BLASCHKE INDUCTIVE LIMITS OF UNIFORM ALGEBRAS

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BLASCHKE INDUCTIVE LIMITS OF UNIFORM ALGEBRAS

S. A. GRIGORYAN and T. V. TONEV (Received 16 February 2001)

Abstract.We consider and study Blaschke inductive limit algebras A(b), defined as inductive limits of disc algebrasA(D)linked by a sequenceb= {B_k}^∞_k=1of finite Blaschke products. It is well known that big G-disc algebras AG over compact abelian groupsG with ordered dualsΓ=G⊂Qcan be expressed as Blaschke inductive limit algebras. Any Blaschke inductive limit algebraA(b)is a maximal and Dirichlet uniform algebra. Its Shilov boundary∂A(b)is a compact abelian group with dual group that is a subgroup ofQ. It is shown that a bigG-disc algebraA_Gover a groupGwith ordered dualG⊂Ris a Blaschke inductive limit algebra if and only ifG⊂Q. The local structure of the maximal ideal space and the set of one-point Gleason parts of a Blaschke inductive limit algebra differ dras- tically from the ones of a bigG-disc algebra. These differences are utilized to construct examples of Blaschke inductive limit algebras that are not bigG-disc algebras. A neces- sary and sufficient condition for a Blaschke inductive limit algebra to be isometrically isomorphic to a bigG-disc algebra is found. We consider also inductive limitsH^∞(I)of algebrasH^∞, linked by a sequenceI= {I_k}^∞_k=1of inner functions, and prove a version of the corona theorem with estimates for it. The algebraH^∞(I)generalizes the algebra of bounded hyper-analytic functions on an open bigG-disc, introduced previously by Tonev.

2000 Mathematics Subject Classification. 46J15, 46J20, 30H05.

1. Introduction. LetT= {z∈C:|z| =1}denote the unit circle and letD= {z∈C:

|z| ≤1}be the closed unit disc inC. Consider an inductive sequence A

T1

ⁱ²1

→A T2

ⁱ³2

→A T3

ⁱ⁴3

→ ··· (1.1)

of disc algebras A(Tk)=A(T) linked by homomorphisms i^k_k⁺¹:A(Tk)→A(Tk+1).

Every conjugate mapping(i^k+1_k )^∗:ᏹk←ᏹk+1maps the maximal ideal spaceᏹk+1≈D¯ ofA(Tk+1)into the maximal ideal spaceᏹk≈D¯ofA(Tk). Sincei^k+1_k (f )=f◦(i^k+1_k )^∗∈ A(Tk+1)for everyf∈A(Tk), the mapping(i^k_k⁺¹)^∗is an analytic function preserving the unit disc. The inverse limit

D¯1 (i²₁)^∗

←D¯2 (i³₂)^∗

← D¯3 (i⁴₃)^∗

←D¯4 (i⁵₄)^∗

←··· ← Ᏸ (1.2) is the maximal ideal space of the inductive limit algebra

lim→

k→∞

A(Tk), i^k+1_k

, (1.3)

where the closure is taken inC(Ᏸ). In general, the mappings(i^k+1_k )^∗are not obliged to map the unit circleTk+1onto itself. The most interesting situations, though, are

the ones when they do, and this is what we will assume in the sequel. In effect, the mappings(i^k+1_k )^∗become finite Blaschke products

Bk(z)=e^iθn_k s=1

z−zs,k

1−z¯s,kz

, 0<zs,k<1 (1.4) on ¯D. The inductive limit algebra[lim→^k→∞{A(Tk), i^k+1_k }]in this case is called aBlaschke inductive limit algebra. Note that all algebrasi^k+1_k (A(Tk))are algebraic extensions of the disc algebra that are isometrically isomorphic to the disc algebra itself. Indeed, letAbe an algebra and letA[x]be the algebra of polynomials inxoverA. For a given unital polynomialp(x)=xⁿ+a1xⁿ⁻¹+···+an, aj∈AinA[x]the setp(x)A[x]is an ideal in the algebraA[x]. Recall that thealgebraic extensionofAbyp(x)is the algebra

Ap=A[x]/

p(x)A[x]

. (1.5)

Apis isometrically isomorphic toA(T)if and only if the diagram

A(T) ⁱ

id

Ap

π

A(T) ^j A(T)

(1.6)

is commutative, whereiis the natural embeddingi:A(T)→Ap, andπ:Ap→A(T) is an isomorphism. In this case the homomorphismj=π◦i:A(T)→A(T)coincides with the composition operatorCB=f◦Bdefined by a finite Blaschke productB, that is,(j(f (z)))=(CB(f ))(z)=f (B(z)).

LetGbe a compact abelian group, whose dual groupGis isomorphic to a subgroup Γ of R. Denote byAG thebig G-disc algebra generated by Γ, that is,AG is the uni- form algebra onGgenerated by the semigroup of characters{χ^a∈G:a∈Γ₊}, where Γ₊= {a∈Γ:a≥0}is thepositive partofΓ. The elements inAGare referred to asgener- alizedG-analytic functionsonG. InSection 2we review some results on finite Blaschke products and generalizedG-analytic functions. InSection 3, we show that Blaschke inductive limit algebras share many properties with bigG-disc algebras. We give also necessary and sufficient conditions on a groupΓ ⊂Rso that the bigG-disc algebra AG,G=Γ can be expressed as the inductive limit of a Blaschke sequence of (algebraic extensions of) disc algebras. InSection 4, we study annulus type Blaschke inductive limit algebras. The local structure of Blaschke inductive limit algebras is studied in Section 5. We construct Blaschke inductive sequences of disc algebras whose limits are not bigG-disc algebras. InSection 6, we describe the one-point Gleason parts in the maximal ideal space of a Blaschke inductive limit algebra. This description plays a crucial role inSection 7, where we find necessary and sufficient conditions for a Blaschke inductive limit algebra to be expressed as a bigG-disc algebra. InSection 8, we consider inductive limits of algebrasH^∞that are linked by inner functions, and prove the corona theorem for them.

2. Preliminaries. Here we state several basic results on finite Blaschke products and generalizedG-analytic functions, we will need further. Given a uniform algebra A,ᏹA and∂Awill denote the maximal ideal space and Shilov boundary ofAcorre- spondingly. Any homomorphismϕ:A→Bbetween two uniform algebras naturally generates a conjugate mapϕ^∗:ᏹA←ᏹBbetween their maximal ideal spaces. If, in addition,ϕis an isometry, that is, if

ϕ(g)_B= gA (2.1)

for everyg∈A, thenϕis called anembeddingofAintoB.

Lemma2.1. LetAandBbe uniform algebras. A homomorphismϕ:A→Bgenerates an embedding ofAintoBif and only ifϕ^∗(∂B)⊃∂A.

Proof. Note that for everyg∈Awe have

m∈maxϕ^∗(∂B)

m(g)=max

s∈∂B

ϕ^∗(s)

(g)=max

s∈∂B

s

ϕ(g)=ϕ(g)_B. (2.2) Ifϕ^∗(∂B)⊃∂A, thengA=max_m∈∂A|m(g)| = ϕ(g)B. Henceϕis an isometry.

On the other hand, ifϕis an isometry, then

m∈ϕmax^∗(∂B)

m(g)=ϕ(g)_B= (g)A (2.3)

implies that the setϕ^∗(∂B)is a boundary forA. Thereforeϕ^∗(∂B)⊃∂A.

Note that every embeddingj:A(T)→A(T)of the disc algebra into itself generates an isometric isomorphism betweenA(T)andj(A(T)). Hencej^∗:ᏹj(A(T))→ᏹA(T)D¯ is a homeomorphism andj^∗∂(j(A(T)))=∂A(T)=T. If, in addition,ᏹj(A(T))=D¯and

∂(j(A(T)))=T, thenj^∗(T)=T, and hence the functionj^∗is a finite Blaschke product (see [6], Chapter I, 2). Consequently, for any isometryj:A(T)→A(T)with the above properties there is a Blaschke product

B(z)=e^iθn k=1

z−zk

1−z¯kz

, 0<zk<1, (2.4) such that

(j◦f )(z)=f◦j^∗(z)=f B(z)

∀f∈A(T). (2.5)

Recall thatz0∈Dis acritical point forBifB(z0)=0, that is, if card(B⁻¹(z0)) <

ordB. By the Brower’s fixed point theorem,Balways has a fixed point, that is,B(z0)= z0for somez0∈D¯. If the order ofB is greater than 1 then by the Schwartz lemma the fixed point ofBis unique.

We will need in the sequel the following result, which is probably well known.

Lemma2.2. IfBis a finite Blaschke product with a single critical pointz0∈D, then B(z)= τθ(z)^m+B

z0

1+B

z0

τθ(z)^m, (2.6)

wherem=ordBandτθ=e^iθ(z−z0)/(1−z¯0z)for someθ,0≤θ <2π.

Proof. The restriction of B on D\{z0} generates a holomorphic covering from D\{z0}onto D\{B(z0)}. Ifϕ(z)=(z−B(z0))/(1−B(z0)z), then the composition ϕ◦B generates an unramifiedm-sheeted holomorphic covering fromD\{z0}onto D\{0}. Consequently, there exists a biholomorphic map σ :D\{z0} →D\{0}, such that (ϕ◦B)(z)=σ (z)^m (cf. [6]). Clearly, σ =τθ for some θ: 0≤θ <2π, that is, ϕ(B(z))=(τθ(z))^m. Hence

B(z)=ϕ⁻¹

τθ(z)m

= τθ(z)^m+B z0 1+B

z0

τθ(z)^m. (2.7)

LetG be a compact abelian group. We assume that its dualGis isomorphic to a subgroup Γ ofR. The big G-disc ∆¯G over G is the compact set obtained from the Cartesian product[0,1]×Gby identifying the points in the fiber{0} ×G. The group G≈ {1}×G⊂∆¯Gis the topological boundary of ¯∆G. IfΓ=Z, thenG=Z=T, and the bigG-disc algebraAGcoincides with the classical disc algebra. We list here some of the basic properties ofAG.

(a)AGis a maximal Dirichlet algebra.

(b) The maximal ideal spaceᏹA_G ofAGis homeomorphic to the closed big disc ¯∆G. (c) The Gelfand transformationχ^a of a characterχ^a,a∈Γ₊on ¯∆G is the function

χ^a(r g)=χ^a(g)r^a, wherer g∈∆¯G.

(d) The originO=({0}×G)/({0}×G)in∆Gis a one-point Gleason part forAG. (e) The groupG=b∆Gis the Shilov boundary ofAG.

(f) Any automorphismτ ofAG,G≠ Tis generated by a pair(g, ϕ) such thatg∈ G and ϕ:Γ →Γ is an automorphism that preserves Γ₊, that is, τ =τ(g,ϕ), where τ(g,ϕ)(χ^a)=(χ^ϕ(a)(g))χ^ϕ(a). The automorphisms ofAG in the case whenG=Tare the Möbius transformations of the unit disc.

3. Blaschke inductive limit algebras. Let Λ= {dk}^∞_k=1 be a sequence of natural numbers. Suppose thatmk=k

l=1dl,m0=1, and denote byΓΛthe abelian subgroup ofQ, that is, generated by the numbers 1/mk,k∈N. The groupΓΛcan be expressed as the inductive (direct) limit of groupsZ, namely

Z1 ζ₁²

→Z2 ζ³₂

→Z3 ζ₃⁴

→Z4 ζ₄⁵

→ ··· →ΓΛ, (3.1) whereζ_k^k+1(mk)=dk·mk, mk∈Zk =Z. The corresponding dual groups form an inverse (projective) sequence of unit circles, whose limit is the compact abelian group GΛ=ΓΛ, that is,

T1 τ₁²

← T2 τ₂³

← T3 τ₃⁴

← T4 τ₄⁵

←··· ← GΛ. (3.2) HereTk=Tare unit circles, andτ_k^k+1(z)=(ζ_k^k+1)^∗(z)=z^dk. Indeed,τ_k^k+1(e^itm)= e^{itζk+1k (m)}=e^itdkm=(e^itm)^dkfor everye^itm∈Tk=Zk.

There arises a conjugated inductive system{A(Tk), i^k_k⁺¹}^∞_k=1of disc algebrasA(T) linked by homomorphismsi^k+1_k =C_τk+1

k :A(Tk)→A(Tk+1):i^k+1_k (f )=f◦τ_k^k+1, that is, (i^k+1_k (f ))(z)=f (z^dk)forz∈Tk+1.

Consider the extensionsτ_k^k+1(z)=z^dk on ¯Dk. The limit of the inverse sequence {D¯k, τ_k^k+1}, lim←^k→∞{D¯k, τ_k^k+1}is the bigGΛ-disc ¯∆G_Λ=([0,1]×GΛ)/({0}×GΛ)over the

groupGΛ=ΓΛ. There arises an analogous inductive system{A(D¯k), i^k+1_k }^∞1 of algebras A(D)¯ A(T) and connecting homomorphisms i^k+1_k :A(D¯k)→A(D¯k+1) defined as before by

i^k+1_k =C_zdk, that is,

i^k+1_k (f ) (z)=

f (z)d_k

. (3.3)

The elements of the component algebrasA(D¯k)can be interpreted as continuous functions on ¯∆G_Λ. The uniform closure

lim→

k→∞

AD¯k

, C_zdk

(3.4)

inC(∆¯G_Λ)of the inductive limit of the system{A(D¯k), C_zdk}^∞k=1, as well as the cor- responding restriction algebra[lim→^k→∞{A(Tk), C_zdk}]is isometrically isomorphic to the bigGΛ-disc algebraAG_Λ, that is, to the algebraA(∆G_Λ)of generalizedGΛ-analytic functions on the bigGΛ-disc ¯∆G_Λ (see [10]).

In a similar way, if{Kl}^∞_l=1is a sequence of compact subsets in the complex planeC withτ_l^l+1(K_l+1)=Klfor everyl∈Z, then the closure of the inductive limit lim→^l→∞{A(Kl), C_zdk}inC(᏷)is the algebra of generalizedGΛ-analytic functionsA(᏷)on the compact set᏷=lim←^l→∞{Kl, τ_l^l+1}in the bigG-planeCG_Λ over the groupGΛ(see [9]).

Consider an inductive sequence of disc algebras A(T1) ⁱ

21

→A(T2) ⁱ

32

→A(T3) ⁱ

43

→ ···, (3.5)

that are linked by the embeddingsi^k_k⁺¹:A(Tk)→A(Tk+1). We have thatᏹ_ik+1 k (A(Tk))= D¯, and also∂(i^k_k⁺¹(A(Tk)))=T. According to the remarks followingLemma 2.1there are finite Blaschke productsBk:D→Dsuch thati^k+1_k =CB_kfor everyk∈N, that is,

i^k+1_k (f )=CB_k(f )=f◦Bk, (3.6) whereBk(z)is a finite Blaschke product.

Letb= {Bk}^∞k=1be the sequence of Blaschke products corresponding toi^k+1_k , that is,CB_k(f )=i^k_k⁺¹(f ).

Consider the sequence Λ= {dk}^∞_k=1of orders of Blaschke products {Bk}^∞_k=1 and let ΓΛ ⊂ Qbe the group generated by the numbers 1/mk, mk=k

l=1dl, m0=1, k=0,1,2, . . . .By᐀kwe denote the standarddk-sheeted lifting of the unit circleTin the Riemann surface᏾kof the functionz^1/dk. Clearly᐀kT, and the diagram

᐀k π_k

᐀k+1 B˜_k

π_k+1

T _B T

k

(3.7)

commutes for everyk=0,1,2, . . ., whereπkbe the natural covering mappingπk:᐀k→ T. The inverse sequence of circles

T1 B₁

← T2 B₂

← T3 B₃

← T4 B₄

←··· ←Ᏻb (3.8)

is isomorphic to the inverse sequence

᐀1 B˜₁

←᐀2

˜B₂

←᐀3 B˜₃

←᐀4 B˜₄

←···, (3.9)

whereBk’s is the natural lifting ofBkto᐀k.

Let againτ_k^k+1(z)=z^dk, andτ_k^k+1(z)be the natural lifting ofτ_k^k+1to᐀k. Clearly, the diagram

᐀k π_k

᐀k+1

˜ τ_k^k+1

π_k+1

T T

τ_k^k+1(z)=z^dk

(3.10)

commutes for everyk∈N. The inverse sequence (3.9) is (topologically) isomorphic to the sequence

᐀1

˜ τ₁²

←᐀2

˜ τ³₂

←᐀3

˜ τ₃⁴

←᐀4

˜ τ⁵₄

←···, (3.11)

which on its own is isomorphic to the sequence T1

τ₁²

← T2 τ₂³

← T3 τ₃⁴

← T4 τ₄⁵

←··· ← GΛ. (3.12) Consequently, the setᏳb from (3.8) is homeomorphic to the groupGΛ. For the dual sequence we get

Z1 Bˆ₁

→Z2 Bˆ₂

→Z3 ˆB₃

→ ··· →GΛ=ΓΛ⊂Q. (3.13) We have obtained the following result.

Lemma3.1. The inverse limitlim←^k→∞{Tk, Bk|Tk} =Ᏻbin (3.11) can be equipped with the structure of a compact abelian group isomorphic toGΛ, whereΓΛ=GΛ⊂Q.

Consider an inverse sequence D¯1

B1

←D¯2 B2

←D¯3 B3

←D¯4 B4

←···, (3.14)

whereb= {Bk}^∞_k=1is a sequence of finite Blaschke products. The inverse limitᏰb= lim←^k→∞{D¯k, Bk}is a Hausdorff compact space. The limit of the adjoint system{A(D¯k), CB_k}^∞1 of disc algebrasA(D¯k)linked by the homomorphisms

CB_k:AD¯k

→AD¯k+1 :

CB_k(f ) z_k+1

=f Bk

z_k+1

(3.15) is an algebra of functions onᏰb, and its closure

A(b)=

lim→

k→∞

AD¯k , CB_k

(3.16)

inC(Ᏸb)is called theBlaschke inductive limit algebracorresponding to the sequence b= {Bk}^∞_k=1of Blaschke products. Note thatA(b)is isometrically isomorphic to the restriction algebra[lim→^k→∞{A(Tk), CB_k}].

Proposition 3.2. Letb= {Bk}^∞_k=1be a sequence of finite Blaschke products and letA(b)=[lim→^k→∞{A(Tk), CB_k}]be the corresponding inductive limit of disc algebras.

Then

(i) A(b)is a uniform algebra on the compact setᏰb=lim←^k→∞{D¯k, Bk}. (ii) The maximal ideal space ofA(b)isᏰb.

(iii) A(b)is a Dirichlet algebra.

(iv) A(b)is a maximal algebra.

(v) The Shilov boundaryᏳb⊂Ᏸb ofA(b)is a group isomorphic to the groupGΛ, whose dual groupGΛis isomorphic to the groupΓΛ_∞

k=0(1/mk)Z⊂Q, wheremk= k

l=1dl,m0=1, anddk=ordBk.

Indeed, under our hypothesisBkmapsTk+1ontoTkandDk+1ontoDk. Since the Shilov boundary of every component algebraA(Dk)is the unit circleTk, and the max- imal ideal space is the disc ¯Dk, then the properties (i)–(iii) follow from the general results of inductive limits of uniform algebras (e.g., [7]). The maximality ofA(B)is a consequence from the following result.

Proposition3.3. Every inductive limit of maximal algebras is a maximal algebra.

Proof. LetA=[lim→^σ∈Σ{A^σ, i^τ_σ}], where A^σ are maximal algebras. If ᏹσ is the maximal ideal space ofA^σ, then by (i)ᏹA=lim

←^σ{ᏹσ, (i^τ_σ)^∗}. Fixh∈C(ᏹ⁾\Aand sup- pose that the algebraA[h]generated byAandhdiffers fromC(ᏹA). Clearly,A[h]= [lim→^σ{A^σ[hσ], (i^τ_σ)^∗∗}]. Letg∈lim→^σ{A^σ[hσ], (i^τ_σ)^∗∗}\A, and consider the algebra A[g]⊂A[h]. We have thatg= {{g^σ}σ∈Σ, gσ∈C(ᏹσ)} ∈lim

→^σ∈Σ{C(ᏹσ), (i^τ_σ)^∗∗}\A⊂ C(ᏹA)\A. Sincei^τ_σ(A^σ)⊂A^τ andg∈A, it follows thatg^σ∈A^σ for everyσ∈Σ. By the maximality we have thatA^σ[g^σ]=C(ᏹσ),σ ∈Σ. Consequently,A[h]⊃A[g]= [lim→^σ{A^σ[g], (i^τ_σ)^∗∗}]=[lim→^σ{C(ᏹσ), (i^τ_σ)^∗∗}]=C(ᏹA). This shows thatAis a max- imal algebra.

We end this section with the following property of bigG-disc algebras.

Theorem3.4. LetGbe a compact abelian group whose dual groupGis isomorphic to a subgroupΓ of R. The bigG-disc algebraAGcan be expressed as a Blaschke inductive limit of disc algebras if and only if Γ is isomorphic to a subgroup ofQ.

Proof. The first part of the theorem follows fromProposition 3.2. LetGΓ⊂Q and let{ai}^∞i=1be an enumeration ofΓ. Without loss of generality, we can assume that a1=1. LetΓ¹=Z,Γ²=Z+a2Z,Γ³=Z+a2Z+a3Z, and so forth. SinceZ⊂Γ^kandΓ^kis isomorphic toZ, there is amk∈N, such thatΓ^k=(1/mk)Z. ByΓ^k⊂Γ^k+1we have that d_k+1=(m_k+1)/mk∈Z. The inclusioni^k+1_k :Γ^kΓ^k+1generates a mappingi^k+1_k :Z→Z such thati^k_k⁺¹(1)=d_k+1, thusi^k_k⁺¹(n)=d_k+1·n,n∈Zk. Clearly, the group

Γ

∞ k=1

1

mkZ=lim→

k→∞

Γ^k,i^k+1_k

⊂Q (3.17)

is generated by the numbers 1/mk,k∈N. As we saw at the beginning of this section, the Blaschke inductive limit[lim→^k→∞{A(Tk), C_zdk}]corresponding to the sequenceΛ= {dk}^∞1 coincides with the bigGΛ-disc algebraAG_Λ.

Theorem3.5. Letb= {Bk}^∞_k=1be a sequence of finite Blaschke products onD¯ with no more than one critical pointz^(k)₀ and such thatBk(z^(k₀⁺¹⁾)=z₀^(k)fornbig enough.

Then the algebraA(b)is isometrically isomorphic to the bigGΛ-disc algebraAG_Λ, where Λ= {dk}^∞k=1,dk=ordBk.

Proof. Without loss of generality, we can suppose that the hypotheses hold for everyn∈N.Lemma 2.2implies that for every Möbius mapϕkon ¯Dwithϕk(z₀^(k))=0 there exist another Möbius mapϕ_k+1on ¯Dsuch that the diagram

D¯

ϕ_k

D¯

B_k k+1

D¯ D¯

z^dk

(3.18)

becomes commutative. Hence,ϕk◦Bk=(ϕ_k+1)^dk andϕk(z₀^(k))=0. Takeϕ0 to be the identity on ¯D.Lemma 2.2allows us to define inductively a sequence{ϕk}^∞_k=1of Möbius maps on ¯D. Everyϕkgenerates an isometric automorphismCϕ_konA(D)¯ such that the conjugate diagram

A(D) ^CBk A(D)

A(D)

C_ϕk

Czdk

A(D)

C_φk

+1 (3.19)

commutes, that is,CB_k◦Cϕ_k=Cϕ_k+1◦C_zdk. Therefore, the inductive sequences A(D¯) ^CB1→A(D¯) ^CB2→A(D¯) ^CB3→ ··· →A(b), (3.20) whereCB_k(f )=f◦Bk, and

A(D¯) ^Czd1→A(D¯) ^Czd2→A(D¯) ^Czd3→ ···→ AG_Λ, (3.21) whereC_zdk(f )=f (z^dk), are isomorphic. Consequently,

A(b)=

lim→

k→∞

AD¯k , CB_k

=

lim→

k→∞

AD¯k , C_zdk

=AG_Λ. (3.22)

Corollary3.6. If there is a Möbius transformationτ, such that(τ⁻¹◦Bk◦τ)(z)= z^dkϕk(z),k=1,2,3, . . ., whereϕk are Möbius transformations anddk>1, then the algebraA(b)is isometrically isomorphic to the bigG-disc algebraAG, whereGis the group generated by the numbers1/mk,mk=k

l=1dl,m0=1,k=0,1,2, . . . .

Corollary3.7. If every Blaschke productBkinTheorem 3.5is a Möbius transfor- mation, then the algebraA(b)is isometrically isomorphic to the disc algebraA_Z=A(T).

Indeed,Theorem 3.5implies that in this caseA(b)=AG_Λ withΛ= {1,1, . . .}. There- foreΓΛ=ZandGΛ=T.

As Theorems3.4and3.5show, certain classes of algebras ofG-generalized analytic functions can be expressed as inductive limits of disc algebras. Actually, any algebra of generalizedG-analytic functions can be expressed as inductive limit of an, in general not necessarily countable, inductive spectrum of disc algebras.

4. Annulus type Blaschke algebra A(b)^{[r ,1]}. Let D^{[r ,1]} = {z∈C:r ≤ |z| ≤1}, andbD^{[r ,1]}= {z∈C:|z| =r or|z| =1}. Denote byA(D^{[r ,1]})the uniform algebra of continuous functions onD^{[r ,1]}that are analytic in the interior. Note thatA(D^{[r ,1]})= R(D^{[r ,1]}), the algebra of continuous rational functions onD^{[r ,1]}. By a well-known result of Bishop, the Shilov boundary ofA(D^{[r ,1]})isbD^{[r ,1]}, and the restriction ofA(D^{[r ,1]}) onbD^{[r ,1]}is a maximal algebra with codim Re(A(D^{[r ,1]})|bD^{[r ,1]})=1. These results have been extended to the generalizedG-analytic case in [5]. Namely, letGbe a compact abelian group whose dual group is isomorphic to a subgroupΓofR. Let∆^{[r ,1]}_G =[r ,1]× G, 0< r <1 be ther-annulus in the bigG-disc∆G, and letR(∆^{[r ,1]}_G )be the uniform algebra on∆^{[r ,1]}_G , generated by the functionsχ^a,a∈Γ, defined inSection 2. Then

(a)∆^{[r ,1]}_G is the maximal ideal space ofR(∆^{[r ,1]}_G ).

(b)b∆^{[r ,1]}_G = {r ,1}×G=({r}×G)∪({1}×G)is the Shilov boundary ofR(∆^{[r ,1]}_G ).

(c)R(∆^{[r ,1]}_G )is a maximal algebra with codim Re(R(∆^{[r ,1]}_G )|_b∆^{[r ,1]}

G )=1.

Consequently, the algebraR(∆^{[r ,1]}_G )coincides with the algebraA(∆^{[r ,1]}_G )of continu- ous functions on∆^{[r ,1]}_G that are locally approximable by generalizedG-analytic func- tions in the interior of∆^{[r ,1]}_G .

LetΛ= {dk}^∞_k=1be a sequence of natural numbers andτ_k^k+1(z)=z^dk. Fixr∈(0,1]

and for everyk∈Nconsider the sets Ek=D^[r1/mk,1]=

z∈C:r^1/mk≤ |z| ≤1

=

τ₁²◦τ₂³◦···◦τk^k+1

−1 D^{[r ,1]}

, (4.1) wheremk=k

l=1dl,m0=1. Hence, there arises an inverse sequence D^{[r ,1]}←^τ21 E1

τ₂³

←E2 τ₃⁴

←E3 τ₄⁵

←··· (4.2)

of compact subsets of ¯D. Consider the conjugate inductive sequence A

D^{[r ,1] C}zd1

→A E1

^Czd2

→A E2

^Czd3

→ ···, (4.3) where the embeddingsC_zdk:A(E_k−1)→A(Ek)are the composition operators byz^dk, namely,

C_zdk◦f

(z)=f z^dk

. (4.4)

LetGdenote the compact abelian group whose dual groupΓΛ=Gis the subgroup of Qgenerated by the numbers 1/mk,mk=k

l=1dl,m0=1,k=0,1,2, . . . .

Lemma4.1. The uniform algebra[lim→^k→∞{A(Ek), C_zdk}]is isomorphic to the algebra A(∆^{[r ,1]}_G )ofG-analytic functions on∆^{[r ,1]}_G .

Proof. Let ak = 1/mk, where as before mk = k

l=1dl, m0 = 1. Consider the algebras A^k(∆^{[r ,1]}_G ) = {g◦χ^ak : g ∈ A(Ek)} ⊂ A(∆^{[r ,1]}_G ), k = 0,1,2, . . . . Clearly, A^k(∆^{[r ,1]}_G )⊂A^k+1(∆^{[r ,1]}_G )andA(∆^{[r ,1]}_G )=[_∞

k=0A^k(∆^{[r ,1]}_G )]. There arises an inductive

sequence A⁰

∆^{[r ,1]}_G ^j10

A¹

∆^{[r ,1]}_G ^j1²

A²

∆^{[r ,1]}_G ^j2³

··· A

∆^{[r ,1]}_G

, (4.5) where j_k^k+1 is the natural inclusion of A^k(∆^{[r ,1]}_G ) into A^k+1(∆^{[r ,1]}_G ). The inductive sequences (4.3), and (4.5) are isomorphic. Indeed, χ^ak maps ∆^{[r ,1]}_G ontoEk, and the mapping ϕk defined by ϕk(g◦χ^ak) = g maps isometrically and isomorphically A^k(∆^{[r ,1]}_G )onto A(Ek). In addition, i^k_k+1⁺²◦ϕk =ϕ_k+1|_A

k(∆^{[r ,1]}_G )=ϕ_k+1◦j_k+1^k+2, that is, the diagram

A^k

∆^{[r ,1]}_G ^j_k+1^k+2

ϕ_k

A^k+1

∆^{[r ,1]}_G

ϕ_k+1

A Ek

^ik+2k+1

A Ek+1

(4.6)

commutes. Therefore (4.3) and (4.5) are two isomorphic sequences, and thus A

∆^{[r ,1]}_G

= _∞

k=0

Ak

∆^{[r ,1]}_G

=

lim→

k→∞

Ak

∆^{[r ,1]}_G , j^k+2_k+1

lim→

k→∞

A

Ek

, i^k+2_k+1 . (4.7) Let nowb= {Bk}^∞_k=1be a sequence of finite Blaschke products on ¯Dand letdk= ordBk. Define inductively the sets

Fn=B_n⁻¹ F_n−1

=

z∈C:Bn(z)∈F_n−1

=

B1◦B2◦···◦Bn

₋1 D^{[r ,1]}

, F0=D^{[r ,1]}. (4.8) Consider the following conjugate sequences

D^{[r ,1]}←^B1 F1 B₂

←F2 B₃

←F3 B₄

←··· ← Ᏸ^{[r ,1]}_b ⊂Ᏸb, (4.9) A

D^{[r ,1] CB}1→A

F1 ^CB2→A

F2 ^CB3→ ···, (4.10) where(CB_k◦f )(z)=f (Bk(z)).

Theorem 4.2. If the Blaschke products Bn do not have critical points onFn for any n∈N, thenᏰ^{[r ,1]}_b ≈∆^{[r ,1]}_G and the algebraA(b)^{[r ,1]} =[lim→^n→∞{A(Fn), Bn}] is isometrically isomorphic to the algebraA(∆^{[r ,1]}_G ).

For the proof we need the following version of a well-known result about Riemann surfaces.

Lemma4.3. Suppose that thedk-sheeted holomorphic coveringBk:Fk→F_k−1does not have critical points, and there exist a biholomorphic mappingψ_k−1fromF_k−1onto Ek−1. Then there exist a biholomorphic mappingψk:Fk→Eksuch that the diagram

Fk−1 ψ_k−1

Fk B_k

ψ_k

E_k−1 Ek z^dk

(4.11)

is commutative, that is,ψk−1◦Bk=(ψk)^dk, wheredk=ordBk.

Proof. The functionz^dk generates a bijection z^dk fromEk onto thedk-sheeted coveringE_k−1overE_k−1. Likewise, the mapψ_k−1◦Bk:Fk→E_k−1generates a bijection (ψ_k−1◦Bk)fromFktoE_k−1. Therefore the mapψk=(z^dk)⁻¹◦(ψ_k−1◦Bk)is a bijec- tion fromFk ontoEk. Since all component mappings ofψkare locally holomorphic, so isψk.

Proof ofTheorem4.2. Letψ0be the identity map onD^{[r ,1]}=E0=F0.Lemma 4.3 allows us to define inductively biholomorphic mappingsψk:Fk→Ekfor everyk∈N such thatψ_k−1◦Bk=(ψk)^dk. Consequently,∆^{[r ,1]}_G =lim←^n→∞{En, z^dn} ≈lim←^n→∞{Fn, Bn}

=Ᏸ^{[r ,1]}_b ⊂Ᏸb. The conjugate mapCψ_kmaps the algebraA(Ek)isometrically and iso- morphically ontoA(Fk). Hence the inductive sequences (4.3) and (4.10) are isomorphic, and therefore,

A(b)[r ,1]=

lim→

k→∞

A Fk

, CB_k

=

lim→

k→∞

A Ek

, z^dk A

∆^{[r ,1]}_G

. (4.12)

In the setting ofTheorem 4.2the listed below properties of the algebraA(b)^{[r ,1]}

follow directly fromTheorem 4.2,Proposition 3.3, and the results in [6].

(a) The maximal ideal space of the algebraA(b)^{[r ,1]} is homeomorphic to the set

∆^{[r ,1]}_G .

(b) The Shilov boundary ofA(b)^{[r ,1]}is the setb∆^{[r ,1]}_G = {r ,1}×G.

(c)A(b)^{[r ,1]}is a maximal algebra on its Shilov boundary.

(d) codim Re(A(b)^{[r ,1]}|_b∆^{[r ,1]}

G )=1.

(e) One-point Gleason parts ofA(b)^{[r ,1]}belong to the Shilov boundaryb∆^{[r ,1]}_G . 5. Local structure of Blaschke inductive limit algebras. LetF be a closed subset of the unit discD. Denote byA(F )the algebra of all continuous functions onF that are analytic in the interior ofF. Recall thatA(F )coincides with the uniform closure onF of the restrictions of Gelfand transforms of the elements inA(T)onF. That is, A(F )=A(D)| F.

Letb= {B1, B2, . . . , Bn, . . .}be a sequence of finite Blaschke products onDand let 0< r <1. Consider the following compact subsets of ¯D:D^{(r )}n =B⁻_n¹(D^{(r )}_n−1), forn≥1, D^{(r )}₀ =D^{[0,r ]}= {z∈D:|z| ≤r}. There arises an inverse sequence

D^{[0,r ]}←^B1 D₁^{(r )}←^B2 D₂^{(r )}←^B3 D^{(r )}₃ ←^B4 ··· (5.1) of subsets ofD. The inductive limit

A(b)^{[0,r ]}=

lim→

n→∞

A D_n^{(r )}

, CB_n+1

(5.2)

is again a uniform algebra on its maximal ideal space lim←^k→∞{D^{(r )}n , B_n+1|_D^{(r )}_n } =Ᏸ^{[0,r ]}_b ⊂ Ᏸb. Every Blaschke product

B(z)=e^iθn k=1

z−zk

1−z¯kz

, zk<1, (5.3)