BOURGAIN ALGEBRAS, MINIMAL ENVELOPES, MINIMAL SUPPORT SETS, AND SOME APPLICATIONS

CARROLL J. GUILLORY

Abstract. We explicitly compute certain Douglas algebras that are invariant under both the Bourgain map and the minimal envelope map. We also compute the Bourgain algebra and the minimal envelope of the maximal subalgebras of a certain singly generated Douglas algebra.

1. Introduction

Let H^{∞} be the Banach algebra of bounded analytic functions on the open unit disk D. By considering boundary functions, we may regard H^{∞} an an essentially supremum norm-closed subalgebra of L^{∞} = L^{∞}(∂D). A closed subalgebra between H^{∞} and L^{∞} is called a Douglas algebra. H^{∞} + C is the smallest Douglas algebra, where C is the space of continuous functions on ∂D. The Chang-Marshall theorem [Ch, Ma] says that every Douglas algebra B is generated by H^{∞} and complex conjugates of interpolating Blaschke products ψ with ψ̄ ∈ B. We denote by M(B) the maximal ideal space of B. Then we may consider that M(L^{∞}) ⊂ M(B) ⊂ M(H^{∞}) and M(L^{∞}) is the Shilov boundary for every Douglas algebra B. For a point x in M(H^{∞}), there is a representing measure μ_{x} on M(L^{∞}); f(x) = ∫_{M(L^{∞})} f dμ_{x} for every f ∈ H^{∞}. A Douglas algebra B with A ⊊ B is called a minimal superalgebra of A if there are no Douglas algebras B′ such that A ⊊ B′ ⊊ B. In [GI1], the authors proved that if A and B are Douglas algebras with A ⊊ B, then B is a minimal superalgebra of A if and only if supp μ_{x} = supp μ_{y} for every x, y ∈ M(A) \ M(B).

Let Y be a Banach algebra with identity and let B be a closed subalgebra of Y. The Bourgain algebra B_{b} of B relative to Y is defined by the set of f in Y such that ∥ff_{n} + B∥ → 0 for every sequence {f_{n}}_{n} in B with f_{n} → 0 weakly. In this note, we consider Y = L^{∞} and B a Douglas algebra. In [CJY], Cima, Janson, and Yale proved that H_{b}^{∞} = H^{∞} + C. Gorkin, Izuchi, and Martini [GIM] studied Bourgain algebras of Douglas algebras and proved that (B_{b})_{b} = B_{b}. We denote by B_{m} the smallest Douglas algebra which contains all minimal superalgebras of B. We call B_{m} the minimal envelope of B. We have B ⊆ B_{b} ⊆ B_{m}.

1991 Mathematics Subject Classification. 46J15,46J20.

Key words and phrases. Maximal subalgebras, Douglas algebras, Bourgain algebras, minimal envelope, interpolating Blaschke product, minimal support points, locally thin points, support sets, Gleason parts.

A sequence {z_{n}}_{n} in D is called interpolating if for every bounded sequence {a_{n}}_{n} there exists f in H^{∞} such that f(z_{n}) = a_{n} for every n. A Blaschke product

ψ(z) = ∏_{n=1}^{∞} \frac{−z̄_{n}}{|z_{n}|} \frac{z − z_{n}}{1 − z̄_{n}z} , z ∈ D

is called interpolating if its zeroes {z_{n}}_{n} is interpolating. For a function f in H^{∞} we put Z(f) = {x ∈ M(H^{∞}+C) : f(x) = 0}. For x ∈ M(H^{∞}) we denote by supp μ_{x} the support set for the representing measure μ_{x}. For a subset E of M(L^{∞}) we denote by cl E the closure of E in M(L^{∞}). Put E = ⋃{supp μ_{x} : x ∈ M(H^{∞} + C), |q(x)| < 1} where q is an interpolating Blaschke product. Then N(q̄) = cl E. A closed set E in M(L^{∞}) is called a peak set for a Douglas algebra B if there is a f ∈ B such that f = 1 on E and |f| < 1 on M(L^{∞}) \ E. A closed set E in M(L^{∞}) is called a weak peak for B if E is the intersection of some family of peak sets. If E is a weak peak set for B, then the set

B_{E} = {f ∈ L^{∞} : f|_{E} ∈ B|_{E}}

is a Douglas algebra. The sets supp μ_{x} and N(q̄) are weak peak sets for H^{∞}, hence H_{supp μ_{x}}^{∞} and H_{N(q̄)}^{∞} are Douglas algebras. A point x ∈ Z(q), where q is an interpolating Blaschke product, is called a minimal element for H^{∞}[q̄] if there is no y ∉ M(H^{∞}[q̄]) such that supp μ_{y} ⊊ supp μ_{x}; that is, if y ∉ M(H^{∞}[q̄]), then either supp μ_{y}∩supp μ_{x} = ∅ or supp μ_{x} ⊆ supp μ_{y}. We put

E_{x} = {λ ∈ M(H^{∞}) : supp μ_{λ} = supp μ_{x}}

and call E_{x} the level set of x. For x, y in M(H^{∞}), we put

ρ(x, y) = sup{|f(y)| : f ∈ H^{∞}, ∥f∥_{∞} ≤ 1, f(x) = 0} .

The set

P (x) = {λ ∈ M(H^{∞}) : ρ(λ, x) < 1}

is called the Gleason part containing x. We have P (x) ⊂ E_{x}, and if x ∈ M(B) then E_{x} ⊂ M(B). The map that assigns the algebra B_{b} to a Douglas algebra B is called the Bourgain map and the map that assigns B_{m} to B is called the minimal envelope map.

A point x ∈ M(H^{∞}) is called locally thin if there is an interpolating Blaschke product q such that q(x) = 0 and

(1 − |z_{n(α)}|^{2})|q′(z_{n(α)})| → 1

whenever z_{n(α)} is a subnet of the zero sequence {z_{n}}_{n} of b in D converging to x. We say that q is locally thin at x.

We say that a minimal support point x of a Douglas algebra B corresponds to the maximal subalgebra A_{x} if it satisfies the equation

M(A_{x}) = M(B) ∪ P_{x} or M(A_{x}) = M(B) ∪ E_{x} .

We put QC = (H^{∞} + C) ∩ \overline{(H^{∞} + C)}. Then

QC = {f ∈ H^{∞} + C : f|_{supp μ_{x}} is constant for every x ∈ M(H^{∞} + C)} .

For x ∈ M(L^{∞}), the set

Q = {y ∈ M(L^{∞}) : f(y) = f(x) for every f ∈ QC}

is called a QC-level set. For every y ∈ M(H^{∞} + C), there is a QC-level set Q_{y} such that supp μ_{y} ⊂ Q_{y}. For any interpolating Blaschke product q set

m_{q} = {x ∈ M(H^{∞} + C) : x is a minimal support point of H^{∞}[q̄]} .

In this paper we give examples (Theorem 1 and Theorem 2) of Douglas algebras that show that the Bourgain algebra of the arbitrary intersection of Douglas algebras does not equal the intersection of their corresponding Bourgain algebras. Theorems 1 and 2 also show that this is true for the minimal envelopes. We also compute the Bourgain algebra and the minimal envelope of certain maximal subalgebras. Finally we give two applications of minimal support points of certain interpolating Blaschke products.

All of the work in this paper was done while the author was at the Mathematical Science Research Institute. The author thanks the Institute for its support during this period.

Lemma 1. Let q be an interpolating Blaschke product that is of type G, and such that Z(q) ∩ P_{x} is a finite set for all x ∈ Z(q). Then each x ∈ Z(q) is a locally thin point.

Proof. Since q is of type G and the set Z(q) ∩ P_{x} is finite there is a factor q_{0} of q such that Z(z_{0}) ∩ P_{x} = {x}. Hence H_{supp μ_{x}}^{∞}[q̄_{0}] is a minimal superalgebra of H_{supp μ_{x}}^{∞}. Hence H_{supp μ_{x}}^{∞} ⊂ (H_{supp μ_{x}}^{∞})_{b} and so by Theorem 5 of [MY] we have that x is a locally thin point.

Lemma 2. Let B = H_{supp μ_{y}}^{∞}, where y is a trivial point. If B ⊂ B_{m}, then there is an interpolating Blaschke product q and an x_{0} ∈ Z(q) such that supp μ_{x_{0}} = supp μ_{y}.

Proof. If B ⊂ B_{m}, then by Theorem D of [GI2] there is an interpolating Blaschke product q such that B[ψ̄] is a minimal superalgebra of B. Hence supp μ_{y} is a minimal support set of H^{∞}[ψ̄]. By Theorem 2 of [Gu] there is an x_{0} ∈ Z(ψ) such that supp μ_{x_{0}} = supp μ_{y}.

It would be nice if we could prove that the converse of Lemma 2 is true. I have been unable to do so. There are special cases when the converse holds.

Theorem 1. Let q be a sparse interpolating Blaschke product and set B = H_{N(q̄)}^{∞}. Then B = B_{b} = B_{m}.

Proof. First we have by Theorem C of [GI3] that there is an interpolating Blaschke product ψ (if B ⊂ B_{b}) such that Z(ψ) ∩ M(B) = {x}, and M(B) = M(B[ψ̄]) ∪ P_{x} for some x in M(B). Since N(q̄) = N_{0}(q̄), by Proposition 1 of [GI1] we have that B = ⋂_{y∈Z(q)} H_{supp μ_{y}}^{∞}. We will show that there is a y ∈ Z(q) with y ∈ P_{x}. Once this is done then we have that P_{x} = E_{x} for all x ∈ M(B) and all ψ with Z(ψ)∩M(B) = {x}. Thus both B_{b} and B_{m} are generated by the same minimal superalgebras and we have B_{b} = B_{m} (see Theorem D of [GI3]).

To see that such a y exists, note that if x ∈ M(B) then by [Ga], p. 39, supp μ_{x} ⊂ N(q̄). By Theorem 1 of [Iz] there is a y ∈ Z(q) such that supp μ_{x} ⊂ Q_{y}. We show that y ∈ P_{x}. Since q is sparse we have that y is unique (Lemma 4 of [Iz]), supp μ_{y} ⊂ Q_{y}, and supp μ_{y} is a maximal support set. Thus we have three possibilities:

(1) supp μ_{x} ⊊ supp μ_{y}, (2) supp μ_{x} ∩ supp μ_{y} = ∅, and (3) supp μ_{x} = supp μ_{y}. We show that (3) holds.

Suppose (1) holds. Then by Theorem 2 of [GI2] there is an uncountable set Γ ⊂ Z(ψ) such that supp μ_{m} ⊂ supp μ_{y} for all m ∈ Γ, and supp μ_{m} ∩ supp μ_{n} = ∅ for n, m ∈ Γ for n ≠ m. This implies that Z(ψ) ∩ M(B) is an infinite set, which is a contradiction. So (1) cannot hold.

Now suppose (2) holds. Then supp μ_{x} ∩ supp μ_{y} = ∅, hence ψ̄ ∈ H_{supp μ_{y}}^{∞}. But Z(ψ) ∩ M(B) = {x} implies that ψ̄ ∈ H_{supp μ_{m}}^{∞} for all m ∈ Z(q). This implies that ψ̄ ∈ B, another contradiction. So (3) must hold. Since q is sparse this implies that x ∈ P_{y} (or y ∈ P_{x}).

Since such a y exists we have that B ⊂ B[ψ̄] ⊂ B[q̄]. We’re going to show that B[ψ̄] cannot be a minimal superalgebra of B. Consider the algebras B[q̄] and H^{∞}[q̄]. Theorem 3.2 of [GI4] shows that any subalgebra A of H^{∞}[q̄] is the intersection of a family of maximal subalgebras of H^{∞}[q̄]. Since the set

⋃_{y∈Z(q)} P_{y}

is the set of minimal support points of both B[q̄] and H^{∞}[q̄], we can use the same proof of Theorem 3.2 to show that any subalgebra A of B[q̄] is also the intersection of a family of maximal subalgebras of B[q̄]. For m ∈ Z(q) set B_{m} = B[q̄] ∩ H_{supp μ_{m}}^{∞}. Then we have that

B = ⋂_{m∈Z(q)} B_{m}

and if H_{m}^{∞} = H^{∞}[q̄] ∩ H_{supp μ_{m}}^{∞}, then

H^{∞} + C = ⋂_{m∈Z(q)} H_{m} .

Hence if y ∈ Z(q) such that y ∈ P_{x}, then M(B) = M(B[ψ̄]) ∪ P_{y}, and we get that

B[ψ̄] = ⋂_{\begin{matrix}m∈Z(q) \\ m≠y \end{matrix}} B_{m} .

Thus if B[ψ̄] is a minimal superalgebra of B, then the algebra

A^{∗} = ⋂_{\begin{matrix}m∈Z(q) \\ m≠y \end{matrix}} H_{m}^{∞}

is a minimal superalgebra of H^{∞}+C. This is impossible since H^{∞}+C has no minimal subalgebra ((H^{∞} +C)_{b} = H^{∞} +C). Hence B[ψ̄] is not a minimal superalgebra for B. We get B = B_{b} = B_{m}.

Theorem 2. Let q be an interpolating Blaschke product and set m̃_{q} = {y ∈ m_{q} : supp μ_{y} = supp μ_{t}, t is a trivial point}. Set T = ⋂_{y∈m̃_{q}} H_{supp μ_{y}}^{∞}. Then T = T_{b}. If m̃_{q} = m_{q} then T = T_{m}.

Proof. Suppose T ≠ T_{b}. Then by Theorem C of [GI3] there is an interpolating Blaschke product ψ such that Z(ψ) ∩ M(T) = {y_{0}}. By Theorem 1 of [GI1] we have M(T) = M(T [ψ̄]) ∪ P_{y_{0}} and y_{0} is a minimal support point of H^{∞}[ψ̄]. By Proposition 6 of [MY], (H_{supp μ_{y_{0}}}^{∞})_{b} = H_{supp μ_{y_{0}}}^{∞}[ψ̄]. So by Theorem 5 of [MY], y_{0} is a locally thin point. We show that this is not the case by showing that there is an x_{0} ∈ m̃_{q} with supp μ_{x_{0}} = supp μ_{y_{0}}. Suppose that supp μ_{y_{0}} ≠ supp μ_{x_{0}} for all x_{0} ∈ m̃_{q}. Then for each x_{0} ∈ m̃_{q} one of the following can occur: (i) supp μ_{x_{0}} ⊊ supp μ_{y_{0}}, (ii) supp μ_{x_{0}} ∩ supp μ_{y_{0}} = ∅, or (iii) supp μ_{y_{0}} ⊊ supp μ_{x_{0}}. We will show that none of these can actually happen. If (i) is true, then |ψ(x_{0})| = 1 implies that ψ̄ ∈ H_{supp μ_{x_{0}}}^{∞} since y_{0} is a minimal support point. If (ii) is true for all x_{0} ∈ m̃_{q} then again ψ̄ ∈ H_{supp μ_{x_{0}}}^{∞}. So if (i) or (ii) happens for all x_{0} ∈ m̃_{q}, then ψ̄ ∈ T, which implies that T [ψ̄] = T. So T_{b} = T here. Now if (iii) happens, then by Theorem 2 of [GI2] there is an uncountable set Γ ⊂ Z(ψ) such that supp μ_{α} ⊊ supp μ_{x_{0}} for all α ∈ Γ and supp μ_{α} ∩ supp μ_{β} = ∅ if α ≠ β and α, β ∈ Γ. Since x_{0} ∈ M(T), each α ∈ Γ is also in M(T). Thus Γ ⊂ Z(ψ)∩M(T). This implies that Z(ψ)∩M(T) ≠ {y_{0}}, which leads to a contradiction. So (iii) cannot hold for any x_{0} ∈ m̃_{q}. So if there is a y_{0} ∈ M(T)∩Z(ψ) such that T [ψ̄] is a minimal superalgebra for T, then there is an x_{0} ∈ m̃_{q} with supp μ_{y_{0}} = supp μ_{x_{0}}. This implies that (H_{supp μ_{y_{0}}}^{∞})_{b} = (H_{supp μ_{y_{0}}}^{∞})_{b} = H_{supp μ_{x_{0}}}^{∞}[ψ̄]. By the remark following Theorem 5 of [MY] we have that x_{0} is not a locally thin point. So no such y_{0} exist and we have T = T_{b}.

To show that T = T_{m} if m̃_{q} = m_{q} we proceed as follows. By the argument above, if T ≠ T_{m} there is a x_{0} ∈ m̃_{q} such that {λ ∈ M(T) : |ψ(λ)| < 1} = E_{x_{0}}, y_{0} ∈ E_{x_{0}}, and {y_{0}} = Z(ψ) ∩ M(T). We show that this is a contradiction by showing the set Z(ψ) ∩ m̃_{q} contains an uncountable set. This will suffice since m̃_{q} ⊂ M(T). Without loss of generality we can assume that if x, y ∈ m̃_{q}, x ≠ y, then supp μ_{x} ∩supp μ_{y} = ∅. Let {z_{n}}_{n=1}^{∞} be the zero sequence of q in D. Then there is a subnet {z_{n_{α}}}_{α∈A} such that z_{n_{α}} → x_{0}. Let ϵ > 0 and set B = A \ {n_{α} : |ψ(z_{n_{α}})| > 1 − ϵ}. Since |ψ(x_{0})| < 1 and the subnet {|ψ(z_{n_{α}})|} converges to |ψ(x_{0})|, there is an α_{0} ∈ A such that if α ≥ α_{0}, we have n_{α} ∈ B. Hence if α_{1} > α_{0} we have that |ψ(z_{n_{α}})| < 1 − ϵ for all α ≥ α_{1}. Take the subnet \{z_{n_{α}}\}_{α∈B}. Then on the set \overline{\{z_{n_{α}}\}_{α∈B}} \ \{z_{n_{α}}\}_{α∈B} we have that |ψ| < 1 − ϵ. Take a subsequence {z_{n_{k}}} of the subnet {z_{n_{α}}}_{α∈B}. Let q_{0} be the factor of q with zero sequence \{z_{n_{k}}\}. Then Z(q_{0}) = \overline{\{z_{n_{k}}\}} \ \{z_{n_{k}}\} and |ψ(u)| < 1 for all u ∈ Z(q_{0}). By Theorem 2 of [GI1] there is an uncountable set Γ ⊂ Z(q_{0}) such that Γ ⊂ m̃_{q}. By Theorem 2 of [GI1] we can assume that if α, β ∈ Γ, α ≠ β, then supp μ_{α} ∩ supp μ_{β} = ∅. For each α ∈ Γ there is an x_{α} ∈ Z(ψ) such that supp μ_{x_{α}} ⊆ supp μ_{α}. This implies that x_{α} ∈ M(T) since M(T) = \overline{⋃_{y∈m̃_{q}} M(H_{supp μ_{y}}^{∞})}. Thus we have that x_{α} ∈ Z(ψ) ∩ M(T). So the set Z(ψ) ∩ M(T) is uncountable. This shows that T = T_{m}.

Both B_{b} and B_{m} are generated by a special type of minimal superalgebras (determined by the character of the minimal support point).

Under certain conditions we can determine the Bourgain algebras and the minimal envelope algebras of maximal subalgebras of a Douglas algebra B (here B will have a maximal subalgebra).

Theorem 3. Let B be a Douglas algebra such that B_{b}[q̄] = B[q̄] or B_{m}[q̄] = B[q̄] for some interpolating Blaschke product q. Let A_{x} be the maximal subalgebra of B[q̄] for which x is the corresponding minimal support point of B[q̄]. Then either: (i). A_{x} ⊂ (A_{x})_{b} and (A_{x})_{b} = (A_{x})_{m} = B[q̄], or (ii). A_{x} = (A_{x})_{b} and (A_{x})_{m} = B[q̄].

Proof. By using Theorem 3 of [MY], Theorem 3 of [GIM], and Theorems 4 and 5 of [GI3], our hypothesis implies that B[q̄] = (B[q̄])_{b} = (B[q̄])_{m}.

Now let A_{x} be any maximal subalgebra of B[q̄] associated with the minimal support point x. First we assume that x is a locally thin point (see Theorem 5 of [MY]). Then the maximal ideal space of B[q̄] and A_{x} are related by the equation

M(A_{x}) = M(B[q̄]) ∪ P_{x} .

Then B[q̄] is a minimal superalgebra of A_{x}, hence B[q̄] ⊆ (A_{x})_{b}. Now, using Theorem 3 of [MY] again, we get

B[q̄] = (B[q̄])_{b}
= (A_{x}[q̄])_{b}
= (A_{x})_{b}[q̄]
= (A_{x})_{b} since q̄ ∈ (A_{x})_{b} .

Similarly B[q̄] = (A_{x})_{m} = (A_{x})_{b} if x is a locally thin point. This proves (i).

Now suppose that x is not a locally thin point (for example, P_{x} is not a homeomorphic disk, see [Ga]). Then M(A_{x}) and M(B[q̄]) are related by

M(A_{x}) = M(B[q̄]) ∪ E_{x}

with P_{x} ⊊ E_{x} (or Z(q) ∩ P_{x} has infinitely many points). Using Theorem 4 of [GI3] we have

B[q̄] = (B[q̄])_{m}
= (A_{x}[q̄])_{m}
= (A_{x})_{m}[q̄] by Theorem 4 of [Gu]
= (A_{x})_{m} since q̄ ∈ (A_{x})_{m} .

Now, for any Douglas algebra A we have that A ⊆ A_{b} ⊆ A_{m}. Thus

A_{x} ⊆ (A_{x})_{b} ⊊ B[q̄] = (A_{x})_{m} .

Since A_{x} is a maximal subalgebra of B[q̄], we have that A_{x} = (A_{x})_{b} if x is not locally thin. This proves (ii).

Corollary 1. Let q be any interpolating Blaschke product and set B = H^{∞}[q̄]. Let A_{x} be any maximal subalgebra of B that corresponds to the minimal support point of B. Then either (i). A_{x} ⊂ (A_{x})_{b} and (A_{x})_{b} = (A_{x})_{m} = B, or (ii). A_{x} = (A_{x})_{b} and (A_{x})_{m} = B.

Corollary 2. Let A be any Douglas algebra such that A = A_{b} or A = A_{m}, and let q be any interpolating Blaschke product such that q̄ ∉ A. Set B = A[q̄] and let B_{x} be any maximal subalgebra of B corresponding to the minimal support point of A[q̄]. Then either (i). B_{x} ⊂ (B_{x})_{b} and (B_{x})_{b} = (B_{x})_{m} = B, or (ii). B_{x} = (B_{x})_{b} and (B_{x})_{m} = B.

Theorem 4. Let B be a Douglas algebra that has a maximal subalgebra A_{x}, where x is the minimal support point of B corresponding to A_{x}. Then (A_{x})_{m} = B_{m}.

Proof. By Theorem 4 of [GI3] we have that (A_{x})_{m} ⊆ B_{m} since A_{x} ⊆ B. Since A_{x} is a maximal subalgebra of B there is an x_{0} ∈ M(A) \ M(B) and an interpolating Blaschke product ψ_{0} such that

M(A_{x}) = M(A[ψ̄_{0}]) ∪ E_{x_{0}}
= M(B) ∪ E_{x_{0}} .

So by Theorem D of [GI3] we have B ⊆ (A_{x})_{m}. If B_{0} is another minimal superalgebra containing A_{x}, then there is some y_{0} ∈ M(A_{x}) such that M(A_{x}) = M(B_{0}) ∪ E_{y_{0}}. Hence we have that B_{0} ⊆ (A_{x})_{m} and y_{0} ∈ M(B), otherwise E_{y_{0}} = E_{x_{0}}. To show that B_{m} ⊆ (A_{x})_{m}, let ψ be any interpolating Blaschke product such that ψ̄ ∈ B_{m}. Then by Theorem D of [GI3] we can assume that

{λ ∈ M(B) : |ψ(λ)| < 1} = E_{x}

for some x ∈ M(B). But

{m ∈ M(A_{x}) : |ψ(m)| < 1} = E_{x} ∪ {m ∈ M(A_{x}) : |ψ(m)| < 1} ∩ E_{x_{0}}.

The set on the right hand side is either E_{x} or E_{x} ∪ E_{x_{0}}. Hence by Theorem 4 of [GI3] we have that ψ̄ ∈ (A_{x})_{m}. Hence B_{m} ⊆ (A_{x})_{m}.

For (A_{x})_{b} we have the following special result if we assume an additional assumption.

Theorem 5. Let B be a Douglas algebra that has a maximal subalgebra A_{x}, where x is the minimal support point of B corresponding to A_{x}. Assume that P_{x} is a nonhomeomorphic disk. Then (A_{x})_{b} ⊊ B_{b}.

Proof. Since B ⊆ B_{b} and (A_{x})_{b} ⊆ B_{b}, it suffices to show that B ⊈ (A_{x})_{b}. Since A_{x} is a maximal subalgebra of B corresponding to x, by Theorem 1 of [GI1] we have

M(A_{x}) = M(B) ∪ E_{x} .

Note that P_{x} ⊂ E_{x}. Hence if ψ is any interpolating Blaschke product, we have by Corollary 1.5 of [GLM] the set M(A_{x}) ∩ Z(ψ) ⊇ P_{x} ∩ Z(ψ) is an infinite set. By Theorem 2 of [GIM], ψ̄ ∉ (A_{x})_{b}. Hence B ⊈ (A_{x})_{b}.

The following two propositions on minimal support points seem to indicate that the sets given in them are smaller in some sense than the set in the following two well-known facts.

Fact 1. Let B be any Douglas algebra. Then an interpolating Blaschke product q is invertible in B if and only if Z(q) ∩ M(B) = ∅.

Fact 2. For any Douglas algebra B we have

B = ⋂_{x∈M(B)} H_{supp μ_{x}}^{∞} .

Proposition 1. An interpolating Blaschke product q in invertible in a Douglas algebra B if and only if M(B) ∩ m_{q} = ∅.

Proof. By Theorem 2 of [GI1] we have that m_{q} ⊆ Z(q), hence if m_{q} ∩ M(B) ≠ ∅ then q̄ ∉ B.

To prove the converse, suppose q̄ ∉ B. Then by Fact 1, Z(q) ∩ M(B) ≠ ∅. By the proof of Theorem 2 of [GI1], there is a y_{0} ∈ Z(q) such that supp μ_{y_{0}} ⊆ supp μ_{x} for any x ∈ Z(q) ∩ M(B) and y_{0} ∈ m_{q}. Since x ∈ M(B) we have that M(H_{supp μ_{x}}^{∞}) ⊂ M(B). Since M(H_{supp μ_{x}}^{∞}) = M(L^{∞}) ∪ {λ ∈ M(H^{∞} + C) : supp μ_{λ} ⊆ supp μ_{x}} we have that y_{0} ∈ M(B). Hence m_{q} ∩ M(B) ≠ ∅.

Let B be any Douglas algebra and set m_{q}(B) = m_{q} ∩ M(B). Let

M_{B} = ⋃{m_{q}(B) : q is an interpolating Blaschke product, q̄ ∉ B} .

Let Γ(B) = {x_{α}}_{α∈Λ} be the family of all minimal support points from M_{B} such that supp μ_{x_{α}} ∩ supp μ_{x_{β}} = ∅ if α ≠ β. Then

Proposition 2. B = ⋂_{x_{α}∈Γ(B)} H_{supp μ_{xα}}^{∞}.

Proof. If x ∈ M_{B}, then there is an x_{α} ∈ Γ(B) such that supp μ_{x_{α}} = supp μ_{x}. So it suffices to show that

B = ⋂_{x∈M_{B}} H_{supp μ_{x}}^{∞} .

Set B_{0} = ⋂_{x∈M_{B}} H_{supp μ_{x}}^{∞}. Since M_{B} ⊆ M(B), by Fact 2 we have that B ⊆ B_{0}. Suppose B ⊊ B_{0}. Then by the Chang-Marshall Theorem [Ch, Ma] there is an interpolating Blaschke product q such that q̄ ∈ B_{0} but q̄ ∉ B. Hence there is a y ∈ M(B) such that q(y) = 0. Hence q̄ ∉ H_{supp μ_{y}}^{∞}. By Theorem 2 of [GI1] (or Proposition 1) there is a y_{0} ∈ Z(q) such that supp μ_{y_{0}} ⊆ supp μ_{y} and y_{0} ∈ M_{B}. This implies that q̄ ∉ H_{supp μ_{y_{0}}}^{∞}, so q̄ cannot be invertible in B_{0}. Thus B_{0} = B. We are done.

References

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[GI1] Guillory, C. J. and Izuchi, K., “Maximal Douglas subalgebras and minimal support points”, Proc. Amer. Math. Soc., 116 (1992), 477–481.

[GI2] Guillory, C. J. and Izuchi, K., “Interpolating Blaschke products and nonanalytic sets”, J. Complex Variable, 23 (1993), 163–175.

[GI3] Guillory, C. J. and Izuchi, K., “Minimal envelopes of Douglas algebras and Bourgain algebras”, Houston J. Math., 19 (1993), 201–222.

[GI4] Guillory, C. J. and Izuchi, K., “Interpolating Blaschke products of type G”, to appear in J. Complex Variables.

[Iz] Izuchi, K., “QC-level sets and quotients of Douglas algebras”, J. Funct. Anal., 65 (1986), 293–308.

[MY] Mortini, R. and Younis, R., “Douglas algebras which are invariant under the Bourgain map”, Arch. Math., vol. 59 (1992), 371–378.

[Ma] Marshall, D., “Subalgebras of L^{∞} containing H^{∞}”, Acta. Math., 137 (1976), 91–98.

Department of Mathematics, University of Southwestern Louisiana, Lafayette, LA 70504

Current address: Mathematical Sciences Research Institute, 100 Centennial Drive, Berkeley, CA 94720

E-mail address: cjg@msri.org