ALGEBRAS ASSOCIATED WITH BLASCHKE PRODUCTS OF TYPE G

CARROLL GUILLORY AND KIN Y. LI

Abstract. Let Ω and Ω_{fin} be the sets of all interpolating Blaschke products of type G and of finite type G, respectively. Let E and E_{fin} be the Douglas algebras generated by H^{∞} together with the complex conjugates of elements of Ω and Ω_{fin}, respectively. We show that the set of all invertible inner functions in E is the set of all finite products of elements of Ω , which is also the closure of Ω among the Blaschke products. Consequently, finite convex combinations of finite products of elements of Ω are dense in the closed unit ball of the subalgebra of H^{∞} generated by Ω. The same results hold when we replace Ω by Ω_{fin} and E by E_{fin}.

1. Introduction

Let D be the open unit disk and T be the unit circle on the complex plane. Let H^{∞} be the Banach algebra of bounded analytic functions on the open unit disk D. Via radial limits we can consider H^{∞} as a closed subalgebra of L^{∞}, where L^{∞} is the family of all essential bounded measurable functions on T. Any function h in H^{∞} with |h| = 1 a.e. on T is called an inner function. Let {z_{n}} be a sequence in D with ∑_{n}(1 − |z_{n}|) < ∞. Then the function

b(z) = ∏_{n} \frac{z̄_{n}}{|z_{n}|} \frac{z_{n} − z}{1 − z̄_{n}z} for z ∈ D,

is called a Blaschke product with roots {z_{n}}. Let

δ(b) = inf_{k} ∏_{n≠k} |\frac{z_{k} − z_{n}}{1 − z̄_{n}z_{k}} | .

If δ(b) > 0, then b and {z_{n}} are called interpolating. By [Ca], if δ(b) > 0, then for every bounded sequence {a_{n}}, there exists f in H^{∞} such that f(z_{n}) = a_{n} for every n. If

lim_{k→∞} ∏_{n≠k} |\frac{z_{k} − z_{n}}{1 − z̄_{n}z_{k}} | = 1,

then b and {z_{n}} are called thin or sparse.

We denote by M(H^{∞}) the maximal ideal space of H^{∞}. A closed subalgebra B between H^{∞} and L^{∞} is called a Douglas algebra. We denote by M(B) the maximal ideal space of the Douglas algebra B. For an interpolating Blaschke product b, we denote by H^{∞}[b̄] the Douglas algebra generated by H^{∞} and the complex conjugate of b. For a function f in H^{∞}, let

Z(f) = {x ∈ M(H^{∞}) : f(x) = 0}

and for 0 < c ≤ 1,

{|f| < c} = {x ∈ M(H^{∞}) : |f(x)| < c}.

For a point x in M(H^{∞}), there is a representing measure μ_{x} on M(L^{∞}), that is,

f(x) = ∫_{M(L^{∞})} f dμ_{x}

for every f ∈ H^{∞}. We denote by supp μ_{x} the support set for the representing measure μ_{x}.

By the Corona Theorem, D can be considered as a dense subset of M(H^{∞}). For points x, y in M(H^{∞}), let

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

and put

P (x) = {m ∈ M(H^{∞}) : ρ(m, x) < 1}.

The set P (x) is called the Gleason part containing x. For z, w ∈ D, we have

ρ(z, w) = |\frac{z − w}{1 − w̄z} |

and P (0) = D. We call x ∈ M(H^{∞}) a trivial part if P (x) = {x}. Let

G = ⋃{P (x) : x ∈ M(H^{∞}), P (x) ≠ {x}}.

Then G is an open subset of M(H^{∞}).

By Hoffman’s work [Ho], Z(b) ⊂ G for every interpolating Blaschke product b and for each x in G, there exists an interpolating Blaschke product b such that b(x) = 0. Also by [Ho], for each x ∈ G there exists a one-to-one and onto map L_{x} : D → P (x) such that f ∘ L_{x} ∈ H^{∞} for every f ∈ H^{∞}. The map L_{x} is given as follows. Let {z_{α}} be a net in D with z_{α} → x and let L_{z_{α}}(z) = (z + z_{α})/(1 + z̄_{α}z). Then

(f ∘ L_{x})(z) = lim_{α}(f ∘ L_{z_{α}})(z) for f ∈ H^{∞} and z ∈ D.

A Blaschke product b is of type G if it is interpolating and {|b| < 1} ⊂ G. It is of finite type G if it is of type G and for every x ∈ Z(b) the set Z(b) ∩ P (x) is finite. A Blaschke product is locally thin if for each x ∈ Z(b) there is an interpolating Blaschke product q such that

lim_{α}(1 − |z_{n_{α}}|^{2})|q′(z_{n_{α}})| = 1

whenever {z_{α}} is a subnet of the root sequence {z_{n}} of q that converges to x. Note q may be different from b. In fact, by [Go-Li-Mo], if b = q for every x ∈ z(b), then b is a thin Blaschke product. Blaschke products of type G, finite type G and locally thin Blaschke products are very important in the studies of Douglas algebras (see for example, [Go-Li-Mo], [Gu], [Gu-Iz-1] and [Gu-Iz-2]).

Let Ω be the family of all interpolating Blaschke products of finite type G and A be the closed subalgebra of H^{∞} generated by Ω. Let B = [A, Ā] be the smallest (closed) C^{∗}-subalgebra of L^{∞} containing A. That is, B is generated by the ratio of interpolating Blaschke products of type G. Then E = [H^{∞}, Ā] will be the Douglas algebra generated by H^{∞} and the complex conjugate of elements of Ω.

Our main results are that every inner function u in E is a finite product of interpolating Blaschke products of type G, from which we are able to identify the closure of the interpolatingBlaschke products of type G among the Blaschke products. As a consequence, we get B = C_{E}, where C_{E} denotes the C^{∗}-subalgebra of L^{∞} generated by the invertible inner functions in E and their complex conjugates. Another consequence is that the finite products of interpolating Blaschke products of type G are the only inner functions that are in B ∩ H^{∞}. Hence by Theorem 4.1 of [Ch-Ma], A = B ∩H^{∞} and finite convex combinations of finite products of interpolating Blaschke products of type G are dense in the closed unit ball of A. For Blaschke product of finite type G, we obtain similar results. In obtaining these results, we follow the approach in [He], but our proofs rely heavily on the results about type G and finite type G developed in [Gu-Iz-1] and [Gu-Iz-2].

Both authors would like to thank MSRI for support and hospitality.

2. Results For Type G

We begin with a few useful propositions concerning basic properties of interpolating Blaschke product of type G.

Proposition 1. If B is of type G and b is a subproduct of B, then b is of type G. If b_{1}, b_{2} are of type G and b_{1}b_{2} is an interpolating Blaschke product, then b_{1}b_{2} is of type G. If b is of type G and b_{λ} = (b − λ)/(1 − λ̄b) is an interpolating Blaschke product, then b_{λ} is of type G. The statements also hold if type G is replaced by finite type G.

Proof. For type G, the first statement follows from {|b| < 1} ⊂ {|B| < 1} ⊂ G. The second statement follows from {|b_{1}b_{2}| < 1} = {|b_{1}| < 1} ∪ {|b_{2}| < 1} ⊂ G. The third statement follows from {|b_{λ}| < 1} = {|b| < 1} ⊂ G.

For finite type G, the first statement follows from Z(b)∩P (m) ⊂ Z(B)∩P (m). The second statement follows from Z(b_{1}b_{2})∩P (m) = (Z(b_{1})∩P (m))∪(Z(b_{2})∩P (m)). The third statement follows from Theorem 3.2 (iii) of [Gu-Iz-2] because H^{∞}[b̄_{λ}] = H^{∞}[b̄] by considering their maximal ideal spaces.

We remark that not all of the statements in Proposition 1 are true for the family of thin Blaschke products.

Proposition 2. Suppose b is an interpolating Blaschke product of type G with roots {z_{n}} in D. Let q be an interpolating Blaschke product with roots {w_{n}} in D such that ρ(w_{n}, z_{n}) ≤ r for all n and for some r < 1. Then q is of type G.

Proof. Suppose 0 < λ < 1 and z ∈ D such that |q(z)| < λ. Then, by Lemma 1.4 and Corollary 1.3 on page 4 of [Gar],

|b(z)| = ∏_{n=1}^{∞} ρ(z, z_{n}) ≤ ∏_{n=1}^{∞}(\frac{ρ(z, w_{n}) + r}{1 + rρ(z, w_{n})}) ≤ \frac{|q(z)| + r}{1 + r|q(z)|} < \frac{λ + r}{1 + rλ} = λ′ < 1.

So we have

{|q| < 1} = ⋃_{0<λ<1}{|q| < λ} ⊂ ⋃_{0<λ′<1}{|b| < λ′} = {|b| < 1} ⊂ G.

Proposition 3. Let ℱ = {x ∈ M(H^{∞}) : x is in the closure of some interpolating sequences in D whose Blaschke products are of type G}. Then ℱ is the union of a family of nontrvial Gleason parts.

Proof. By the definition of ℱ, every point in ℱ belongs to a nontrivial Gleason part. So let m_{0} ∈ ℱ and m ∈ P (m_{0}). Then m_{0} ∈ \overline{\{z_{n}\}} for some interpolating sequence {z_{n}} whose Blaschke product b(z) is of type G. So there is a subnet {z_{α}} of {z_{n}} converging to m_{0}. Since m ∈ P (m_{0}), there is ζ ∈ D such that

lim_{α} L_{z_{α}}(ζ) = L_{m_{0}}(ζ) = m.

Let

ζ_{n} = L_{z_{n}}(ζ) = \frac{ζ + z_{n}}{1 + z̄_{n}ζ} .

Then ρ(ζ_{n}, z_{n}) = |ζ| < 1 for all n. By Corollary 1.6 on page 407 of [Gar], there is a factorization b = b_{1}b_{2} ⋯ b_{k} with

δ(b_{j}) > \frac{2|ζ|}{1 + |ζ|^{2}} for j = 1, 2, …, k.

By Lemma 5.3 on page 310 of [Gar], each Z(b_{j}) ∩ D = {z_{j,n}} is interpolating. Since

\overline{\{z_{n}\}} = ⋃_{j=1}^{k} \overline{Z(b_{j})}

and the Z(b_{j})’s have disjoint closures [Gar, p. 422], it follows that m_{0} ∈ \overline{Z(b_{j})} = \overline{\{z_{j,n}\}} for some j and the net {z_{α}} is eventually in Z(b_{j}) because

M(H^{∞}) \ ⋃_{i≠j} \overline{Z(b_{i})}

is an open neighborhood of \overline{Z(b_{j})} and m_{0} ∈ \overline{Z(b_{j})}.

By Proposition 1, each b_{j}(z) is of type G. By Proposition 2 (and the above argument), the Blaschke product with roots {ζ_{j,n}} is of type G. Finally,

lim_{α} ζ_{j,α} = lim_{α} L_{z_{j,α}}(z) = L_{m_{0}}(ζ) = m

and our assertion follows.

Proposition 4. An interpolating Blaschke product b of type G has modulus 1 on those Gleason parts of M(H^{∞}) that do not contain a point in Z(b). Since Z(b) equals the closure of Z(b) ∩ D and ℱ is the union of a family of Gleason parts, b has in particular modulus 1 on M(H^{∞}) \ ℱ.

Proof. By Lemma 1.1 of [Gu-Iz-2] (or Theorem 1 of [Gu-Iz-1]) we have

{|b| < 1} = ⋃_{m∈Z(b)} P (m).

Thus

|b| = 1 on m(H^{∞}) \ ⋃_{m∈Z(b)} P (m).

The second statement follows from the proof of Proposition 3.

Corollary 5. M(E) = M(H^{∞}) \ ℱ.

Proof. By Theorem 1.3 on page 375 of [Gar],

M(E) = {m ∈ M(H^{∞}) : |b(m)| = 1 for all b ∈ Ω}.

Now the results follows immediately from Proposition 4.

Corollary 6. A is a proper subalgebra of H^{∞}.

Proof. This follows because M(H^{∞}) \ (ℱ ∪ M(L^{∞})) is not empty.

Theorem 7. Every invertible inner function in E is a finite product of interpolating Blaschke products of type G.

Proof. Let u be an arbitrary invertible inner function in E, then ū = u^{−1} in E ⊂ L^{∞}. So H^{∞}[ū] ⊂ E. By Corollary 5 above and Theorem 1.3 on page 375 of [Gar]

M(H^{∞}) \ ℱ = M(E) ⊂ M(H^{∞}[ū]) = {|u| = 1}.

Hence {|u| < 1} ⊂ ℱ ⊂ G. This implies Z(u) cannot contain any trivial parts. By Corollary 24 of [McD-Su], u = b_{1}b_{2} ⋯ b_{n}, where each b_{j} is an interpolating Blaschke product. Finally, for each j,

{|b_{j}| < 1} ⊂ ⋃_{k=1}^{n}{|b_{k}| < 1} = {|u| < 1} ⊂ G.

Corollary 8. B = C_{E}, the C^{∗}-subalgebra of L^{∞} generated by the inner functions invertible in E and their complex conjugates.

Corollary 9. Let b be a finite product of interpolating Blaschke products of type G. If f ∈ H^{∞} is such that ∥f∥_{∞} < 1 and f̄b equals on H^{∞} a function g almost everywhere on T, then the function

b_{f}(z) = \frac{b(z) − f(z)}{1 − g(z)}, for z ∈ D,

is a finite product of interpolating Blaschke products of type G.

Proof. Just observe that b_{f} is an invertible inner function in E.

In [Ch-Ma], Chang and Marshall showed that for an arbitrary Douglas algebra J, the closed unit ball of H^{∞}∩C_{J} is the norm-closed convex hull of the Blaschke products in H^{∞} ∩ C_{J}, where C_{J} is the C^{∗}-subalgebra of L^{∞} generated by the invertible inner functions in J and their complex conjugates. They also showed that J = H^{∞} + C_{J} and that D is dense in the maximal ideal space of H^{∞} ∩ C_{J}. In our case J = E, C_{J} = B and we have the following corollary. (Note that an inner function in B ∩ H^{∞} is invertible in E.)

Corollary 10. (a) A = B ∩ H^{∞}, and finite convex combinations of finite products of interpolating Blaschke products of type G are dense in the closed unit ball of A.

(b) E = H^{∞} + B.

(c) D is dense in the maximal ideal space M(A) of A.

3. Results For Finite Type G

Next we will turn to the main results for interpolating Blaschke product of finite type G analogous to those established in section 2. Let Ω_{fin} be the family of all interpolating Blaschke products of finite type G and A_{fin} be the closed subalgebra of H^{∞} generated by Ω_{fin}. Let E_{fin} = [H^{∞}, Ā_{fin}], then E_{fin} is the Douglas algebra generated by H^{∞} and the complex conjugate of elements of Ω_{fin}. We will show that Theorem 7 holds if E is replaced by E_{fin}.

Theorem 11. Every invertible inner function in E_{fin} is a finite product of interpolating Blaschke products of finite type G.

Proof. We will first show that the analog of Proposition 2 is true for interpolating Blaschke products of finite type G. Let b be an interpolating Blaschke product of finite type G with zeros {z_{n}} in D, and let q be an interpolating Blaschke product with zeros {w_{n}} in D such that ρ(z_{n}, w_{n}) ≤ r for all n and some r < 1. The proof of Proposition 2 shows that {|q| < 1} ⊂ {|b| < 1}. Since b is of finite type G, by Theorem 2.1 of [Gu-Iz-2], there is a subproduct b_{0} of b such that {|b_{0}| < 1} = {|q| < 1}. We will show that q is of finite type G.

For x ∈ Z(q), |b_{0}(x)| < 1. By Lemma 1.1 of [Gu-Iz-2], there is an x_{0} ∈ Z(b_{0}) such that x ∈ P (x_{0}). Suppose the set Z(q) ∩ P (x) = Z(q) ∩ P (x_{0}) is infinite. Then, by Theorem 3.1(i) of [Gu-Iz-2], there exist y and y_{0} in Z(q) such that supp μ_{y} ⊊ supp μ_{y_{0}}. Hence there are m and m_{0} in Z(b_{0}) such that y ∈ P (m) and y_{0} ∈ P (m_{0}), but then supp μ_{m} ⊊ supp μ_{m_{0}} (because by page 143 of [Gam], supp μ_{y} = supp μ_{m} and supp μ_{y_{0}} = supp μ_{m_{0}}). Since b_{0} is of finite type G, this contradicts Theorem 3.2(ii) of [Gu-Iz-2]. Thus Z(q) ∩ P (x_{0}) = Z(q) ∩ P (x) must be finite.

Next we remark that the analogs of Propositions 3,4 and Corollaries 5,6 for finite type G also hold by the same reasoning because of the analog of Proposition 2 for finite type G.

Now let u be an invertible inner function in E_{fin} ⊂ E. By Theorem 7, u = u_{1}u_{2} ⋯ u_{m}, where each u_{i} is of type G. Observe that if ℱ_{fin} is the analog of ℱ for finite type G, then

Z(u_{i}) ⊂ \{|u| < 1\} ⊂ ℱ_{fin} = ⋃_{b∈Ω_{fin}}\{|b| < \frac{1}{2}\}.

Let δ_{i} = inf{ρ(w, z) : w, z ∈ Z(u_{i}) ∩ D, w ≠ z} > 0. Since Z(u_{i}) is compact,

Z(u_{i}) ⊂ ⋃_{j=0}^{n_{i}}\{|b_{j}| < \frac{1}{2}\},

for some b_{1}, b_{2}, …, b_{n_{i}} of finite type G. Let

S_{ij} = Z(u_{i}) ∩ \{|b_{j}| < \frac{1}{2}\} ∩ D,

then

Z(u_{i}) ∩ D = ⋃_{j=1}^{n_{i}} S_{ij}.

By removing overlapping elements, we may assume the S_{ij}’s are disjoint. Since b_{j} is of type G, by Lemma 2.1 of [Gu-Iz-2], there is δ < 1 such that

S_{ij} ⊂ \{|b_{j}| < \frac{1}{2}\} ⊂ \{z ∈ D : ρ(z, \{z_{j,n}\}) ≤ δ\},

where {z_{j,n}} is the root sequence of b_{j} in D. For each disk B(z_{j,n}, δ) = {z ∈ D : ρ(z, z_{j,n}) ≤ δ}, there is at most k_{i} elements of S_{ij} in B(z_{j,n}, δ), where k_{i} depends only on δ_{i}. So S_{ij} is the union of at most k_{i} sequences, each of which has at most one element in each B(z_{j,n}, δ). By the analog of Proposition 2 for finite type G, proved above, the Blaschke product with root sequence S_{ij} is a product of at most k_{i} interpolating Blaschke product of finite type G. So u_{i} is a finite product of at most n_{i}k_{i} interpolating Blaschke product of finite type G. Therefore, u is a finite product of interpolating Blaschke product of finite type G.

In general, it is a difficult problem to determine the closure of an infinite set of interpolating Blaschke products among the family of all Blaschke products (see for example [Li]). However, for Blaschke products of type G and finite type G, their closures can be identified because of Proposition 1 and Theorems 7, 11.

Theorem 12. Let ℬ be the family of all Blaschke products with essential sup-norm. The closure of all interpolating Blaschke products of type G in ℬ is the set of all finite products of interpolating Blaschke products of type G. Also, the closure of all interpolating Blaschke products of finite type G in ℬ is the set of all finite products of interpolating Blaschke products of finite type G.

Proof. Suppose B is in the closure of type G products. Take B of type G such that ∥B −b∥_{∞} = ∥1−Bb̄∥_{∞} < 1. It follows that Bb̄ is invertible in E and so is B = (Bb̄)b. By Theorem 7, B is a finite product of interpolating Blaschke products of type G.

For the converse, it suffices to show for B_{1}, B_{2} of type G and ε > 0, there is B of type G such that ∥B_{1}B_{2} − B∥_{∞} < ε. Let the root sequences of B_{1} and B_{2} be {z_{n}} and {w_{n}}, respectively. Since each of these sequences are separated,

δ_{I} = inf{ρ(z_{m}, z_{n}), ρ(w_{m}, w_{n}) : m ≠ n} > 0.

By Hoffman’s lemma (see Lemma 1.4 on pages 404-5 of [Gar]), there are δ_{0}, ε_{0} < δ_{I}/3 such that

V_{n} ⊂ {z ∈ D : ρ(z, z_{n}) < ε_{0}},
W_{n} ⊂ {z ∈ D : ρ(z, w_{n}) < ε_{0}}

and

∥z − \frac{z − (δ_{0}/2)}{1 − (δ_{0}/2)z} ∥_{∞} < ε,

where V_{n} and W_{n} are the components of {z ∈ D : |B_{1}(z)| < δ_{0}} and {z ∈ D : |B_{2}(z)| < δ_{0}} containing z_{n} and w_{n}, respectively. Since 3ε_{0} < δ_{I}, we have

ρ(V_{n}, V_{m}) > δ_{I}/3 and ρ(W_{n}, W_{m}) > δ_{I}/3 forn ≠ m.

Factor B_{2} = B_{3}B_{4} so that w_{n} ∈ Z(B_{3}) if ρ(w_{n}, Z(B_{1})) ≥ δ_{0}/4 and w_{n} ∈ Z(B_{4}) if ρ(w_{n}, Z(B_{1})) < δ_{0}/4. Let

B_{5} = \frac{B_{4} − (δ_{0}/2)}{1 − (δ_{0}/2)B_{4}}

and B = B_{1}B_{3}B_{5}, then ∥B_{1}B_{2} − B∥_{∞} = ∥B_{4} − B_{5}∥_{∞} < ε.

Next we will show B is an interpolating Blaschke product. Since ρ(Z(B_{1}), Z(B_{3})) ≥ δ_{0}/4, B_{1}B_{3} is an interpolating Blaschke product. By Lemma 1.4 on pages 404-5 of [Gar], B_{5} is an interpolating Blaschke product.

To see ρ(Z(B_{1}B_{3}), Z(B_{5})) > 0, let w ∈ Z(B_{5}) and z ∈ Z(B_{1}B_{3}). Then w ∈ W_{n} for some n and w_{n} ∈ Z(B_{4}). If m ≠ n, then

\frac{δ_{0}}{2} = ρ(B_{5}(w), B_{5}(w_{n})) ≤ ρ(w, w_{n}) < ε_{0} < ρ(w, w_{m}).

Since w_{n} ∈ Z(B_{4}), we have ρ(w_{n}, z_{k}) < δ_{0}/4 for some z_{k} ∈ Z(B_{1}).

In the case z ∈ Z(B_{1}) and ρ(z, Z(B_{4})) < ε_{0}/4, there is w_{m} ∈ Z(B_{4}) such that ρ(z, w_{m}) < δ_{0}/4 and so

ρ(w, z) ≥ ρ(w, w_{m}) − ρ(w_{m}, z) ≥ \frac{δ_{0}}{2} − \frac{δ_{0}}{4} = \frac{δ_{0}}{4} .

In the case z ∈ Z(B_{1}) and ρ(z, Z(B_{4})) ≥ δ_{0}/4, we have z ≠ z_{k}, hence ρ(z, z_{k}) ≥ δ_{I}. So

ρ(w, z) ≥ ρ(z, z_{k}) − ρ(z_{k}, w_{n}) − ρ(w_{n}, w) ≥ δ_{I} − \frac{δ_{0}}{4} − ε_{0} ≥ \frac{δ_{I}}{2} .

In the case z ∈ Z(B_{3}), we have

ρ(w, z) ≥ ρ(z, w_{n}) − ρ(w_{n}, w) ≥ δ_{I} − ε_{0} ≥ \frac{2δ_{I}}{3} .

So ρ(Z(B_{1}B_{3}), Z(B_{5})) > 0. Therefore, B = B_{1}B_{3}B_{5} is an interpolating Blaschke product.

Finally since B_{1}, B_{2} are of type G, by Proposition 1, B is of type G. This completes the proof of the statement for the closure of type G. For the closure of finite type G, use Theorem 11 instead of Theorem 7 and repeat the above proof.

4. Questions

Let A_{loc} be the closed subalgebra of H^{∞} generated by the locally thin Blaschke product and let E_{loc} = [H^{∞}; Ā_{loc}].

1. Does Theorem 7 hold if E is replaced with E_{loc}?

2. Does E = E_{loc} or does E_{loc} = E_{fin}, or neither? If we let A^{∗} be the closed subalgebra of H^{∞} generated by the thin Blaschke products, and set E^{∗} = [H^{∞}; Ā^{∗}],the main result of Hedenmalm [He, Theorem 2.6] asserts that Theorem 7 holds for E^{∗}. By Proposition 1.1 and Example 3.1 of [Gu-Iz-2], there exists a Blaschke product b of finite type G, which is not a finite product of thin Blaschke products. It follows that E^{∗} ⊊ E_{fin} because otherwise E_{fin} = E^{∗} would contain H^{∞}[b̄] forcing b to be a finite product of thin Blaschke products by [He, Theorem 2.6]. Also, by the proof of Theorem 2 of [Gu-Iz-1], there exists a Blaschke product of type G, but not of finite type G. So we have E_{fin} ⊊ E. By Lemma 1 of [Gu], it can be shown that E_{fin} ⊆ E_{loc}, but it is not clear whether E_{loc} ⊆ E or E_{loc} ⊇ E.

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Carroll Guillory, Department of Mathematics, University of Southwestern Lou9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999