) ⊆ ∂ A (A\A−1 ) ⊆ A\A−1 , and so the result follows.
•
Theorem 6.2 and Lemma 6.3 now imply: Corollary 6.4. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Then ∂σ(a) ⊆ ∂B (a) ⊆ S∂ (a), for all a ∈ A, where ∂B (a) is smallest when B = C, in which case ∂B (a) = ∂σ(a), and largest when B = A, in which case ∂B (a) = S∂ (a). Using (2.1) as well, we obtain: Corollary 6.5. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Then ∂σB (a) ⊆ ∂σA (a) ⊆ ∂B (a) ⊆ ∂A (a) ⊆ σA (a) ⊆ σB (a), for all a ∈ B. Analogously to (6.1), for any closed subalgebra B of a Banach algebra A such that 1 ∈ B and a ∈ A, we define ηB (a) := {λ ∈ C : a − λ ∈ η B (A\A−1 )}.
(6.6)
Theorem 6.7. Let A be a complex Banach algebra with unit 1 and let a ∈ A. Then 1. (a) ησ(a) ⊆ ηC (a). (b) If a ∈ C, then ηC (a) = ησ(a). (c) If a ∈ A\C, then ηC (a) = C. 2. ηA (a) = ε(a). Proof. 1. (a) From Theorem 5.13 and (6.6) we have that η C σ(a) ⊆ ηC (a), so that the result follows from (5.3).
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(b) By Proposition 5.12 C\{0} ⊆ CompC (1, A−1 ). Therefore if a ∈ C, then, by (6.6) and (5.4), λ ∈ ηC (a) if and only if a − λ ∈ / CompC (1, A−1 ) if and only if λ = a. It follows that ηC (a) = {a} = ησ(a). (c) By (a) ησ(a) ⊆ ηC (a). In order to prove that C\ησ(a) ⊆ ηC (a) as well if a ∈ / C, let λ0 ∈ C\ησ(a), C C −1 i.e. λ0 ∈ Comp (∞, C\σ(a)), and let G = Comp (a − λ0 , A ). Then, by Proposition 5.10 (2), a − G = CompC (λ0 , C\σ(a)). Since a ∈ / C, it follows that 1 ∈ / G, so that a − λ0 ∈ / CompC (1, A−1 ). C −1 Therefore a − λ0 ∈ η (A\A ), so that λ0 ∈ ηC (a). We have proven that C = ησ(a) ∪ C\ησ(a) ⊆ ηC (a), and therefore ηC (a) = C. 2. It follows directly from (6.6) and (5.4) that ηA (a) = {λ ∈ C : a − λ ∈ / CompA (1, A−1 )}, and since A −1 Exp A = Comp (1, A ), the result follows. • Note that, in Theorem 6.7 (1(c)), we have also shown that, for all a ∈ A\C and for all λ ∈ C, a − λ ∈ / CompC (1, A−1 ). Therefore A\C ⊆ A\CompC (1, A−1 ), so that CompC (1, A−1 ) ⊆ C\{0}. Together with Proposition 5.12 it follows that CompC (1, A−1 ) = C\{0}. In addition, Theorem 6.7 (1(c)) shows that the inclusion in (5.14) is in general proper. Lemma 6.8. Let A be a complex Banach algebra with unit 1 and let B and C be closed subalgebras of A such that 1 ∈ C ⊆ B ⊆ A. If a ∈ A, then σ(a) ⊆ ηB (a) ⊆ ηC (a), for all a ∈ A. Proof. It follows from (5.4) and Corollary 5.6 that A\A−1 ⊆ η A (A\A−1 ) ⊆ η B (A\A−1 ) ⊆ η C (A\A−1 ), and so the result follows.
•
Now Theorem 6.7 and Lemma 6.8 imply: Corollary 6.9. Let A be a complex Banach algebra with unit 1, B a closed subalgebra of A such that 1 ∈ B and a ∈ A. 1. If a ∈ C, then ε(a) = ηB (a) = ησ(a) = {a}. 2. If a ∈ / C, then ε(a) ⊆ ηB (a) ⊆ C, where ηB (a) is largest when B = C, in which case ηB (a) = C, and smallest when B = A, in which case ηB (a) = ε(a). Combining Corollaries 6.4 and 6.9 and using (2.2) and (2.3), we obtain the following sets of equalities and inclusions: Corollary 6.10. Let A be a complex Banach algebra with unit 1, B a closed subalgebra of A such that 1 ∈ B and a ∈ A. 1. If a ∈ C, then ∂σ(a) = ∂C (a) = ∂B (a) = ∂A (a) = S∂ (a) = σ(a) = ε(a) = ηA (a) = ηB (a) = ηC (a) = ησ(a) = {a}.
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2. If a ∈ / C, then ∂σ(a) = ∂C (a) ⊆ ∂B (a) ⊆ ∂A (a) = S∂ (a) ⊆ σ(a) ⊆ ε(a) = ηA (a) ⊆ ηB (a) ⊆ ηC (a) = C. We note that there is a type of duality between the boundary spectrum S∂ (a) and the exponential spectrum ε(a), in the sense that S∂ (a) = ∂A (a)
and
ε(a) = ηA (a).
The “connected hull” corresponding to ∂σ(a) via this duality is ησ(a) = {a} if a ∈ C and C if a ∈ A\C. In the following example, let Γ denote the circle with centre 0 and radius 1 and D the closed disk with centre 0 and radius 1 in C. Example 6.11. Let A = C(Γ), the Banach algebra of complex-valued functions which are continuous on Γ, and B = A(D), the closed subalgebra of A consisting of all functions which are continuous on D and analytic on its interior. Let a = z : λ → λ be the identity function on D. Then ∂B (a) = Γ and ηB (a) = D. Proof. Since, by ([9], Problem 9, p. 399), σA (a) = Γ, it follows that ∂σA (a) = Γ. By Corollary 6.10 ∂σA (a) ⊆ ∂B (a) ⊆ σA (a), and so ∂B (a) = Γ. Since, by ([9], Problem 9, p. 399), σB (a) = D, it follows that ησB (a) = D. By (2.3) σB (a) ⊆ εB (a) ⊆ ησB (a) and so εB (a) = D. But by Corollary 6.10 εB (a) = ηB (a), and so ηB (a) = D. • In conclusion, we note that certain results that are known to hold for the boundary spectrum S∂ (a) = ∂A (a) of an element a ∈ A can be generalised by replacing A by B, with B a closed subalgebra of A containing the unit of A. For instance, considering ([5], Theorem 3.13), we note that the conclusion can be rephrased as ∂A (f (a)) = f (∂A (a)). Theorem 6.15 will show that the result remains true if A is replaced by B. We first prove the following lemma: Lemma 6.12. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Let a ∈ ∂ B (A−1 ) and d ∈ A−1 . If d ∈ B, then {ad, da} ⊆ ∂ B (A−1 ). Proof. If a ∈ ∂ B (A−1 ) and d ∈ A−1 , then Proposition 3.7 implies that for each > 0 there exist elements b1 , b2 ∈ B such that ||b1 || < ||d|| , ||b2 || < ||d|| , a − b1 ∈ / A−1 and a − b2 ∈ A−1 . Let b1 = b1 d and b2 = b2 d. If d ∈ B, then b1 , b2 ∈ B and ||b1 || < , ||b2 || < . Moreover, ad − b1 ∈ / A−1 and ad − b2 ∈ A−1 . It follows that if U ∈ NbdB (0), then (ad − U ) ∩ A\A−1 = ∅ and (ad − U ) ∩ A−1 = ∅, so that ad ∈ ∂ B (A−1 ). Similarly, da ∈ ∂ B (A−1 ). • The following theorem and lemma are generalisations of Theorem 3.12 and Lemma 3.10, respectively, in [5]. The same proofs apply. Theorem 6.13. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Let a ∈ B and let f be a complex valued function which is analytic and one-to-one on a neighbourhood G of σB (a). If λ0 ∈ G and β = f (λ0 ), then there exists an invertible element y ∈ B such that β − f (a) = (λ0 − a)y. For a ∈ B in the following lemma, the set of all complex valued functions which are analytic and one-to-one on a neighbourhood of σB (a) is indicated by H1 (a). We recall, using the open mapping property
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of analytic functions, that if f ∈ H1 (a), then there exists g ∈ H1 (a) such that g ◦ f ≡ z and f ◦ g ≡ z in a neighbourhood of a and f (a), respectively. Lemma 6.14. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Let ω : B → 2C be any mapping such that ω(a) ⊆ σB (a) for all a ∈ B. Then the following statements are equivalent: 1. ω(f (a)) ⊆ f (ω(a)) for all a ∈ B and all f ∈ H1 (a). 2. f (ω(a)) ⊆ ω(f (a)) for all a ∈ B and all f ∈ H1 (a). Finally, we have: Theorem 6.15. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Let a ∈ B and let f be a complex valued function which is analytic and one-to-one on a neighbourhood of σB (a). Then ∂B (f (a)) = f (∂B (a)). Proof. Let β = f (λ0 ), where λ0 ∈ ∂B (a). Since ∂B (a) ⊆ σB (a), it follows from Theorem 6.13 that f (a) −β = (a − λ0 )y for some invertible element y in B. By (6.1) we have a − λ0 ∈ ∂ B (A\A−1 ) and therefore, by Lemma 6.12, f (a) − β ∈ ∂ B (A\A−1 ), so that β ∈ ∂B (f (a)). This proves the inclusion f (∂B (a)) ⊆ ∂B (f (a)). The result then follows from Lemma 6.14. • Since Exp is an upper semi-regularity we have that ε(f (a)) ⊆ f (ε(a)) (see [7,8]). Then a slight generalisation of ([5], Lemma 3.10) gives Theorem 6.15 with B = A and “boundary” replaced by “hull”. If B = C and “boundary” is replaced by “hull” in this result, the statement become trivial. However, we have not been able to establish whether “boundary” can be replaced by “hull” in Theorem 6.15 in general. Here it seems relevant that ∂B (a) (see (6.1)), although related to one, is not actually a topological boundary itself, and similarly for ηB (a). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
B. Aupetit, A Primer on Spectral Theory, Springer, New York, 1991. R.E. Harte, The exponential spectrum in Banach algebras, Proc. Amer. Math. Soc. 58 (1976) 114–118. R.E. Harte, A.W. Wickstead, Boundaries, hulls and spectral mapping theorems, Proc. R. Ir. Acad. 81A (1981) 201–208. S. Mouton, On the boundary spectrum in Banach algebras, Bull. Aust. Math. Soc. 74 (2006) 239–246. S. Mouton, Mapping and continuity properties of the boundary spectrum in Banach algebras, Illinois J. Math. 53 (2009) 757–767. S. Mouton, A condition for spectral continuity of positive elements, Proc. Amer. Math. Soc. 137 (2009) 1777–1782. V. Müller, Axiomatic theory of spectrum III: semiregularities, Studia Math. 142 (2) (2000) 159–169. V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Birkhäuser, Basel, 2007. A.E. Taylor, D.C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York, 1958.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Linking the boundary and exponential spectra via the restricted topology ✩ Sonja Mouton a,∗ , Robin Harte b a
Department of Mathematical Sciences, Private Bag X1, Stellenbosch University, Matieland 7602, South Africa b School of Mathematics, Trinity College, Dublin, Ireland
a r t i c l e
i n f o
Article history: Received 6 December 2016 Available online 11 May 2017 Submitted by M. Mathieu
a b s t r a c t We build a chain, based on subalgebras, connecting the boundary spectrum/exponential spectrum duality with the duality between the usual boundary and the connected hull. © 2017 Elsevier Inc. All rights reserved.
Keywords: Boundary spectrum Exponential spectrum Boundary Hull
1. Introduction If a is an element of a complex Banach algebra A with unit 1, then the boundary spectrum S∂ (a) = {λ ∈ C : a − λ1 ∈ ∂A−1 } of a (see [4]) is a compact set in C that lies between the usual topological boundary of the spectrum and the spectrum σ(a) itself, i.e. ∂σ(a) ⊆ S∂ (a) ⊆ σ(a). It therefore seems natural to view ∂σ(a), on the one hand, and S∂ (a), on the other hand, as the “thin” and “fat” boundaries of σ(a), respectively. In this paper we show that, using closed subalgebras B, it is possible to define a topology on A (different from the norm-topology) in such a way that a whole range of “boundaries” can be obtained, with B = C giving the “thin” boundary and B = A the “fat” boundary of σ(a) — see Corollary 6.4.
✩
The first author was supported by the National Research Foundation (NRF) of South Africa.
* Corresponding author. E-mail addresses: [email protected] (S. Mouton), [email protected] (R. Harte). http://dx.doi.org/10.1016/j.jmaa.2017.05.020 0022-247X/© 2017 Elsevier Inc. All rights reserved.
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From [3] we recall a certain duality between the boundary ∂σ(a) and the connected hull ησ(a) of the spectrum of a. Bearing in mind the fact that the exponential spectrum ε(a) of a lies between σ(a) and ησ(a), i.e. σ(a) ⊆ ε(a) ⊆ ησ(a), we also investigate the “connected hulls” accompanying these new “boundaries”, and show that B = A yields ε(a), making ε(a) the “little” connected hull of σ(a). In addition it is shown that in trivial cases, the “big” connected hull of σ(a) is ησ(a), while otherwise it is actually the whole of C (see Corollary 6.9). In the following section we provide all the relevant notation and terminology. In Section 3, given an additive topological group A and a subgroup B of A, we define a new topology on A, called the B-topology or the restricted topology, via the closure operation and show that, for elements and subsets of A lying inside B, topological concepts in the B-topology coincide with those in the relative topology (see Theorem 3.4). We also obtain a number of basic properties relating to the B-topology and provide a number of examples. In Section 4 we investigate the correspondence between subsets Hω of A and certain mappings ω from A into 2B relying on some special element e of B. (The motivation for this is the relationship between the set of all invertible elements of a Banach algebra A and the function that maps an element a ∈ A onto its spectrum, where B = C and e = 1.) Section 5 is devoted to finding the relationships between the restricted boundaries and between the restricted connected hulls of ω and A\Hω , with the main results contained in Theorems 5.2 and (its partial analogue) 5.13. Finally, in Section 6, we return to Banach algebras. Using the concepts of the restricted boundary and the restricted connected hull of the set of all non-invertible elements of a Banach algebra A, we define a range of “boundaries” and “connected hulls” contained in C (see (6.1) and (6.6)) and thereby arrive at our main results Theorems 6.2 and 6.7. In conclusion, we show that certain known results about the boundary spectrum can be generalised by using our new “boundary” concept (see, in particular, Theorem 6.15). 2. Preliminaries Let X be a topological space and let t be an element of X. Then we denote the set of all neighbourhoods of t in X by NbdX (t). In addition, if K is a subset of X, then the (topological) closure, interior and boundary of K in X will be denoted by clX (K), IntX (K) and ∂X (K), respectively. / K, then the connected hull In some sense dual to the topological boundary is the connected hull. If t ∈ ηt (K) relative to t of K is defined by X\ηt (K) = CompX (t, X\K), where CompX (t, H) ⊆ X is the (connected) component of t in H. When in particular X is a normed linear space and K is a bounded set in X, we shall also write ηK := η∞ K for the complement of the unique unbounded component of X\K. If instead X is a topological ring with identity e, and e ∈ / S ⊆ A, we shall prefer ηS := ηe S. If, in turn, 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 λ1 in A will be denoted by λ. If a is an element of A, then the spectrum {λ ∈ C : a − λ ∈ / A−1 } of a in A will be denoted by σ(a) (or by σA (a), if necessary to avoid confusion). Referring to the notation given above, we will use the symbols ∂σ(a) := ∂C σ(a) and (as already mentioned) ησ(a) := η∞ σ(a) for the boundary and the connected hull, respectively, of σ(a). It is well known that if B is a closed subalgebra of A containing 1, then σA (a) ⊆ σB (a) (see, for instance, [1], Theorem 3.2.13).
and ∂σB (a) ⊆ ∂σA (a)
(2.1)
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The boundary spectrum of a in A (which plays an interesting role regarding spectral continuity if A is an ordered Banach algebra — see [6]) was introduced in [4] as S∂ (a) := {λ ∈ C : a − λ ∈ ∂A (A\A−1 )}. It is easy to see ([4], Proposition 2.1) that ∂σ(a) ⊆ S∂ (a) ⊆ σ(a)
(2.2)
and that S∂ (a) is a closed set. Therefore the boundary spectrum of a is a non-empty compact subset of the complex plane, for every a ∈ A. Due to the first inclusion in (2.2) the boundary spectrum is sometimes called the “fat boundary”. Let Exp A denote the set {ea1 . . . eak : k ∈ N, a1 , . . . , ak ∈ A}. Then Exp A equals the component CompA (1, A−1 ) of 1 in A−1 (see, for instance, [1], Theorem 3.3.7). The exponential spectrum of a in A was introduced in [2] as ε(a) := {λ ∈ C : a − λ ∈ / Exp A}. The exponential spectrum of a is another non-empty compact subset of the complex plane, for every a ∈ A, and it satisfies σ(a) ⊆ ε(a) ⊆ ησ(a)
(2.3)
(see [2], Theorem 1). 3. The restricted closure and the B-topology If Y ⊆ X is a subset of a topological space then the relative topology of Y induced by the topology of X is familiar: in terms of the closure operation we set, for arbitrary K ⊆ Y , clY (K) = Y ∩ clX (K).
(3.1)
We now extend this idea to more general subsets K ⊆ X, when X and Y ⊆ X are topological abelian groups, by making the following Definition 3.2. Let A be an additive topological group and let B be a subgroup of A. If K ⊆ A is arbitrary then its restricted closure in A relative to B is given by clB (K) = {a ∈ A : ∀ U ∈ NbdB (0) : (a − U ) ∩ K = ∅}. By (separate) continuity of addition, NbdA (a) = a − NbdA (0), for all a ∈ A, and therefore clA (K) = clA (K),
(3.3)
i.e., if B = A then the restricted closure of K in A coincides with the topological closure of K in A. Moreover, if K ⊆ B, then the restricted closure of K in A relative to B coincides with the relative closure (3.1) of K in B: Theorem 3.4. Let B be a subgroup of a topological group A and let K ⊆ B. Then clB (K) = clB (K).
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Proof. Since clB (K) = B ∩clA (K), by (3.3) it suffices to prove that clB (K) = B ∩clA (K). So let a ∈ clB (K) and let U ∈ NbdA (0). Then V = B ∩ U ∈ NbdB (0), so that (a − V ) ∩ K = ∅, say v ∈ V and y = a − v ∈ K. Since (a − V ) ∩ K ⊆ (a − U ) ∩ K, it follows that (a − U ) ∩ K = ∅ and so a ∈ clA (K). Also, since K ⊆ B, we have that y ∈ B and since v ∈ B, it follows that a = y + v ∈ B. Conversely, let a ∈ B ∩ clA (K) and let U ∈ NbdB (0). Then U = B ∩ V with V ∈ NbdA (0). Hence (a − V ) ∩ K = ∅, say v ∈ V and y = a − v ∈ K. Since K ⊆ B, we have that y ∈ B and since a ∈ B, it follows that v = a − y ∈ B. Therefore v ∈ U , so that (a − U ) ∩ K = ∅ and so a ∈ clB (K). • In order to verify that the restricted closure does in fact define a topology, we check the Kuratowski conditions: Theorem 3.5. If B is a subgroup of a topological group A and if K, H ⊆ A then 1. 2. 3. 4. 5.
clB (∅) = ∅, K ⊆ H =⇒ clB (K) ⊆ clB (H), K ⊆ clB (K), clB clB (K) ⊆ clB (K), clB (K ∪ H) ⊆ clB (K) ∪ clB (H).
Proof. Each of (1) and (2) is clear. Towards (3), if a ∈ K and U ∈ NbdB (0) then a ∈ (a − U ) ∩ K. Towards (4), suppose a ∈ A is in the left hand side and U ∈ NbdB (0) is arbitrary. Then there are bU ∈ U and then cU ∈ U for which a − bU ∈ clB (K) , a − bU − cU ∈ K. By joint continuity of addition there is also V ∈ NbdB (0) for which V + V ⊆ U . Hence b U = bV + cV ∈ U and a − b U ∈ (a − U ) ∩ K. Finally, for (5), suppose that a ∈ A is in the left hand side and not in the closure of H. For arbitrary U ∈ NbdB (0) we have (a − U ) ∩ (K ∪ H) = ∅. There is however at least one V ∈ NbdB (0) for which (a − V ) ∩ H = ∅.
(3.6)
In particular, intersecting U and V , (a − (U ∩ V )) ∩ (K ∪ H) = ∅; but (3.6) gives (a − (U ∩ V )) ∩ H = ∅. It now follows that (a − U ) ∩ K ⊇ (a − (U ∩ V )) ∩ K = (a − (U ∩ V )) ∩ (K ∪ H) = ∅.
•
We will refer to the topology induced on an additive topological group A by the restricted closure in A relative to a subgroup B as the restricted topology or the B-topology. In order to compile a number of basic results associated with the restricted topology, let A denote a topological group and let B denote a subgroup of A. Knowing that the restricted closure defines a topology, our first proposition follows immediately from the facts that IntB (K) = A\clB (A\K) and ∂ B (K) = clB (K) ∩ clB (A\K) = clB (K)\IntB (K):
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Proposition 3.7. If K ⊆ A then IntB (K) = {a ∈ A : ∃ U ∈ NbdB (0) , a − U ⊆ K} and ∂ B (K) = {a ∈ A : ∀U ∈ NbdB (0) : (a − U ) ∩ K = ∅ and (a − U ) ∩ A\K = ∅}. Neighbourhoods in the B-topology can be described as follows: Proposition 3.8. If a ∈ A then NbdB (a) = {U ⊆ A : a − U ∈ NbdB (0)}. Proof. If U ∈ NbdB (a), then there exists a subset V of U , open in the B-topology, containing a. By Proposition 3.7 there exists a W ∈ NbdB (0) such that a − W ⊆ V . It follows that W ⊆ a − V ⊆ a − U , so that a − U ∈ NbdB (0). Conversely, suppose that a − U ∈ NbdB (0). Then a − U = B ∩ V for some V ∈ NbdA (0), so that U = a −(B ∩V ), with V containing an open set V1 in A which contains 0. It follows that a ∈ a −(B ∩V1 ) ⊆ U . If b ∈ B ∩ V1 , then B ∩ V1 ∈ NbdB (b), so that (B ∩ V1 ) − b ∈ NbdB (0). Hence, if b ∈ B ∩ V1 , then U = (B ∩ V1 ) − b ∈ NbdB (0) with a − b − U ⊆ a − (B ∩ V1 ), so that a − (B ∩ V1 ) is open in the B-topology by Proposition 3.7. It follows that U ∈ NbdB (a). • If C denotes another subgroup of A, we have the following properties: Lemma 3.9. If C ⊆ B ⊆ A and K ⊆ A, then clC (K) ⊆ clB (K). Proof. Let a ∈ clC (K) and V ∈ NbdB (0). Then V = W ∩ B for some W ∈ NbdA (0), so that U = W ∩ C ∈ NbdC (0). Hence (a − U ) ∩ K = ∅. Since (a − U ) ∩ K ⊆ (a − V ) ∩ K, it follows that (a − V ) ∩ K = ∅, so that a ∈ clB (K). • Corollary 3.10. If C ⊆ B ⊆ A, then the C-topology of A is stronger than the B-topology of A. Hence the B-topology of A is stronger than the (original) topology of A. Corollary 3.11. Let C ⊆ B ⊆ A and G ⊆ A. If G is connected in the C-topology, then G is connected in the B-topology. Corollary 3.12. Let C ⊆ B ⊆ A and K ⊆ A. Then: 1. IntB (K) ⊆ IntC (K) ⊆ clC (K) ⊆ clB (K) 2. ∂ C (K) ⊆ ∂ B (K) Proof. 1. If a ∈ IntB (K), then there exists a U ∈ NbdB (0), i.e. U = W ∩ B for some W ∈ NbdA (0), such that a − U ⊆ K. Let V = W ∩ C. Then V ∈ NbdC (0) and a − V ⊆ a − U ⊆ K, so that a ∈ IntC (K). The result now follows from Lemma 3.9. 2. Follows from (1). •
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For the purposes of illustrating some of these concepts and providing examples, we will consider the following Banach spaces and subsets: the space CC [0, 1] of all continuous complex valued functions on [0, 1] with the supremum norm, the subset Cx of CC [0, 1] of all complex valued homogeneous polynomials on [0, 1] of degree 1, the space CR [0, 1] of all continuous real valued functions on [0, 1] with the supremum norm and the subset PR [0, 1] of CR [0, 1] of all real valued polynomials on [0, 1]. Example 3.13. If A = CC [0, 1], B = C and K = Cx, then clB (K) = K = clA (K) and IntA (K) = ∅ = IntB (K). Proof. If f ∈ clA (K), then there exists a sequence (pn ) in K, say pn (x) = αn x (x ∈ [0, 1]) such that supx∈[0,1] |αn x − f (x)| = ||pn − f || → 0. Therefore αn → α where α = f (1), so that αn x → αx, but since αn x → f (x), it follows that f (x) = αx, for all x ∈ [0, 1], i.e. f ∈ K. We have shown that clA (K) = K, and therefore, by Corollary 3.12 (1), clB (K) = K. Suppose that f ∈ IntB (K). We may assume that f ∈ K, say f (x) = α0 x, for all x ∈ [0, 1]. Then, by Proposition 3.7, there exists 0 > 0 such that to every λ ∈ C with modulus less than 0 there corresponds αλ ∈ C such that (α0 − αλ )x = λ for all x ∈ [0, 1]. But the case x = 0 forces λ to be zero, which yields IntB (K) = ∅. By Corollary 3.12 (1), IntA (K) = ∅ as well. • Example 3.14. If A = CR [0, 1], B = R and K = PR [0, 1], then clB (K) = K, clA (K) = A, IntA (K) = ∅ and IntB (K) = K. Proof. By the Weierstrass Approximation Theorem clA (K) = A, so suppose that f ∈ clB (K). Then given > 0, by Definition 3.2 there exist λ ∈ (− , ) and a polynomial p ∈ PR [0, 1] such that f − λ = p, i.e. f (x) = λ + p(x) for all x ∈ [0, 1], so that f ∈ PR [0, 1] = K. Therefore clB (K) = K. Suppose that p ∈ IntA (K). Then, since p is a continuous real-valued function on [0, 1], p can be approximated by continuous piecewise linear functions. Hence there is no -neighbourhood of p consisting only of polynomials, which contradicts p ∈ IntA (K). Therefore IntA (K) = ∅. Now let p ∈ K, say p(x) = α0 + α1 x + · · · + αn xn , take any 0 > 0 and let U = (− 0 , 0 ). For λ ∈ U , define qλ (x) := (α0 − λ) + α1 x + · · · + αn xn . Then qλ ∈ PR [0, 1] and p(x) − λ = qλ (x). We have shown that p − U ⊆ PR [0, 1], and so p ∈ IntB (K), by Proposition 3.7. Therefore IntB (K) = K. • It follows from the example above that the set of all real-valued polynomials on [0, 1] is both open and closed in the R-topology and therefore the set of continuous real-valued functions on [0, 1] is not connected in the R-topology. u the space of all 2 × 2complex upper triangular matrices, B = C, K = Example 3.15. Let A = M2 (C), w z w z : z, w ∈ C . Then : z ∈ C, w ∈ Q + iQ and S = 0 w 0 w
clB (K) = S = clA (K) and IntA (K) = ∅ = IntB (K).
w z Proof. It is easy to check that cl (K) = S, so let ∈ S and let > 0. Take w ∈ Q + iQ such that 0 w w z λ 0 w z |w − w | < and let λ = w − w . Then ∈ K, so that − = 0 w 0 w 0 λ A
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w 0
z w
− B(0, )
∩ K = ∅
(where B(0, ) indicates the open ball with centre 0 and radius in C (considered as a subset of A)), i.e. w z ∈ clB (K). We have shown that S ⊆ clB (K), and since clA (K) = S, it follows from Corol0 w lary 3.12 (1) that clB (K) = S. The density of the set R\Q + iR\Q in C forces the interiors to be empty.
•
Finally, we show that if A is a locally connected topological group and B is a connected subgroup of A, then A is locally connected in the B-topology (Proposition 3.21). We need some preliminary results: Lemma 3.16. If V is a connected subset of A which is contained in B, then V is connected in the B-topology (i.e., in the relative topology). Proof. Suppose that V ⊆ B and V is not connected in the relative topology, say V = U1 ∪ U2 with U1 and U2 non-empty sets such that U1 ∩ clB (U2 ) = ∅ = U2 ∩ clB (U1 ). Then, since U1 ⊆ V ⊆ B, we have ∅ = U1 ∩ clB (U2 ) = U1 ∩ B ∩ clA (U2 ) = U1 ∩ clA (U2 ) and similarly U2 ∩ clA (U1 ) = ∅. Hence V is not connected in A.
•
Lemma 3.17. If a ∈ A and G ⊆ A, then clB (a − G) = a − clB (G), and hence if G is closed in A in the B-topology, so is a − G. Proof. It suffices to prove that clB (a − G) ⊆ a − clB (G) for all a ∈ A and all G ⊆ A. So let c ∈ clB (a − G) and let W ∈ NbdB (0). Then W = −W ∈ NbdB (0), so that (c + W ) ∩ (a − G) = ∅, say x ∈ W , y ∈ G are such that c + x = a − y. It follows that if b = a − c, then b − x = a − c − x = y ∈ (b − W ) ∩ G = ∅, so that b ∈ clB (G), and so c = a − b ∈ a − clB (G). • Corollary 3.18. If a ∈ A and H ⊆ A is open in A in the B-topology, then so is a − H. Corollary 3.19. If a ∈ A and G ⊆ A is connected in the B-topology, then so is a − G. Proof. Suppose that a −G = H1 ∪H2 with H1 and H2 non-empty sets with H1 ∩clB (H2 ) = ∅ = H2 ∩clB (H1 ). Then G = (a − H1 ) ∪ (a − H2 ) with a − H1 and a − H2 non-empty sets, and it follows from Lemma 3.17 that (a − H1 ) ∩ clB (a − H2 ) = ∅ = (a − H2 ) ∩ clB (a − H1 ). • Corollary 3.20. Suppose that B is connected in A. If V is a connected neighbourhood of 0 in A, then U = a − (B ∩ V ) is a connected neighbourhood of a in the B-topology. Proof. If V is a connected neighbourhood of 0 in A, then since B ∩ V ⊆ B, it follows from Lemma 3.16 that B ∩ V is connected in the relative topology. Then it follows from Corollary 3.19 that U = a − (B ∩ V ) is connected in the B-topology, and U is a neighbourhood of a in the B-topology, by Proposition 3.8. • Using the previous observations, we can now prove: Proposition 3.21. Suppose that A is locally connected and B is connected in A. Then A is locally connected in the B-topology.
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Proof. Let B0 be a neighbourhood base of 0 consisting of open connected sets in A and let a ∈ A. Then by Corollaries 3.18 and 3.20 BaB = {a − (B ∩ V ) : V ∈ B0 } is a neighbourhood base of a consisting of open connected sets in the B-topology. Indeed, if U ∈ NbdB (a), then a − U ∈ NbdB (0), by Proposition 3.8, and hence a − U = B ∩ W for some W ∈ NbdA (0). It follows that there exists a V ∈ B0 such that V ⊆ W , and so a − (B ∩ V ) ⊆ U . • 4. The ω-spectrum Let B denote a subgroup of a topological group A. In the presence of a distinguished element e ∈ B there is a correspondence ω↔G between subsets of A and certain mappings from A into subsets of B, where we denote elements of the form be in B by b: Definition 4.1. If G ⊆ A and if ω : A → 2B satisfies ∀ (a, b) ∈ A × B : ω(a − b) = ω(a) − b ⊆ B,
(4.2)
set Hω = {a ∈ A : 0 ∈ / ω(a)} ⊆ A and B
G (a) = {b ∈ B : a − b ∈ / G} ⊆ B.
Evidently B ω(·) = G (·) ⇐⇒ G = Hω .
We note that if x ∈ B, then x∈ / ω(a) if and only if a − x ∈ Hω .
(4.3)
The most important situation is when A is a ring, e ∈ A is an identity and B is a subring, in particular a field. For a good spectral theory for ω the subset G = Hω should be a regularity [8]. When A is a ring, in particular a linear algebra over a field B, the most fundamental “spectrum” is derived from the invertible group: σ = A−1 ; Hσ = A−1 . In the remainder of this section, let B be a subgroup of a topological group A and let ω : A → 2B be a mapping satisfying (4.2). We now establish the following basic properties relating to the map ω and the set Hω . These properties will be used in the next section. Lemma 4.4. If ω(a) is closed in A in the B-topology for all a ∈ A, then Hω is open in A in the B-topology.
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Proof. Suppose that ω(a) is closed in A in the B-topology for all a ∈ A. If a ∈ Hω , then 0 ∈ / ω(a), so that there exists a W ∈ NbdB (0) such that W ⊆ A\ω(a). It follows from (4.3) that a − W ⊆ Hω and hence Hω is open in A in the B-topology. • If G ⊆ X ⊆ A, then we will denote the closure of G in X in the B-topology by clB (G, X); so clB (G, X) = cl (G) ∩ X. B
Lemma 4.5. If G ⊆ B\ω(a), then a − clB (G, B\ω(a)) = clB (a − G, Hω ∩ (a − B)). Proof. Let b ∈ clB (G, B\ω(a)). Then b ∈ B\ω(a), so that a − b ∈ Hω , by (4.3). Let W ∈ NbdB (0). Then W = −W ∈ NbdB (0), so that (b + W ) ∩ G = ∅, say x ∈ W , y ∈ G are such that b + x = y. It follows that a − b − x = a − y ∈ (a − b − W ) ∩ (a − G) = ∅. Hence a − b ∈ clB (a − G, Hω ∩ (a − B)). Now let c ∈ clB (a − G, Hω ∩ (a − B)). Then c = a − b with b ∈ B and c ∈ Hω , so that b ∈ B\ω(a), by (4.3). If W ∈ NbdB (0), then, as before, (c + W ) ∩ (a − G) = ∅, say x ∈ W , y ∈ G are such that c + x = a − y. It follows that b − x = a − c − x = y ∈ (b − W ) ∩ G = ∅. Hence b ∈ clB (G, B\ω(a)), so that c ∈ a − clB (G, B\ω(a)). • Corollary 4.6. If G ⊆ B\ω(a) is closed in B\ω(a) in the B-topology, then a − G is closed in Hω ∩ (a − B) in the B-topology. Lemma 4.7. Suppose that B is closed in A. Then Hω ∩ (a − B) is closed in Hω in the B-topology. Proof. If B is closed in A, then B is closed in A in the B-topology, since the latter is a stronger topology. Then the result follows from Lemma 3.17. • 5. The restricted boundary and connected hull The closure in B of the “spectrum” ω(a) determines the restricted closure of the “non singulars” Hω ⊆ A: Theorem 5.1. Let B be a subgroup of a topological group A. If a ∈ A, e, b ∈ B and ω : A → 2B satisfies (4.2), then b∈ / clB ω(a) ⊆ B ⇐⇒ a − b ∈ intB Hω ⊆ A. Proof. Simply observe that for arbitrary U ∈ NbdB (b) ω(a) ∩ U = ∅ ⇐⇒ a − U ⊆ Hω
•
Theorem 5.1 relates the (restricted) topological boundaries of ω and Hω : Theorem 5.2. Let B be a subgroup of a topological group A. If a ∈ A, e ∈ B and ω : A → 2B satisfies (4.2), then ∂ B ω(a) = {b ∈ B : a − b ∈ ∂ B (A\Hω ) ⊆ A}. Proof. We first note that if b ∈ B, then by Theorem 5.1, b ∈ clB ω(a) if and only if a − b ∈ A\IntB Hω = clB (A\Hω ) and, similarly, b ∈ clB (B\ω(a)) if and only if a − b ∈ A\IntB (A\Hω ) = clB (Hω ). The result then follows by observing that
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∂ B ω(a) = clB ω(a) ∩ clB (B\ω(a)) and ∂ B (A\Hω ) = clB (A\Hω ) ∩ clB (Hω ).
•
Having discussed the boundaries of ω and Hω , we now turn to the (somewhat dual) concept of the connected hull. Similarly to the situation in general topological spaces, if A is a topological group, B is a subgroup of A and t ∈ / K ⊆ A, then the restricted connected hull ηtB (K) relative to t of K is given by A\ηtB (K) = CompB (t, A\K), and if K ⊆ B, then B\ηtB (K) = CompB (t, B\K). In particular, if A is a normed algebra, e = 1 and B is a closed subalgebra of A with 1 ∈ B, then if, for some a ∈ A, ω(a) is bounded (where ω : A → 2B satisfies (4.2)), then η B ω(a) = B\CompB (∞, B\ω(a)),
(5.3)
where CompB (∞, B\ω(a)) indicates the unique unbounded component of B\ω(a) relative to the B-topology, and if 1 ∈ Hω , then η B (A\Hω ) = A\CompB (1, Hω ),
(5.4)
where CompB (1, Hω ) is the component of Hω containing 1 in the B-topology. With A a topological group and B and C subgroups of A, we first note, directly from Corollary 3.11, the following: / K. Then Corollary 5.5. Let C ⊆ B ⊆ A and K ⊆ A such that t ∈ CompC (t, A\K) ⊆ CompB (t, A\K). Corollary 5.6. Let C ⊆ B ⊆ A and K ⊆ A such that t ∈ / K. Then ηtB (K) ⊆ ηtC (K). In order to obtain a partial analogue of Theorem 5.2 for the connected hull (Theorem 5.13), we need a number of preliminary results relating to components in the B-topology. In the following results, let B be a subgroup of a topological group A and let ω : A → 2B be a mapping satisfying (4.2). We first note the following: Proposition 5.7. Suppose that A is locally connected, B is closed and connected in A and ω(a) is closed in B for all a ∈ A. If G is a component in the B-topology of B\ω(a), then a − G is open in A in the B-topology. Proof. Since ω(a) is closed in B, we have that B\ω(a) is open in A in the B-topology (note that B is open in A in the B-topology). It follows from Proposition 3.21 that G is open in A in the B-topology. Hence the result follows from Corollary 3.18. •
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Directly from Corollary 4.6 and Lemma 4.7 we have: Corollary 5.8. Suppose that B is closed in A and b ∈ B\ω(a). If G = CompB (b, B\ω(a)), then a − G is closed in Hω in the B-topology. The previous corollary, together with results from Sections 3 and 4, yields: Proposition 5.9. Suppose that A is locally connected, B is closed and connected in A, ω(a) is closed in B for all a ∈ A and b ∈ B\ω(a). If G = CompB (b, B\ω(a)), then a − G is closed in CompB (a − b, Hω ) in the B-topology. Proof. If B is closed in A and ω(a) is closed in B for all a ∈ A, then it follows from Lemma 4.4 and Proposition 3.21 that CompB (a − b, Hω ) is open in Hω in the B-topology. By Corollary 3.19 and (4.3) we have a − G ⊆ CompB (a − b, Hω ) ⊆ Hω . The result then follows from Corollary 5.8. • Now Propositions 5.7 and 5.9 imply: Proposition 5.10. Suppose that A is locally connected, B is closed and connected in A, ω(a) is closed in B for all a ∈ A and b ∈ B\ω(a). 1. If G = CompB (b, B\ω(a)), then a − G = CompB (a − b, Hω ). 2. If G = CompB (a − b, Hω ), then a − G = CompB (b, B\ω(a)). Proof. 1. By (4.3) a − b ∈ a − G ⊆ Hω , and a − G is connected in the B-topology, by Corollary 3.19. Therefore ∅ = a − G ⊆ CompB (a − b, Hω ) ⊆ A. By Proposition 5.7 a − G is open in A, and hence in CompB (a − b, Hω ), in the B-topology, and by Proposition 5.9 a − G is closed in CompB (a − b, Hω ) in the B-topology. Hence the result follows from the connectedness in the B-topology of CompB (a − b, Hω ). 2. Follows from (1). • The following proposition provides a sufficient condition for a function f : [0, 1] → A to be continuous in the B-topology: Proposition 5.11. Suppose that f : [0, 1] → A is continuous in (the usual topology of) A and f ([0, 1]) ⊆ B. Then f is continuous in the B-topology. Proof. Let K ⊆ [0, 1] and let α ∈ cl[0,1] (K). If U ∈ NbdB (0), then U = B ∩ V for some V ∈ NbdA (0), so that f (α) − V ∈ NbdA (f (α)). Since f is continuous in A, we have f (cl[0,1] (K)) ⊆ clA (f (K)) and hence f (α) ∈ clA (f (K)), so that (f (α) − V ) ∩ f (K) = ∅, say b ∈ V and μ ∈ K are such that f (α) − b = f (μ). Then f (α) − f (μ) ∈ V , so that it follows from the assumption that f (α) − f (μ) ∈ U . Hence f (μ) ∈ f (α) − U , so that (f (α) − U ) ∩ f (K) = ∅. Hence f (α) ∈ clB (f (K)). We have proven that f (cl[0,1] (K)) ⊆ clB (f (K)) for all K ⊆ [0, 1], and so f is continuous in the B-topology.
•
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Proposition 5.11 guarantees the unboundedness of certain components in the B-topology: Proposition 5.12. Let A be a complex normed algebra with unit 1 and B a closed subalgebra of A such that 1 ∈ B. Let ω : A → 2B be a mapping satisfying (4.2) with the property that ω(λ) = λω(1) for all λ ∈ C. If 1 ∈ Hω , then C\{0} ⊆ CompB (1, Hω ) and, therefore, CompB (1, Hω ) is unbounded. Proof. If λ1 , λ2 ∈ C\{0}, then there exists a continuous function g : [0, 1] → C such that g(0) = λ1 , g(1) = λ2 and g([0, 1]) ⊆ C\{0}. The function f : [0, 1] → A, defined by f (μ) = g(μ) (:= g(μ)1), is then continuous in (the usual topology of) A and f ([0, 1]) ⊆ C\{0} ⊆ B, so that f is continuous in the B-topology, by Proposition 5.11. Also, f (0) = λ1 and f (1) = λ2 , so that C\{0} is pathwise connected, hence connected, in the B-topology. Using (4.3), we have that 0 ∈ / ω(1), since 1 ∈ Hω , and therefore C\{0} ⊆ Hω , since ω(λ) = λω(1). This yields the result. • Now, using Propositions 5.10 and 5.12, we obtain the following result, which is partially analogous to Theorem 5.2: Theorem 5.13. Let A be a complex normed algebra with unit 1 and B a closed subalgebra of A such that 1 ∈ B. Let ω : A → 2B be a mapping satisfying (4.2), such that ω(a) is bounded and closed in B for all a ∈ A and with the property that ω(λ) = λω(1) for all λ ∈ C. If 1 ∈ Hω then η B ω(a) ⊆ {b ∈ B : a − b ∈ η B (A \ Hω ) ⊆ A}.
(5.14)
Proof. Suppose that b ∈ B and a − b ∈ CompB (1, Hω ). Then b ∈ / ω(a). Now suppose that b is in a bounded component G (in the B-topology) of B\ω(a). Let H = a − G. Then H is bounded, and H = CompB (a − b, Hω ), by Proposition 5.10 (1). But since a − b ∈ CompB (1, Hω ), it follows that H = CompB (1, Hω ), so that CompB (1, Hω ) is bounded, which contradicts Proposition 5.12. Therefore b ∈ CompB (∞, B\ω(a)) (the unique unbounded component of B\ω(a) in the B-topology). The result now follows from (5.3) and (5.4). • It will be shown in the next section that we do not in general have equality in (5.14). 6. The restricted topology applied to Banach algebras Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that e := 1 ∈ B. For any a ∈ A, we define ∂B (a) := {λ ∈ C : a − λ ∈ ∂ B (A\A−1 )}.
(6.1)
For example, if B = A then we get the “fat boundary”, while if B = C then it gives back the “thin boundary”, of the spectrum of a — i.e., the boundary spectrum S∂ (a) and the usual boundary ∂σ(a) of the spectrum of a, respectively: Theorem 6.2. Let A be a complex Banach algebra with unit 1. Then 1. ∂C (a) = ∂σ(a) and 2. ∂A (a) = S∂ (a), for all a ∈ A.
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Proof. 1. If B = C in Definition 4.1, then ω = σ satisfies (4.2) and Hω = A−1 . Therefore it follows from Theorem 5.2 and (6.1) that ∂C (a) = ∂ C σ(a). By Proposition 3.7 ∂ C σ(a) = ∂σ(a), so that the result follows. 2. This follows directly from (6.1) by recalling the definition of S∂ (a). • Lemma 6.3. Let A be a complex Banach algebra with unit 1 and let B and C be closed subalgebras of A such that 1 ∈ C ⊆ B ⊆ A. If a ∈ A, then ∂C (a) ⊆ ∂B (a) ⊆ σ(a), for all a ∈ A. Proof. It follows from Corollary 3.12 (2) that ∂ C (A\A−1 ) ⊆ ∂ B (A\A−1 ) ⊆ ∂ A (A\A−1 ) ⊆ A\A−1 , and so the result follows.
•
Theorem 6.2 and Lemma 6.3 now imply: Corollary 6.4. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Then ∂σ(a) ⊆ ∂B (a) ⊆ S∂ (a), for all a ∈ A, where ∂B (a) is smallest when B = C, in which case ∂B (a) = ∂σ(a), and largest when B = A, in which case ∂B (a) = S∂ (a). Using (2.1) as well, we obtain: Corollary 6.5. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Then ∂σB (a) ⊆ ∂σA (a) ⊆ ∂B (a) ⊆ ∂A (a) ⊆ σA (a) ⊆ σB (a), for all a ∈ B. Analogously to (6.1), for any closed subalgebra B of a Banach algebra A such that 1 ∈ B and a ∈ A, we define ηB (a) := {λ ∈ C : a − λ ∈ η B (A\A−1 )}.
(6.6)
Theorem 6.7. Let A be a complex Banach algebra with unit 1 and let a ∈ A. Then 1. (a) ησ(a) ⊆ ηC (a). (b) If a ∈ C, then ηC (a) = ησ(a). (c) If a ∈ A\C, then ηC (a) = C. 2. ηA (a) = ε(a). Proof. 1. (a) From Theorem 5.13 and (6.6) we have that η C σ(a) ⊆ ηC (a), so that the result follows from (5.3).
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(b) By Proposition 5.12 C\{0} ⊆ CompC (1, A−1 ). Therefore if a ∈ C, then, by (6.6) and (5.4), λ ∈ ηC (a) if and only if a − λ ∈ / CompC (1, A−1 ) if and only if λ = a. It follows that ηC (a) = {a} = ησ(a). (c) By (a) ησ(a) ⊆ ηC (a). In order to prove that C\ησ(a) ⊆ ηC (a) as well if a ∈ / C, let λ0 ∈ C\ησ(a), C C −1 i.e. λ0 ∈ Comp (∞, C\σ(a)), and let G = Comp (a − λ0 , A ). Then, by Proposition 5.10 (2), a − G = CompC (λ0 , C\σ(a)). Since a ∈ / C, it follows that 1 ∈ / G, so that a − λ0 ∈ / CompC (1, A−1 ). C −1 Therefore a − λ0 ∈ η (A\A ), so that λ0 ∈ ηC (a). We have proven that C = ησ(a) ∪ C\ησ(a) ⊆ ηC (a), and therefore ηC (a) = C. 2. It follows directly from (6.6) and (5.4) that ηA (a) = {λ ∈ C : a − λ ∈ / CompA (1, A−1 )}, and since A −1 Exp A = Comp (1, A ), the result follows. • Note that, in Theorem 6.7 (1(c)), we have also shown that, for all a ∈ A\C and for all λ ∈ C, a − λ ∈ / CompC (1, A−1 ). Therefore A\C ⊆ A\CompC (1, A−1 ), so that CompC (1, A−1 ) ⊆ C\{0}. Together with Proposition 5.12 it follows that CompC (1, A−1 ) = C\{0}. In addition, Theorem 6.7 (1(c)) shows that the inclusion in (5.14) is in general proper. Lemma 6.8. Let A be a complex Banach algebra with unit 1 and let B and C be closed subalgebras of A such that 1 ∈ C ⊆ B ⊆ A. If a ∈ A, then σ(a) ⊆ ηB (a) ⊆ ηC (a), for all a ∈ A. Proof. It follows from (5.4) and Corollary 5.6 that A\A−1 ⊆ η A (A\A−1 ) ⊆ η B (A\A−1 ) ⊆ η C (A\A−1 ), and so the result follows.
•
Now Theorem 6.7 and Lemma 6.8 imply: Corollary 6.9. Let A be a complex Banach algebra with unit 1, B a closed subalgebra of A such that 1 ∈ B and a ∈ A. 1. If a ∈ C, then ε(a) = ηB (a) = ησ(a) = {a}. 2. If a ∈ / C, then ε(a) ⊆ ηB (a) ⊆ C, where ηB (a) is largest when B = C, in which case ηB (a) = C, and smallest when B = A, in which case ηB (a) = ε(a). Combining Corollaries 6.4 and 6.9 and using (2.2) and (2.3), we obtain the following sets of equalities and inclusions: Corollary 6.10. Let A be a complex Banach algebra with unit 1, B a closed subalgebra of A such that 1 ∈ B and a ∈ A. 1. If a ∈ C, then ∂σ(a) = ∂C (a) = ∂B (a) = ∂A (a) = S∂ (a) = σ(a) = ε(a) = ηA (a) = ηB (a) = ηC (a) = ησ(a) = {a}.
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2. If a ∈ / C, then ∂σ(a) = ∂C (a) ⊆ ∂B (a) ⊆ ∂A (a) = S∂ (a) ⊆ σ(a) ⊆ ε(a) = ηA (a) ⊆ ηB (a) ⊆ ηC (a) = C. We note that there is a type of duality between the boundary spectrum S∂ (a) and the exponential spectrum ε(a), in the sense that S∂ (a) = ∂A (a)
and
ε(a) = ηA (a).
The “connected hull” corresponding to ∂σ(a) via this duality is ησ(a) = {a} if a ∈ C and C if a ∈ A\C. In the following example, let Γ denote the circle with centre 0 and radius 1 and D the closed disk with centre 0 and radius 1 in C. Example 6.11. Let A = C(Γ), the Banach algebra of complex-valued functions which are continuous on Γ, and B = A(D), the closed subalgebra of A consisting of all functions which are continuous on D and analytic on its interior. Let a = z : λ → λ be the identity function on D. Then ∂B (a) = Γ and ηB (a) = D. Proof. Since, by ([9], Problem 9, p. 399), σA (a) = Γ, it follows that ∂σA (a) = Γ. By Corollary 6.10 ∂σA (a) ⊆ ∂B (a) ⊆ σA (a), and so ∂B (a) = Γ. Since, by ([9], Problem 9, p. 399), σB (a) = D, it follows that ησB (a) = D. By (2.3) σB (a) ⊆ εB (a) ⊆ ησB (a) and so εB (a) = D. But by Corollary 6.10 εB (a) = ηB (a), and so ηB (a) = D. • In conclusion, we note that certain results that are known to hold for the boundary spectrum S∂ (a) = ∂A (a) of an element a ∈ A can be generalised by replacing A by B, with B a closed subalgebra of A containing the unit of A. For instance, considering ([5], Theorem 3.13), we note that the conclusion can be rephrased as ∂A (f (a)) = f (∂A (a)). Theorem 6.15 will show that the result remains true if A is replaced by B. We first prove the following lemma: Lemma 6.12. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Let a ∈ ∂ B (A−1 ) and d ∈ A−1 . If d ∈ B, then {ad, da} ⊆ ∂ B (A−1 ). Proof. If a ∈ ∂ B (A−1 ) and d ∈ A−1 , then Proposition 3.7 implies that for each > 0 there exist elements b1 , b2 ∈ B such that ||b1 || < ||d|| , ||b2 || < ||d|| , a − b1 ∈ / A−1 and a − b2 ∈ A−1 . Let b1 = b1 d and b2 = b2 d. If d ∈ B, then b1 , b2 ∈ B and ||b1 || < , ||b2 || < . Moreover, ad − b1 ∈ / A−1 and ad − b2 ∈ A−1 . It follows that if U ∈ NbdB (0), then (ad − U ) ∩ A\A−1 = ∅ and (ad − U ) ∩ A−1 = ∅, so that ad ∈ ∂ B (A−1 ). Similarly, da ∈ ∂ B (A−1 ). • The following theorem and lemma are generalisations of Theorem 3.12 and Lemma 3.10, respectively, in [5]. The same proofs apply. Theorem 6.13. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Let a ∈ B and let f be a complex valued function which is analytic and one-to-one on a neighbourhood G of σB (a). If λ0 ∈ G and β = f (λ0 ), then there exists an invertible element y ∈ B such that β − f (a) = (λ0 − a)y. For a ∈ B in the following lemma, the set of all complex valued functions which are analytic and one-to-one on a neighbourhood of σB (a) is indicated by H1 (a). We recall, using the open mapping property
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of analytic functions, that if f ∈ H1 (a), then there exists g ∈ H1 (a) such that g ◦ f ≡ z and f ◦ g ≡ z in a neighbourhood of a and f (a), respectively. Lemma 6.14. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Let ω : B → 2C be any mapping such that ω(a) ⊆ σB (a) for all a ∈ B. Then the following statements are equivalent: 1. ω(f (a)) ⊆ f (ω(a)) for all a ∈ B and all f ∈ H1 (a). 2. f (ω(a)) ⊆ ω(f (a)) for all a ∈ B and all f ∈ H1 (a). Finally, we have: Theorem 6.15. Let A be a complex Banach algebra with unit 1 and let B be a closed subalgebra of A such that 1 ∈ B. Let a ∈ B and let f be a complex valued function which is analytic and one-to-one on a neighbourhood of σB (a). Then ∂B (f (a)) = f (∂B (a)). Proof. Let β = f (λ0 ), where λ0 ∈ ∂B (a). Since ∂B (a) ⊆ σB (a), it follows from Theorem 6.13 that f (a) −β = (a − λ0 )y for some invertible element y in B. By (6.1) we have a − λ0 ∈ ∂ B (A\A−1 ) and therefore, by Lemma 6.12, f (a) − β ∈ ∂ B (A\A−1 ), so that β ∈ ∂B (f (a)). This proves the inclusion f (∂B (a)) ⊆ ∂B (f (a)). The result then follows from Lemma 6.14. • Since Exp is an upper semi-regularity we have that ε(f (a)) ⊆ f (ε(a)) (see [7,8]). Then a slight generalisation of ([5], Lemma 3.10) gives Theorem 6.15 with B = A and “boundary” replaced by “hull”. If B = C and “boundary” is replaced by “hull” in this result, the statement become trivial. However, we have not been able to establish whether “boundary” can be replaced by “hull” in Theorem 6.15 in general. Here it seems relevant that ∂B (a) (see (6.1)), although related to one, is not actually a topological boundary itself, and similarly for ηB (a). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
B. Aupetit, A Primer on Spectral Theory, Springer, New York, 1991. R.E. Harte, The exponential spectrum in Banach algebras, Proc. Amer. Math. Soc. 58 (1976) 114–118. R.E. Harte, A.W. Wickstead, Boundaries, hulls and spectral mapping theorems, Proc. R. Ir. Acad. 81A (1981) 201–208. S. Mouton, On the boundary spectrum in Banach algebras, Bull. Aust. Math. Soc. 74 (2006) 239–246. S. Mouton, Mapping and continuity properties of the boundary spectrum in Banach algebras, Illinois J. Math. 53 (2009) 757–767. S. Mouton, A condition for spectral continuity of positive elements, Proc. Amer. Math. Soc. 137 (2009) 1777–1782. V. Müller, Axiomatic theory of spectrum III: semiregularities, Studia Math. 142 (2) (2000) 159–169. V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Birkhäuser, Basel, 2007. A.E. Taylor, D.C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York, 1958.