Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 147–165

AN IDEAL BOUNDARY FOR DOMAINS IN n-SPACE

Nathan Soderborg

Ford Motor Company, Scientific Research Laboratory MD 2036, P.O. Box 2053, Dearborn, MI 48121-2053, U.S.A.; nathan@vulcan.srl.ford.com

Abstract. The Royden ideal boundary of a domain is the set of all points in the maximal ideal space of the domain’s Royden algebra that do not lie in the domain. Elements of the Royden ideal boundary can be characterized as nets convergent in both the weak^{∗} and Euclidean topologies that have no subnet which is a sequence. As with other function algebras, boundary fibers can be defined as subsets of all points in the ideal boundary that project onto a single Euclidean boundary point. Quasiconformal mappings have homeomorphic extensions if and only if the adjoint of the corresponding Royden algebra isomorphism maps boundary fibers to boundary fibers. For domains with certain homogeneity properties, all boundary fibers are homeomorphic; and for domains finitely connected on the boundary, fibers are equivalent to prime ends. In any case, each boundary fiber has a complicated topology which contains the complement of the natural numbers in the Stone–Cech compactification of the natural numbers.

Introduction

Lewis [L] showed that two domains, Ω and Ω′, in Euclidean n-space are quasiconformally equivalent if and only if the corresponding Royden algebras A(Ω) and A(Ω′) are isomorphic. His proof extends ideas of Nakai [N] who proved the theorem for domains in two dimensions. It uses methods of functional analysis to characterize the maximal ideal space Ω^{∗} of A(Ω) as a compact space containing Ω. For this reason, Ω^{∗} is called the Royden compactification of Ω, and the set Ω^{∗} \Ω is called the Royden ideal boundary of Ω. Lewis also showed that a quasiconformal mapping between two domains induces a homeomorphism between their Royden compactifications which preserves Royden ideal boundaries. Consequently, one might hope to find necessary conditions for the existence of such quasiconformal mappings by studying these boundaries.

We wish to examine the structure of the Royden ideal boundary from several different points of view and present some of its unusual properties. Borrowing ideas previously applied to the Banach algebra H^{∞} [H], [Ga], we develop a theory of fibers in Ω^{∗} over boundary points of Ω. We show that a quasiconformal mapping f: Ω → Ω′ has a homeomorphic extension from ∂Ω to ∂Ω′ if and only if the adjoint of the corresponding Royden algebra isomorphism maps boundary fibers in Ω^{∗} onto boundary fibers in Ω^{∗}′. We discuss domains quasiconformally homogeneous with respect to the boundary and prove that all fibers in such domains are homeomorphic to each other; but in the special case that Ω is the unit ball, we prove that the Royden boundary is not the natural product of S^{n−1} with any particular fiber. We prove that if a domain is finitely connected on its boundary, then components of fibers correspond to the domain’s prime ends. Finally, to exhibit the complicated topology of boundary fibers, we discuss βN, the Stone– Cech compactification of the natural numbers, and show that each fiber contains a homeomorphic copy of βN \ N.

1991 Mathematics Subject Classification: Primary 30C65; Secondary 46J20.

1. Preliminaries

N denotes the natural numbers; R, the real numbers; R^{n}, Euclidean nspace; and Ω and Ω′, domains in R^{n} for n ≥ 2. B(x, r) and S(x, r) denote the ball and sphere of radius r centered at x, while B^{n} and S^{n−1} denote the unit ball and sphere. In the context of integration, dx and dy designate n-dimensional Lebesgue measure. \overline{R}^{n} denotes R^{n} ∪\{∞\}, the one-point compactification of R^{n}, and ∂Ω refers to the boundary of Ω in \overline{R}^{n}. If \{x_{i}\} is a sequence in Ω, we write x_{i} → ∂Ω whenever {x_{i}} has no cluster point in Ω.

The symbols L_{p}^{1}(Ω) and L_{p,loc}^{1}(Ω) denote the spaces of functions whose weak partial derivatives exist and belong to L^{p}(Ω) and L_{loc}^{p}(Ω) respectively. W_{p}^{1}(Ω) and W_{p,loc}^{1}(Ω) denote the function spaces L^{p}(Ω) ∩ L_{p}^{1}(Ω) and L_{loc}^{p}(Ω) ∩ L_{p,loc}^{1}(Ω) respectively.

Let K ≥ 1 be a constant. Then a homeomorphism f: Ω → R^{n} is K - quasiconformal if

f ∈ W_{n,loc}^{1}(Ω)

and

|Df(x)|^{n} ≤ KJ_{f}(x)

for almost every x ∈ Ω. The references [V1] and [BI] give comprehensive introductions to quasiconformal mappings.

A Banach space A is a real Banach algebra if A is an algebra over R with norm ∥ · ∥ satisfying

∥uw∥ ≤ ∥u∥ ∥w∥

for each u, w ∈ A. If I is a subset of a commutative Banach algebra A, then I is an ideal if uw ∈ I whenever u ∈ A and w ∈ I. An ideal is proper if it is a proper subset of A, and it is maximal if no proper ideal contains it.

If A is a real Banach algebra, the space A′ of all real-valued bounded linear functionals on A is called the dual space of A. If A_{1} and A_{2} are Banach algebras with norms ∥ · ∥_{1} and ∥ · ∥_{2} , and T: A_{1} → A_{2} is a linear transformation, the norm of T is given by

∥T ∥ = sup{∥T v∥_{2} : v ∈ A_{1}, ∥v∥_{1} ≤ 1}.

An ideal boundary for domains in n-space 149

The transformation T is continuous if and only if ∥T ∥ < ∞.

We let A(Ω) denote the Royden algebra (over R) on Ω. This algebra consists of all real functions

u ∈ C(Ω) ∩ L^{∞}(Ω) ∩ L_{n}^{1}(Ω),

with multiplication and addition defined pointwise on Ω, and norm

∥u∥ = ∥u∥_{Ω} = ∥u∥_{∞} + ∥∇u∥_{n,Ω} ,

where

∥∇u∥_{n,Ω} = (∫_{Ω} |∇u|^{n} dx)^{1/n}.

The algebra A(Ω) is a commutative Banach algebra that is regular on Ω, i.e., for each closed set F ⊂ Ω and each x ∈ Ω\F, there is a u ∈ A(Ω) such that u(x) = 1 and u|F ≡ 0. As a result, A(Ω) separates points in Ω. Furthermore, A(Ω) is inverse-closed, which means that if u ∈ A(Ω) and inf{u(x) : x ∈ Ω} > 0 then (1/u) ∈ A(Ω). The papers [N], [L], [L-F] and [S2] contain detailed discussions of the Royden algebra. The following theorem summarizes the relationship between quasiconformal mappings and Royden algebras.

Theorem 1.1 [L, Theorems 3.2, 7.1], [L-F, Theorem 11.3]. Let Ω and Ω′ be domains in R^{n}, n ≥ 2. Then if f: Ω → Ω′ is K -quasiconformal, the transformation f^{∗}: A(Ω′) → A(Ω) defined by f^{∗}v = v ∘ f for each v ∈ A(Ω′) is an algebra isomorphism with ∥f^{∗}∥^{n} ≤ K. Conversely, if T: A(Ω′) → A(Ω) is an algebra isomorphism, then T induces a ∥T ∥^{n} -quasiconformal mapping f: Ω → Ω′ such that f^{∗} = T.

Interpolating sequences will be useful for studying the Royden boundary. A sequence {x_{i}} of points in Ω is an interpolating sequence for A(Ω) if for each bounded sequence of real numbers {r_{i}} there exists u ∈ A such that u(x_{i}) = r_{i} for each i ∈ N.

Theorem 1.2 [S2, Theorem 1.11]. Let {x_{i}} be a sequence of distinct points in Ω such that x_{i} → ∂Ω, then {x_{i}} is an interpolating sequence for A(Ω).

2. The Royden compactification

The Royden compactification Ω^{∗} is the collection of all non-zero, bounded linear functionals χ on A(Ω) satisfying χ(uw) = χ(u)χ(w) for each u, w ∈ A(Ω). In other words, χ: A(Ω) → R is a continuous homomorphism, so Ω^{∗} ⊂ A(Ω)′. The original construction of the Royden compactification is in [N], and some of its important properties are outlined in [S2]. It is not hard to show that ∥χ∥ = 1 for each χ ∈ Ω^{∗}, and that Ω^{∗} is a compact Hausdorff space in the relative weak^{∗} topology generated by A(Ω). In this topology, if {χ_{α}}_{α∈Λ} is a net in Ω^{∗}, then χ_{α} → χ ∈ Ω^{∗} if and only if

lim_{α} |χ_{α}(u) − χ(u)| = 0

for each u ∈ A(Ω).

Definition 2.1. If x ∈ Ω, then x̂ ∈ Ω^{∗} denotes the point evaluation homomorphism defined by x̂(u) = u(x) for each u ∈ A(Ω).

Because A(Ω) separates the points in Ω, the mapping x ↦ x̂ is a homeomorphism of Ω onto its image Ω̂ ⊂ Ω^{∗}. By identifying each x with x̂, it is possible to consider Ω a subset of Ω^{∗}, and we use this convention when convenient. Because Ω̂ is dense in Ω^{∗} in the weak^{∗} topology [L, p. 489], the set Δ = Δ_{Ω} = Ω^{∗} \ Ω̂ constitutes a boundary known as the Royden ideal boundary of Ω.

Definition 2.2. If u ∈ A, then û ∈ C(Ω^{∗}) denotes the mapping defined by û(χ) = χ(u) for each χ ∈ Ω^{∗}. The mapping u ↦ û is an algebra homomorphism of A(Ω) onto a set Â ⊂ C(Ω^{∗}).

By standard methods of functional analysis, it is possible to show that Ω^{∗} can be identified with ℳ , the space of proper maximal ideals of A(Ω). If χ ∈ Ω^{∗}, the explicit correspondence M_{χ} ↔ χ is given by

M_{χ} = {u ∈ A : û(χ) = 0} = χ^{−1}(0) ∈ ℳ ,

see [S2, p. 1154].

If T = f^{∗} is the Royden algebra isomorphism described in Theorem 1.1, the adjoint T^{∗} of T defined by T^{∗}χ = χ∘T is a mapping of Ω into Ω′. Lewis showed that T^{∗} is in fact a homeomorphism.

Theorem 2.3 [L, p. 490]. If f: Ω → Ω′ is quasiconformal and T: A(Ω′) → A(Ω) is the corresponding Royden algebra isomorphism, then T^{∗}: Ω^{∗} → Ω^{′∗} is a homeomorphism such that T^{∗}(Δ) = Δ′ and T^{∗}|Ω = f.

Certain ideals in A(Ω) are useful in establishing a connection between the compactification Ω^{∗} and the algebra A(Ω). For example, if {x_{i}} is a sequence in Ω then I(x_{i}) denotes the ideal of functions tending to 0 on {x_{i}}, i.e.,

I(x_{i}) = I({x_{i}}) = {u ∈ A(Ω) : lim_{i→∞} u(x_{i}) = 0}.

The fact that A(Ω) is inverse-closed easily implies the following lemma.

Lemma 2.4 [S1, Lemma 5.1]. If u ∈ A(Ω), then u is contained in a proper ideal of A(Ω) if and only if u ∈ I(x_{i}) for some sequence {x_{i}} in Ω.

Definition 2.5. Two sequences {x_{i}} and {z_{j}} in Ω are disjoint if there is no sequence that is a subsequence of both, i.e., if there exists an integer m such that

{x_{i} : i ≥ m} ∩ {z_{j} : j ≥ m} = ∅.

Definition 2.6. A subsequence {x_{k(i)}} of {x_{i}} is a proper subsequence if {x_{i}} has a subsequence disjoint from {x_{k(i)}}.

Lemma 2.7. Let {x_{i}} be a sequence in Ω and {x_{k(i)}} a subsequence of {x_{i}}. Then I(x_{i}) ⊂ I(x_{k(i)}). If x_{i} → x ∈ Ω, then I(x_{i}) = I(x_{k(i)}). If x_{i} → ∂Ω, then the inclusion I(x_{i}) ⊂ I(x_{k(i)}) is proper whenever {x_{k(i)}} is a proper subsequence of {x_{i}}.

Proof. I(x_{i}) ⊂ I(x_{k(i)}) because u(x_{k(i)}) → 0 whenever u(x_{i}) → 0. If x_{i} → x ∈ Ω then continuity of u at x implies u(x_{i}) → 0 whenever u(x_{k(i)}) → 0, and so I(x_{k(i)}) ⊂ I(x_{i}).

Suppose x_{i} → ∂Ω and {x_{k(i)}} is a proper subsequence of {x_{i}}. We may assume {x_{k(i)}} is a sequence of distinct points without altering the contents of I(x_{k(i)}). By definition there is a subsequence {x_{m(i)}} of {x_{i}} disjoint from {x_{k(i)}}, and we may assume {x_{m(i)}} is a sequence of distinct points that does not intersect {x_{k(i)}}. Let {y_{i}} denote the alternating sequence

{y_{i}} = {x_{k(1)}, x_{m(1)}, x_{k(2)}, x_{m(2)}, …}.

Since y_{i} → ∂Ω as i → ∞, Theorem 1.2 implies {y_{i}} is an interpolating sequence. Therefore there exists u ∈ A(Ω) such that u(y_{2j+1}) = 0 and u(y_{2j}) = 1 for each j ∈ N. Thus u ∈ I(x_{k(i)}) while u ∉ I(x_{i}).

Theorem 2.8. If {x_{i}} is a sequence in Ω, then I(x_{i}) is a maximal ideal if and only if x_{i} → x ∈ Ω.

Proof. Sufficiency is straightforward. If x_{i} → x ∈ Ω, then I(x_{i}) = x̂^{−1}(0), a maximal ideal in A(Ω).

Necessity follows by proving the contrapositive. If it is not true that x_{i} → x ∈ Ω, then either {x_{i}} has a subsequence x_{k(i)} → ∂Ω or {x_{i}} has distinct cluster points x_{1} , x_{2} in Ω. In the first case, {x_{i}} has infinitely many distinct points and so has a proper subsequence. Lemma 2.7 then implies I(x_{i}) is not maximal. In the second case, {x_{i}} has disjoint subsequences {x_{k(i)}} and {x_{m(i)}} tending respectively to distinct points x_{1} and x_{2} in Ω. Lemma 2.7 implies I(x_{i}) ⊂ I(x_{k(i)}). But because A(Ω) separates points in Ω, there exists u ∈ A(Ω) such that u(x_{1}) = 0 and u(x_{2}) = 1. Thus u ∉ I(x_{i}) while u ∈ I(x_{k(i)}), and so the inclusion I(x_{i}) ⊂ I(x_{k(i)}) is proper.

We let A_{0}(Ω) denote the ideal of functions that tend to 0 at ∂Ω. To be precise,

A_{0}(Ω) = {u ∈ A(Ω) : if x_{i} → ∂Ω then lim_{i→∞} u(x_{i}) = 0}
= \{u ∈ C(\overline{Ω}) ∩ A(Ω) : u|∂Ω ≡ 0\}.

Clearly, A_{0}(Ω) properly contains the ideal of functions with compact support in Ω. Elements in the Royden boundary Δ can be characterized either by the fact that they annihilate A_{0}(Ω) or by the “size” of their neighborhood bases in the weak^{∗} topology.

Theorem 2.9 [L, Lemma 6.2], [S2, Theorem 4.7]. The following statements are equivalent:

1) χ ∈ Δ.

2) χ has no countable neighborhood basis in Ω^{∗}.

3) χ(u) = 0 for each u ∈ A_{0}(Ω).

4) χ(u) = 0 for each u ∈ A with compact support in Ω.

3. Elements of the Royden boundary characterized as nets

Because elements of Δ have no countable neighborhood bases, we cannot adequately describe convergence in Ω^{∗} to elements of Δ using the theory of sequences. We must rely on the more general but less intuitive theory of nets, which is described well in [K]. Briefly, if Λ is an indexing set directed by a binary relation ≥, then {x_{α}}_{α∈Λ} is a net in a set E if x_{α} ∈ E for each α ∈ Λ. The net {x_{α}} is eventually in a set U if there exists β ∈ Λ such that α ≥ β implies x_{α} ∈ U; it is frequently in U if for each α ∈ Λ there is a β ∈ Λ such that β ≥ α and x_{β} ∈ U. A net in a topological space X converges to x ∈ X if and only if it is eventually in each neighborhood of x, in which case x is a limit point of the net and we write lim_{α} x_{α} = x. A point s ∈ X is a cluster point of the net if the net is frequently in each neighborhood of s. A net {y_{β}}_{β∈B} is a subnet of {x_{α}}_{α∈Λ} if and only if there exists a function g defined on B with values g(β) ∈ Λ such that y_{β} = x_{g(β)} for each β ∈ B, and for each α ∈ Λ there exists γ ∈ B such that if β ≥ γ then g(β) ≥ α. A net {x_{α}} has s as a cluster point if and only if some subnet of {x_{α}} converges to s.

The theory of nets allows us to show that elements of Δ can be characterized as special kinds of nets for which no subnet is a sequence. Before presenting this characterization, however, we define a set of functions that will be useful in proving several results. For each z ∈ R^{n}, we define a “cutoff” of the distance function:

σ_{z}(x) = \{ \begin{matrix}|x − z|, & x ∈ B(z, 1) \\ 1, & x ∈ R^{n} \ B(z, 1). \end{matrix}

For z = ∞, we define

σ_{∞}(x) = \frac{1}{1 + |x|}.

These functions have the properties that σ_{z}(x) → 0 whenever x → z, σ_{z}(x) = 0 if and only if x = z, and σ_{z}(x) ≤ 1 for each x ∈ R^{n}. Furthermore, straightforward calculations show that ∥∇σ_{z}∥_{n,Ω} < ∞, so σ_{z} ∈ A(R^{n}) and σ_{z}|Ω ∈ A(Ω) for each z ∈ \overline{R}^{n}. In general, we will let σ_{z} denote σ_{z}|Ω which, of course, extends continuously to \overline{Ω}.

Theorem 3.1. Let Ω be a domain in R^{n}. Then χ ∈ Δ if and only if there exists a net {z_{α}}_{α∈Λ} in Ω such that

1) ẑ_{α} → χ in Ω^{∗},

2) there exists z ∈ ∂Ω for which z_{α} → z in \overline{Ω},

3) {z_{α}} has no subnet that is a sequence.

Proof. First we show necessity. If χ ∈ Δ, then because Ω̂ is dense in Ω^{∗}, there is a net {ẑ_{α}} in Ω̂ such that ẑ_{α} → χ in Ω^{∗}. Thus (1) holds.

For (2), \overline{Ω} is compact in \overline{R}^{n}, and so \{z_{α}\} has a cluster point z ∈ \overline{Ω} [K, p. 136]. If {z_{α}} does not converge to z, then there is a subnet of {z_{α}} which has no subnet converging to z [K, p. 74]. Because \overline{Ω} is compact, this subnet has a cluster point y ≠ z. Thus {z_{α}} has a subnet on which the function σ_{z} tends to σ_{z}(y) > 0 and a subnet on which σ_{z} tends to 0; in other words, lim_{α} ẑ_{α}(σ_{z}) does not exist. This means {ẑ_{α}} does not converge in Ω^{∗}, a contradiction to (1). Clearly z ∈ ∂Ω; for if not, ẑ_{α} → ẑ ∈ Ω̂, a contradiction to χ ∈ Δ.

For (3), suppose {z_{α}} has a subnet {z_{i}} that is a sequence. Then z_{i} → z [K, p. 74], and Theorem 1.2 implies {z_{i}} has a subsequence of distinct points that is an interpolating sequence. However, {ẑ_{i}} is a subnet of {ẑ_{α}} and thus ẑ_{i} → χ in Ω^{∗}. This means

lim_{i→∞} u(z_{i}) = χ(u)

for each u ∈ A(Ω), a contradiction to {z_{i}} having an interpolating subsequence.

For sufficiency, assume (1), (2), and (3) hold, and suppose χ = x̂ ∈ Ω̂. Property (2) implies lim_{α} σ_{z}(z_{α}) = 0, and so (1) implies x̂(σ_{z}) = 0. However,

x̂(σ_{z}) = σ_{z}(x) > 0,

a contradition. Thus χ ∈ Δ.

An algebraic restatement of part of Theorem 3.1 yields the following corollary.

Corollary 3.2. If χ ∈ Δ and M denotes the maximal ideal χ^{−1}(0) in A(Ω), then there exists a net {z_{α}}_{α∈Λ} in Ω converging to z ∈ ∂Ω for which no subnet is a sequence and

M = {u ∈ A(Ω) : lim_{α} u(z_{α}) = 0}.

Definition 3.3. Let {z_{α}} be a net in Ω. We say {z_{α}} is a Royden net if ẑ_{α} converges to a point χ ∈ Δ. Theorem 3.1 says that each χ ∈ Δ is determined by a corresponding Royden net. A particular χ may have many corresponding Royden nets; however, each must converge to the same point in \overline{Ω}.

Theorem 3.4. If {z_{α}} and {z_{β}} are nets in Ω such that ẑ_{α} and ẑ_{β} converge in Ω^{∗} to a common point χ ∈ Δ, then z_{α} and z_{β} converge in \overline{Ω} to a common point z ∈ ∂Ω.

Proof. By Theorem 3.1, z_{α} → z ∈ ∂Ω and z_{β} → y ∈ ∂Ω. Thus,
0 = lim_{α} ẑ_{α}(σ_{z}) = χ(σ_{z}) = lim_{β} ẑ_{β}(σ_{z}) = lim_{β} σ_{z}(z_{β}) = σ_{z}(y),

and hence z = y.

4. Fibers in the Royden boundary

By definition, each element in Ω̂ corresponds to a unique point in Ω, and we have shown that each element in Δ corresponds to nets in Ω that must converge in the Euclidean topology to a unique point in ∂Ω. These correspondences can be expressed explicitly by a projection mapping π: Ω^{∗} → \overline{Ω}. Let χ ∈ Ω^{∗}; if χ = x̂ ∈ Ω̂, then

π(χ) = x;

if χ ∈ Δ, then

π(χ) = the limit point z ∈ ∂Ω of a Royden net corresponding to χ.

Theorem 3.4 shows that π is well-defined, and it is not hard to see that π|Ω̂ is a bijective mapping of Ω̂ onto Ω with π^{−1}(x) = x̂ for each x ∈ Ω.

Lemma 4.1. If u is continuous on \overline{Ω}, then χ(u) = u(π(χ)).

Proof. If χ = x̂ ∈ Ω̂, then π(χ) = x, so that x̂(u) = u(x) = u(π(χ)). If χ ∈ Δ, Theorem 3.1 implies there is a corresponding Royden net {z_{α}} in Ω such that ẑ_{α} → χ and z_{α} → z ∈ ∂Ω. Then χ = lim_{α} ẑ_{α} and continuity of u implies

χ(u) = lim_{α} ẑ_{α}(u) = lim_{α} u(z_{α}) = u(z) = u(π(χ)).

Theorem 4.2. The projection mapping π is continuous.

Proof. We must show that if χ_{α} → χ in Ω^{∗}, then π(χ_{α}) → π(χ) in \overline{Ω}. Let χ_{α} → χ; then the definition of convergence in the weak^{∗} topology yields

lim_{α}|χ_{α}(σ_{π(χ)}) − χ(σ_{π(χ)})| = 0.

Lemma 4.1 implies

χ(σ_{π(χ)}) = σ_{π(χ)}(π(χ)) = 0;

therefore lim_{α} χ_{α}(σ_{π(χ)}) = 0, which, by Lemma 4.1, implies lim_{α} σ_{π(χ)}(π(χ_{α})) = 0. The definition of σ_{π(χ)} then implies π(χ_{α}) → π(χ).

Theorem 4.3. π|Δ is a surjective mapping of Δ onto ∂Ω.

Proof. Let χ ∈ Δ with {z_{α}} a corresponding Royden net such that z_{α} → z ∈ ∂Ω. By definition, π(χ) = z, and so π(Δ) ⊂ ∂Ω. Furthermore, the fact that π(Ω̂) = Ω implies Ω ⊂ π(Ω^{∗}) ⊂ \overline{Ω}. Because Ω^{∗} is compact, the continuity of π implies π(Ω^{∗}) is a compact subset of \overline{Ω}. But the only compact subset of \overline{Ω} that contains Ω is the set \overline{Ω} itself. Hence π(Ω^{∗}) = \overline{Ω} and π(Δ) = \overline{Ω} \ Ω = ∂Ω.

Definition 4.4. If z ∈ \overline{Ω}, then Φ_{z} denotes the fiber over z, defined as Φ_{z} = π^{−1}(z) = {χ ∈ Ω^{∗} : χ corresponds to a Royden net converging to z}. Continuity of π implies Φ_{z} is a closed subset of Δ and hence compact. Also

Φ_{z} ∩ Φ_{y} = ∅ whenever z ≠ y, and Theorem 4.3 implies Δ = ∪{Φ_{z} : z ∈ ∂Ω}.

For bounded domains there is an equivalent definition of π. If Ω is bounded, then p_{i}, the ith coordinate projection mapping in R^{n} restricted to Ω, belongs to A(Ω). We may define π by

π(χ) = (χ(p_{1}), …, χ(p_{n})) = (p̂_{1}(χ), …, p̂_{n}(χ)) ∈ R^{n}.

For example, if x̂ ∈ Ω̂, then

π(x̂) = (x̂(p_{1}), …, x̂(p_{n})) = (x_{1}, …, x_{n}) = x;

and if χ ∈ Δ, and {z_{α}} is a Royden net corresponding to χ, then

π(χ) = (lim_{α} ẑ_{α}(p_{1}), …, lim_{α} ẑ_{α}(p_{n})) = (lim_{α} p_{1}(z_{α}), …, lim_{α} p_{n}(z_{α})) = z ∈ ∂Ω.

Lemma 4.5. χ ∈ Φ_{z} if and only if χ(σ_{z}) = 0.

Proof. Suppose χ ∈ Φ_{z} with corresponding Royden net {z_{α}}. Then χ = lim_{α} ẑ_{α}, so

χ(σ_{z}) = lim_{α} ẑ_{α}(σ_{z}) = lim_{α} σ_{z}(z_{α}) = 0.

Conversely, suppose χ(σ_{z}) = 0. Then Lemma 4.1 implies σ_{z}(π(χ)) = 0. But σ_{z} vanishes only at z, so π(χ) = z, i.e. χ ∈ Φ_{z}.

Theorem 4.6. Let u ∈ A(Ω), z ∈ ∂Ω, and r ∈ R. There exists χ ∈ Φ_{z} for which χ(u) = r if and only if there exists a sequence {z_{j}} in Ω such that z_{j} → z and u(z_{j}) → r.

Proof. Let χ ∈ Φ_{z} and χ(u) = r. Let {z_{α}} be a Royden net corresponding to χ. Suppose there is no sequence {z_{j}} in Ω for which z_{j} → z and u(z_{j}) → r. Then there exists ε > 0 such that

|u(x) − r| ≥ c > 0

for each x ∈ B(z, ε) ∩ Ω. Consequently |ẑ_{α}(u) − r| ≥ c for each z_{α} ∈ B(z, ε) ∩ Ω, and hence χ(u) ≠ r, a contradiction.

Conversely, let z_{j} → z and u(z_{j}) → r. Then by Zorn’s lemma, there is a maximal ideal M containing I(z_{j}); and by the Gelfand identification, M = χ^{−1}(0) for some χ ∈ Ω^{∗}. But σ_{z} ∈ I(z_{j}) ⊂ M, and so χ(σ_{z}) = 0. Thus Lemma 4.5 implies χ ∈ Φ_{z}. Furthermore, u − r ∈ I(z_{j}), so χ(u − r) = 0, and hence χ(u) = r.

Corollary 4.7. If u ∈ A(Ω) and z ∈ ∂Ω, then the range of û on Φ_{z} is exactly the set of all real numbers r for which there exists a sequence {z_{j}} in Ω with z_{j} → z and u(z_{j}) → r.

Proof. Let r lie in the range of û on Φ_{z}. Then there exists χ ∈ Φ_{z} such that χ(u) = û(χ) = r. So Theorem 4.6 implies there exists a sequence {z_{j}} in Ω such that z_{j} → z and u(z_{j}) → r. Conversely, let {z_{j}} be a sequence in Ω such that z_{j} → z and u(z_{j}) → r. By Theorem 4.6, there exists χ ∈ Φ_{z} for which r = χ(u) = û(χ); hence r is in the range of û on Φ_{z}.

If u ∈ A(Ω), we say u has a limit at z ∈ ∂Ω if there is some r ∈ R such that u converges to r on every sequence in Ω converging to z. A special case of Corollary 4.7 is the following.

Corollary 4.8. If u ∈ A(Ω) and z ∈ ∂Ω, then u has a limit r at z if and only if χ(u) = r for each χ ∈ Φ_{z}.

Proof. Let u have a limit r at z. Then u converges to r on every sequence in Ω converging to z. By Corollary 4.7, {r} is the range of û on Φ_{z}, and so r = û(χ) = χ(u) for each χ ∈ Φ_{z}. Conversely, let r be such that r = χ(u) = û(χ) for each χ ∈ Φ_{z}. If {z_{j}} is a sequence in Ω converging to z, then Corollary 4.7 implies each convergent subsequence of {u(z_{j})} converges to r. But {u(z_{j})} is bounded, and so this means {u(z_{j})} converges to r.

It is difficult to rely on our intuition about sequences to fully understand the behavior of nets. This is illustrated explicitly in the following corollary. Sequences tending to ∂Ω in Ω are interpolating sequences for A(Ω); however, an analog of this fact for nets does not hold.

Corollary 4.9. For each z ∈ ∂Ω there is a net {z_{α}} in Ω such that z_{α} → z in the Euclidean topology and lim_{α} u(z_{α}) exists for each u ∈ A(Ω).

Proof. The surjectivity of π implies Φ_{z} is not empty. If χ ∈ Φ_{z} and {z_{α}} is a corresponding Royden net, then χ = lim_{α} ẑ_{α} in Ω^{∗} and the definition of convergence in the weak^{∗} topology implies

lim_{α} ẑ_{α}(u) = lim_{α} u(z_{α}) = χ(u)

for each u ∈ A(Ω).

5. Homeomorphic boundary extensions of quasiconformal mappings

Because each boundary fiber in a domain’s Royden compactification corresponds explicitly to a particular boundary point, it is possible to use the concept of boundary fibers to characterize the existence of homeomorphic boundary extensions of quasiconformal mappings. Such extensions exist if and only if the induced homeomorphism between Royden compactifications maps boundary fibers onto boundary fibers.

Theorem 5.1. Let f: Ω → Ω′ be a quasiconformal mapping and T = f^{∗}: A(Ω′) → A(Ω) be its corresponding Royden algebra isomorphism. Then f has a homeomorphic extension \overline{f}: \overline{Ω} → \overline{Ω}′ if and only if for each z ∈ ∂Ω there exists y ∈ ∂Ω′ such that T^{∗}(Φ_{z}) = Φ_{y}, in which case y = \overline{f}(z).

Proof. To prove necessity, observe that each χ ∈ Φ_{z} has a corresponding Royden net \{z_{α}\}. Let y_{α} = \overline{f}(z_{α}) and y = lim_{α} y_{α} = \overline{f}(z). Since T^{∗}x̂ = f̂(x) for each x ∈ Ω,

T^{∗}χ = T^{∗}(lim_{α} ẑ_{α}) = lim_{α} T^{∗}ẑ_{α} = lim_{α} ŷ_{α}.

Thus T^{∗}χ has {y_{α}} as a corresponding Royden net, i.e. T^{∗}χ ∈ Φ_{y}. So T^{∗}(Φ_{z}) ⊂ Φ_{y} and a similar argument shows T^{∗^{−1}}(Φ_{y}) ⊂ Φ_{z}. Hence T^{∗}(Φ_{z}) = Φ_{y}.

To prove sufficiency, for each z ∈ ∂Ω define \overline{f}(z) to be that point y ∈ ∂Ω for which T^{∗}(Φ_{z}) = Φ_{y}. We first show \overline{f} is injective by showing that if z and x are distinct points in ∂Ω, then T^{∗}(Φ_{z}) ≠ T^{∗}(Φ_{x}). Let v = T^{−1}σ_{z}. By Lemma 4.5, χ ∈ Φ_{z} implies χ(σ_{z}) = 0 and η ∈ Φ_{x} implies η(σ_{z}) > 0. If χ′ ∈ T^{∗}(Φ_{z}), then there exists χ ∈ Φ_{z} for which T^{∗}χ = χ′. It follows that

χ′(v) = T^{∗}χ(v) = χ(T v) = χ(σ_{z}) = 0.

Similarly, if η′ ∈ T^{∗}(Φ_{x}), then η′(v) = η(σ_{z}) > 0. Hence T^{∗}(Φ_{z}) ≠ T^{∗}(Φ_{x}).

We next show \overline{f} is surjective. Let y ∈ ∂Ω′. Then there is a sequence \{y_{j}\} in Ω′ for which y_{j} → y. Because f is a homeomorphism, f^{−1}(y_{j}) → ∂Ω, and because \overline{Ω} is compact, there is z ∈ ∂Ω for which a subseqence \{z_{k}\} of \{f^{−1}(y_{j})\} converges to z. By hypothesis, there is w ∈ ∂Ω′ for which T^{∗}Φ_{z} = Φ_{w}. We show w = y.

Let u = T σ_{w}. The continuity of σ_{w} at w and Corollary 4.8 imply that for each χ ∈ Φ_{z}

χ(u) = T^{∗}χ(σ_{w}) = 0.

This means, again by Corollary 4.8, that u has the limit 0 at z, and so u(z_{k}) → 0. But

u(z_{k}) = T σ_{w}(z_{k}) = f^{∗}σ_{w}(z_{k}) = σ_{w}(f(z_{k})),

and so σ_{w}(f(z_{k})) → 0. Because f(z_{k}) → y, the fact that σ_{w} vanishes only at w implies w = y.

Finally, we show \overline{f} and \overline{f}^{−1} are continuous. Because f and f^{−1} are homeomorphisms, it suffices to show \overline{f} is continuous on ∂Ω. A similar argument shows \overline{f}^{−1} is continuous on ∂Ω′. Let z_{j} converge in \overline{Ω} to z ∈ ∂Ω. We must show \overline{f}(z_{j}) converges in \overline{Ω}′ to \overline{f}(z).

Suppose this is not true. Then because \overline{Ω}′ is compact, \{\overline{f}(z_{j})\} has a subsequence \{y_{k}\} converging to y ≠ \overline{f}(z), and because f is a homeomorphism, y ∈ ∂Ω′. Let z_{k} denote the sequence \{\overline{f}^{−1}(y_{k})\}. Because \overline{f} is surjective, there exists x ∈ ∂Ω for which T^{∗}(Φ_{x}) = Φ_{y}. Clearly x ≠ z, because T^{∗}(Φ_{z}) = Φ_{\overline{f}(z)} ≠ Φ_{y}. If v = T^{−1}σ_{z}, then for each χ ∈ Φ_{x}

T^{∗}χ(v) = χ(σ_{z}) = σ_{z}(x) > 0.

Because T^{∗} maps Φ_{x} onto Φ_{y}, this inequality means η(v) > 0 for each η ∈ Φ_{y}. We claim the existence of a sequence {w_{k}} in Ω′ such that

|w_{k} − y_{k}| < \frac{1}{k}  and  |v(w_{k})| < 2σ_{z}(z_{k})

for each k. Then w_{k} → y; and since σ_{z}(z_{k}) → 0, it follows that v(w_{k}) → 0 as k → ∞. By Theorem 4.6, there exists η ∈ Φ_{y} for which η(v) = 0, a contradiction. Therefore, our argument is complete if we prove the claim.

We construct {w_{k}} in two steps. First, if y_{k} ∈ Ω′, let w_{k} = y_{k}. Then |w_{k} − y_{k}| = 0 and

v(w_{k}) = T^{−1}σ_{z}(y_{k}) = (f^{−1})^{∗}σ_{z}(y_{k}) = σ_{z}(z_{k}).

Second, if y_{k} ∈ ∂Ω′, then for each ξ ∈ Φ_{y_{k}} there is χ ∈ Φ_{z_{k}} such that T^{∗}χ = ξ. Hence,

ξ(v) = T^{∗}χ(v) = χ(σ_{z}) = σ_{z}(z_{k})

for each ξ ∈ Φ_{y_{k}}. Corollary 4.8 now implies that v has the limit σ_{z}(z_{k}) at y_{k}. Thus, it is possible to choose w_{k} ∈ Ω′ close enough to y_{k} to satisfy the two conditions in the claim.

6. Boundary fibers and homogeneity

As a result of Theorem 5.1, it is possible to show that all boundary fibers of domains satisfying certain homogeneity conditions on the boundary are homeomorphic to each other. To describe this condition precisely, we define the concept of transitivity. If Γ is a family of homeomorphisms of a set X onto itself then Γ acts transitively on a set E ⊂ X if for each a, b ∈ E there exists g ∈ Γ such that g(a) = b. Let Γ(Ω) denote the family of self-homeomorphisms f: \overline{Ω} → \overline{Ω} such that f|Ω is quasiconformal. In this section, we consider the class of domains for which Γ(Ω) acts transitively on ∂Ω. Using terminology from Gehring and Palka [GP], we might say that such domains are quasiconformally homogeneous with respect to the boundary. A typical domain of this type is B^{n}, on which the family of rotations of B^{n} acts transitively. We will show, however, that even for B^{n}, the Royden boundary ΔB^{n} is not the natural product of its boundary S^{n−1} with any particular boundary fiber.

Theorem 6.1. Let Γ(Ω) act transitively on ∂Ω. If y, z ∈ ∂Ω, then Φ_{z} is homeomorphic to Φ_{y}.

Proof. Let f ∈ Γ(Ω) be such that f(z) = y. Then f|Ω is quasiconformal,

(f|Ω)^{∗} = T: A(Ω) → A(Ω)

is a Royden algebra isomorphism, and T^{∗}: Ω^{∗} → Ω^{∗} is a homeomorphism. We only need to show that T^{∗}(Φ_{z}) ⊂ Φ_{y}, for then Theorem 5.1 implies T^{∗}(Φ_{z}) = Φ_{y} and T^{∗}|Φ_{z} is the desired homeomorphism.

Let χ ∈ Φ_{z} with corresponding Royden net {z_{α}} and f(z_{α}) = y_{α} → y. Then

T^{∗}χ = T^{∗}(lim_{α} ẑ_{α}) = lim_{α} T^{∗}ẑ_{α} = lim_{α} ŷ_{α},

so {y_{α}} is a Royden net for T^{∗}χ, i.e. T^{∗}χ ∈ Φ_{y}.

Let e denote the unit vector (1, 0, 0, …, 0) ∈ S^{n−1} , and Δ = ΔB^{n}. Because

Δ = ∪{Φ_{z} : z ∈ S^{n−1}},

and because Δ can be identified with S^{n−1} ×Φ_{e} by rotations of \overline{B^{n}}, Theorem 6.1 leads to the question of whether or not Δ is homeomorphic to S^{n−1} × Φ_{e} in the natural way. In other words, if for each z ∈ S^{n−1} , T_{z}^{∗}: Φ_{e} → Φ_{z} denotes the homeomorphism induced by the rotation of \overline{B^{n}} mapping e to z, is the function h: S^{n−1} × Φ_{e} → Δ given by h(z, η) = T_{z}^{∗}(η) a homeomorphism? Theorem 6.4 shows the answer is no. This is a result of the nonexistence of continuous sections between S^{n−1} and Δ, a fact we prove in Theorem 6.3.

Lemma 6.2. If {χ_{j}} is a convergent sequence in Δ, then {χ_{j}} is eventually in a single fiber Φ_{z}, z ∈ S^{n−1} .

Proof. Let χ_{j} → χ ∈ (B^{n})^{∗}, and let
z_{j} = π(χ_{j}) ∈ S^{n−1}

and z = π(χ). Suppose {z_{j}} has a subsequence {z_{k}} such that for each k, z_{k} ≠ z. Then we consider two cases.

First, suppose some point y ≠ z appears infinitely often in the sequence {z_{k}}. Then the sequence {χ_{j}(σ_{z})} assumes the value |y − z| > 0 an infinite number of times. However, the fact that χ_{j} → χ implies

χ_{j}(σ_{z}) → χ(σ_{z}) = 0,

a contradiction.

Second, suppose each point in the sequence {z_{k}} appears only finitely many times. Then {z_{k}} contains a subsequence of distinct points, again denoted by {z_{k}}. Let {B_{k}} be a sequence of balls in R^{n} with mutually disjoint closures centered at z_{k}. There exist functions w_{k} ∈ A(R^{n}) such that w_{k}(z_{k}) = (−1)^{k}, |w_{k}| ≤ 1, w_{k}|R^{n} \ B_{k} ≡ 0 and

(∫_{R^{n}} |Δw_{k}(x)|^{n} dx))^{1/n} ≤ 2^{−k}

for each k ∈ N, (this follows from the fact that the conformal capacity of a spherical ring with a degenerate boundary component is 0 [G1, p. 138]). Therefore

v(x) = ∑_{i=1}^{∞} w_{k}(x)

is a function in A(R^{n}). Furthermore, because v is the sum of continuous functions with disjoint support, u = v|Ω ∈ C(Ω). Thus u ∈ A(Ω), and u can be continuously extended to each of the boundary points z_{k} such that

u(z_{k}) = w_{k}(z_{k}) = (−1)^{k}.

But Corollary 4.8 implies χ_{k}(u) = u(z_{k}), and so {χ_{k}(u)} is not a convergent sequence, a contradiction to the hypothesis that {χ_{j}} is convergent in (B^{n})^{∗}.

Theorem 6.3. If s: S^{n−1} → Δ is any function for which π ∘ s is the identity on S^{n−1} , then s cannot be continuous.

Proof. Suppose π ∘ s is the identity on S^{n−1} (i.e. s is a section) and s is continuous. Let {z_{j}} be a sequence of distinct points in S^{n−1} with z_{j} → z ∈ S^{n−1} . Then s(z_{j}) → s(z) in Ω^{∗}, and Lemma 6.2 implies there exists an integer m and y ∈ S^{n−1} such that s(z_{j}) ∈ Φ_{y} for each j ≥ m. Hence z_{j} = π(s(z_{j})) = y for each j ≥ m, a contradiction.

Theorem 6.4. Δ is not homeomorphic to S^{n−1} × Φ_{e} in the natural way.

Proof. Suppose the function h: S^{n−1} × Φ_{e} → Δ defined by h(z, η) = T_{z}^{∗}(η) is a homeomorphism. If η is fixed in Φ_{e}, then the function s: S^{n−1} → Δ defined by s(z) = h(z, η) is a continuous function. But π ∘ s is the identity on S^{n−1} because

π ∘ s(z) = π(T_{z}^{∗}(η)) = z

for each z ∈ S^{n−1} . This contradicts Theorem 6.3.

7. Correspondence between prime ends and boundary fibers

Generalizations of the theory of homeomorphic boundary extensions lead naturally to a discussion of prime ends. Theorem 5.1 indicates a possible connection between the boundary fibers and prime ends of Ω. We make this connection explicit by showing that whenever Ω is finitely connected on the boundary, the set of components of boundary fibers and the set of prime ends are one and the same.

A domain Ω is finitely connected at z ∈ ∂Ω if z has arbitrarily small neighborhoods in R^{n} whose intersection with Ω has a finite number of components. If this number is one, Ω is locally connected at z. The domain Ω is finitely connected on the boundary or locally connected on the boundary if these conditions hold for each z ∈ ∂Ω.

The theory of prime ends provides a method for defining connectedness of domains at their boundaries. Caratheodory initiated this theory by showing that each conformal mapping f: B^{2} → Ω has a homeomorphic extension from \overline{B^{2}} onto a compactification Ω̃ of Ω obtained by combining Ω with its set of prime ends. Näkki and Zorich applied this idea to domains in higher dimensions and obtained similar results for quasiconformal mappings of domains in n-space [Nk], [Z]. Väisälä gave a particularly simple description of prime ends for domains finitely connected on the boundary [V2]. We use his description to prove our theorem.

If Ω is a domain in R^{n}, then an endcut of Ω is a path γ: [a, b) → Ω such that γ(t) → z ∈ ∂Ω as t → b. We associate z to γ by the notation z = h(γ). A subendcut of γ is a restriction of γ to a subinterval [a_{1}, b). If U is a neighborhood of h(γ), then there is a unique component P (U, γ) of U ∩Ω containing a subendcut of γ. Two endcuts γ and λ are equivalent if h(γ) = h(λ) and if P (U, γ) = P (U, λ) for each neighborhood U of h(γ). The equivalence class γ̃ of γ is a prime end of Ω, and the collection 𝒫 of all prime ends is the prime end boundary of Ω. The set Ω̃ = 𝒫 ∪ Ω is the prime end compactification of Ω. A natural impression map θ_{Ω}: Ω̃ → \overline{Ω} is defined by θ_{Ω}(γ̃) = h(γ) for γ̃ ∈ 𝒫 and θ_{Ω}(x) = x for x ∈ Ω. If Ω is locally connected at z ∈ ∂Ω, then θ_{Ω}^{−1}(z) is a single point. Thus if Ω is locally connected on the boundary, 𝒫 can be identified with ∂Ω. The following lemmas show that analogous equivalences can be defined for elements in the Royden boundary.

Lemma 7.1. Let Ω be finitely connected on the boundary with z ∈ ∂Ω. If χ ∈ Φ_{z} and U is a neighborhood of z in \overline{R}^{n}, then there is a unique component Q(U, χ) of U ∩Ω such that each Royden net {z_{α}} corresponding to χ is eventually in Q(U, χ).

Proof. Let ẑ_{α} → χ. First we claim that {z_{α}} is eventually in a single component of U ∩ Ω. If not, then since U ∩ Ω has a finite number of components, there must be a component V of U ∩ Ω such that {z_{α}} is both frequently inside and frequently outside V . We define a variant of the function σ_{z} by choosing n small enough so that B(z, 1/n) ⊂ U and letting

u(x) = \{ \begin{matrix}|x − z|, & x ∈ V ∩ B(z, 1/n), \\ 1/n, & x ∈ Ω \ (V ∩ B(z, 1/n)). \end{matrix}

Then u ∈ A(Ω), and the net {ẑ_{α}(u)} has a subnet converging to 0 and a subnet converging to 1/n; hence {ẑ_{α}(u)} does not converge to χ(u), a contradiction.

Furthermore, let {z_{α}} and {z_{β}} be distinct Royden nets corresponding to χ. If there are distinct components V and W of U ∩ Ω such that {z_{α}} is eventually in V and {z_{β}} is eventually in W ⊂ Ω\V , then for u defined as above, we obtain χ(u) = lim_{α} ẑ_{α}(u) = 0 and χ(u) = lim_{β} ẑ_{β}(u) = 1/n, a contradiction.

Theorem 7.2. Two elements χ and η in Φ_{z} are in the same component of Φ_{z} if and only if Q(U, χ) = Q(U, η) for each neighborhood U of z in \overline{R}^{n}.

Proof. Suppose there is a neighborhood U of z for which Q(U, χ) ≠ Q(U, η). If n is chosen small enough so that B(z, 1/n) ⊂ U and

u(x) = \{ \begin{matrix}|x − z|, & x ∈ Q(U, χ) ∩ B(z, 1/n), \\ 1/n, & x ∈ Ω \ (Q(U, χ) ∩ B(z, 1/n)), \end{matrix}

then u ∈ A(Ω), and by Corollary 4.7 the range of û on Φ_{z} is the two point set {0, 1/n}. The fact that û is continuous on Ω^{∗} implies Φ_{z} has at least two components, one of which contains χ and one of which contains η, a contradiction.

Conversely, let Q(U, χ) = Q(U, η) for each neighborhood U of z, and let {z_{α}} and {y_{β}} be nets in Ω such that ẑ_{α} → χ and ŷ_{β} → η. Suppose that χ and η are in different components Ψ and Π of Φ_{z}. Since Φ_{z} is a closed subset of Δ, the component Ψ and the set Φ_{z} \ Ψ are also closed. By the Urysohn Lemma, there is a g ∈ C(Ω^{∗}) such that g|Ψ ≡ 1 and g|(Φ_{z} \ Ψ) ≡ 0. Because Â is dense in C(Ω^{∗}), there is v̂ ∈ Â so that the range of v̂ restricted to Ψ lies in (\frac{2}{3}, \frac{4}{3}) and the range of v̂ restricted to Φ_{z} \ Ψ lies in (−\frac{1}{3}, \frac{1}{3}). Because χ is equivalent to η, Q(B(z, 1/n), χ) = Q(B(z, 1/n), η) = Q_{n} for each positive integer n. Thus \{z_{α}\} is eventually in Q_{n} and v(z_{α}) is eventually greater than \frac{1}{2} , while \{y_{β}\} is eventually in Q_{n} and v(y_{β}) is eventually less than \frac{1}{2} . Because Q_{n} is connected and v is continuous, for each n there exists x_{n} ∈ Q_{n} such that v(x_{n}) = \frac{1}{2} . Corollary 4.7 then implies that \frac{1}{2} lies in the range of v̂ on Φ_{z}, a contradiction.

Corollary 7.3. If Ω is locally connected at z ∈ ∂Ω, then Φ_{z} is connected.

Proof. By Theorem 7.2, it is enough to show that if χ, η ∈ Φ_{z} then Q(U, χ) = Q(U, η) for each neighborhood U of z. Let U be a neighborhood of z. Then because Ω is locally connected on the boundary there exists a neighborhood V of z such that V ⊂ U and V ∩Ω is connected. Thus Q(V, χ) = Q(V, η). By definition, both Q(U, χ) and Q(U, η) are components of U ∩Ω that contain Q(V, χ); therefore they are the same.

Theorem 7.4. Let Ω be finitely connected on the boundary. The set C of components of boundary fibers in Δ is identical to the set 𝒫 of prime ends of Ω.

Proof. We define the function κ: 𝒫 → C by the rule κ(γ̃) = Ψ if and only if h(γ) = π(Ψ) and, for each χ ∈ Ψ and neighborhood U of z in \overline{R}^{n}, P (U, γ) = Q(U, χ). The fact that κ is well-defined and injective follows from Theorem 7.2. We show that κ is surjective and hence a bijection. Let Ψ ∈ C and χ ∈ Ψ; then {Q(B(z, 1/j), χ)} = {Q_{j}} is a sequence of connected open sets in R^{n} with Q_{j+1} ⊂ Q_{j} for each j ∈ N. For each j, let x_{j} be a point in Q_{j}. Since Q_{j} is an open connected set in R^{n}, there is a path γ_{j} joining x_{j} to x_{j+1} in Q_{j}. Let γ be the path defined by joining γ_{1} to γ_{2} to γ_{3} , etc. Then h(γ) = z. If U is a neighborhood of z, then there is m large enough so that Q_{m} ⊂ U. By definition, each Royden net {z_{α}} corresponding to χ is eventually in the connected set Q_{m} ⊂ U, so Q(U, χ) must be the component of U ∩ Ω containing Q_{m}. On the other hand, Q_{m} contains a subendcut of γ, and therefore P (U, γ) is the component of U ∩ Ω containing Q_{m}. Thus P (U, γ) = Q(U, χ).

8. The Royden boundary and Stone–Cech compactification of N

Finally, we examine the topology of individual boundary fibers. Because each sequence tending to ∂Ω is an interpolating sequence for A(Ω), Corollary 4.9 highlights how differently nets converging to a boundary point may behave with respect to A(Ω) than sequences converging to a boundary point. Boundary fibers are determined by nets, so one might guess that whenever z ∈ ∂Ω, the fiber Φ_{z} is a large and complicated set. This is true. If βN denotes the Stone–Cech compactification, then each such fiber contains a homeomorphic copy of βN \ N.

Up to homeomorphism, βN is the unique compactification of N on which each real-valued bounded continuous function on N can be uniquely extended. The space βN can also be characterized as the compactification of N that has the properties outlined in the following theorem.

Theorem 8.1 [Ga, Theorem 1.4, p. 186]. Let Y be a compact Hausdorff space and let τ: N → Y be a continuous mapping. Then the mapping τ has a unique continuous extension \overline{τ}: βN → Y . If τ(N) is dense in Y and if the images of disjoint subsets of N have disjoint closures in Y , then the extension \overline{τ} is a homeomorphism of βN onto Y .

The Stone–Cech compactification of N is a huge space. For example, if Z is any infinite closed subset of βN, then card (Z) = card (I^{I}) where I = [0, 1], [D, p. 244]. In fact, βN can be mapped onto any separable compact Hausdorff space; however, no point of βN \ N can be exhibited concretely [Ga, p. 187]. This is unfortunate for the study of fibers in the Royden boundary because of the following theorem.

Theorem 8.2. For each z ∈ ∂Ω, Φ_{z} contains a homeomorphic image of βN \ N.

164 Nathan Soderborg

Proof. Let {z_{j}} be a sequence of distinct points in Ω such that z_{j} → z ∈ ∂Ω, and let Y be the closure of {ẑ_{j}} in Ω^{∗}. Then as a subspace of Ω^{∗}, Y is compact and Hausdorff. We define τ: N → Y by τ(j) = ẑ_{j} for each j ∈ N, and thus τ(N) = {ẑ_{j}} is dense in Y .

If S = {s_{1}, s_{2}, …} and Q = {q_{1}, q_{2}, …} are disjoint subsets of N, then because {z_{j}} is an interpolating sequence, there exists u ∈ A(Ω) for which u(z_{s_{i}}) = 1 and u(z_{q_{k}}) = 0 for each i and k. This means û(ẑ) = 1 for each ẑ ∈ τ(S); and since û ∈ C(Ω^{∗}), χ ∈ \overline{τ(S)} implies χ(u) = û(χ) = 1. Similarly, η ∈ \overline{τ(Q)} implies η(u) = 0. Thus \overline{τ(S)} ∩ \overline{τ(Q)} = ∅, and by Theorem 8.1, τ has a homeomorphic extension \overline{τ} of βN onto Y . Because z_{j} → z and Y is the closure of \{ẑ_{j}\} minus any finite set, it follows that σ̂_{z} is identically 0 on Y \ {ẑ_{j}}; i.e. χ(σ_{z}) = 0 for each χ ∈ Y \ {ẑ_{j}}, so

\overline{τ}(βN \ N) = Y \ \{ẑ_{j}\} ⊂ Φ_{z}.

Acknowledgement. The author thanks Professors Fred Gehring, Juha Heinonen, and Shanshuang Yang for their helpful suggestions.

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Received 29 January 1993