Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal...
Transcript of Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal...
Analytic structures in maximal idealspaces
Manuel Maestre
Infinite Dimensional Analysis, October 28-30, 2016 Kent State University
RICHARD’S PARTY
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Definition
Given U an open subset of Cn , we will denote by
H(U) the space of all functions f : U → C which are holomorphic on U. It is aFréchet algebra endowed with uniform convergence on compact subsets of U.
For U any open subset of a complex Banach space X , H∞(U) is the Banachalgebra of all functions f : U → C which are holomorphic=Fréchet differentiableand bounded on U with the supremum norm.
Given x ∈ U there exists L : X → C linear AND CONTINUOUS such that
limh→0
f (x + h) − f (x) − L (h)‖h‖
= 0.
In particular, X ∗ ⊂ H(U).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Definition
Given U an open subset of Cn , we will denote by
H(U) the space of all functions f : U → C which are holomorphic on U. It is aFréchet algebra endowed with uniform convergence on compact subsets of U.
For U any open subset of a complex Banach space X , H∞(U) is the Banachalgebra of all functions f : U → C which are holomorphic=Fréchet differentiableand bounded on U with the supremum norm.
Given x ∈ U there exists L : X → C linear AND CONTINUOUS such that
limh→0
f (x + h) − f (x) − L (h)‖h‖
= 0.
In particular, X ∗ ⊂ H(U).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Definition
Given U an open subset of Cn , we will denote by
H(U) the space of all functions f : U → C which are holomorphic on U. It is aFréchet algebra endowed with uniform convergence on compact subsets of U.
For U any open subset of a complex Banach space X , H∞(U) is the Banachalgebra of all functions f : U → C which are holomorphic=Fréchet differentiableand bounded on U with the supremum norm.
Given x ∈ U there exists L : X → C linear AND CONTINUOUS such that
limh→0
f (x + h) − f (x) − L (h)‖h‖
= 0.
In particular, X ∗ ⊂ H(U).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Definition
Given U an open subset of Cn , we will denote by
H(U) the space of all functions f : U → C which are holomorphic on U. It is aFréchet algebra endowed with uniform convergence on compact subsets of U.
For U any open subset of a complex Banach space X , H∞(U) is the Banachalgebra of all functions f : U → C which are holomorphic=Fréchet differentiableand bounded on U with the supremum norm.
Given x ∈ U there exists L : X → C linear AND CONTINUOUS such that
limh→0
f (x + h) − f (x) − L (h)‖h‖
= 0.
In particular, X ∗ ⊂ H(U).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Ball Algebra
Given a complex Banach space X , and its open unit ball BX , we will denote byAu(BX ) the Banach algebra of all functions f : B̄X → C such that are uniformlycontinuous on B̄X and holomorphic=Fréchet differentiable on BX .
Maximal ideal space
For A either H∞(U) or Au(BX ) the maximal ideal space ( spectrum)M(A ) is thecompact set of all non-null linear and multiplicative ϕ : A → C endowed with theweak-star topology w(A ∗,A ).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Ball Algebra
Given a complex Banach space X , and its open unit ball BX , we will denote byAu(BX ) the Banach algebra of all functions f : B̄X → C such that are uniformlycontinuous on B̄X and holomorphic=Fréchet differentiable on BX .
Maximal ideal space
For A either H∞(U) or Au(BX ) the maximal ideal space ( spectrum)M(A ) is thecompact set of all non-null linear and multiplicative ϕ : A → C endowed with theweak-star topology w(A ∗,A ).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Maximal ideal space
The maximal ideal space (spectrum)M(H(U)) is set of all non-null linear andmultiplicative ϕ : H(U)→ C endowed with the weak-star topology w(H(U)∗,H(U)).
Remark
For any U open subset of C
M(H(U)) = {δz : z ∈ U}
Let φ ∈ M(H(U)) anda = φ(z � z).
a ∈ U
If not, then 1z−a ∈ H(U),
1 = φ(1) = φ(z − az − a
) = φ(1
z − a)φ(z − a) = φ(
1z − a
)(φ(z) − a) = 0.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Maximal ideal space
The maximal ideal space (spectrum)M(H(U)) is set of all non-null linear andmultiplicative ϕ : H(U)→ C endowed with the weak-star topology w(H(U)∗,H(U)).
Remark
For any U open subset of C
M(H(U)) = {δz : z ∈ U}
Let φ ∈ M(H(U)) anda = φ(z � z).
a ∈ U
If not, then 1z−a ∈ H(U),
1 = φ(1) = φ(z − az − a
) = φ(1
z − a)φ(z − a) = φ(
1z − a
)(φ(z) − a) = 0.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Maximal ideal space
The maximal ideal space (spectrum)M(H(U)) is set of all non-null linear andmultiplicative ϕ : H(U)→ C endowed with the weak-star topology w(H(U)∗,H(U)).
Remark
For any U open subset of C
M(H(U)) = {δz : z ∈ U}
Let φ ∈ M(H(U)) anda = φ(z � z).
a ∈ U
If not, then 1z−a ∈ H(U),
1 = φ(1) = φ(z − az − a
) = φ(1
z − a)φ(z − a) = φ(
1z − a
)(φ(z) − a) = 0.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Maximal ideal space
The maximal ideal space (spectrum)M(H(U)) is set of all non-null linear andmultiplicative ϕ : H(U)→ C endowed with the weak-star topology w(H(U)∗,H(U)).
Remark
For any U open subset of C
M(H(U)) = {δz : z ∈ U}
Let φ ∈ M(H(U)) anda = φ(z � z).
a ∈ U
If not, then 1z−a ∈ H(U),
1 = φ(1) = φ(z − az − a
) = φ(1
z − a)φ(z − a) = φ(
1z − a
)(φ(z) − a) = 0.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Maximal ideal space
The maximal ideal space (spectrum)M(H(U)) is set of all non-null linear andmultiplicative ϕ : H(U)→ C endowed with the weak-star topology w(H(U)∗,H(U)).
Remark
For any U open subset of C
M(H(U)) = {δz : z ∈ U}
Let φ ∈ M(H(U)) anda = φ(z � z).
a ∈ U
If not, then 1z−a ∈ H(U),
1 = φ(1) = φ(z − az − a
) = φ(1
z − a)φ(z − a) = φ(
1z − a
)(φ(z) − a) = 0.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Maximal ideal space
The maximal ideal space (spectrum)M(H(U)) is set of all non-null linear andmultiplicative ϕ : H(U)→ C endowed with the weak-star topology w(H(U)∗,H(U)).
Remark
For any U open subset of C
M(H(U)) = {δz : z ∈ U}
Let φ ∈ M(H(U)) anda = φ(z � z).
a ∈ U
If not, then 1z−a ∈ H(U),
1 = φ(1) = φ(z − az − a
) = φ(1
z − a)φ(z − a) = φ(
1z − a
)(φ(z) − a) = 0.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
M(Au(D)) = {δx : x ∈ D}
Corona Theorem (Carlesson, 1962)
If we denote by D the open unit disk of C
M(H∞(D)) = {δx : x ∈ D}w∗.
But!βN \ N ⊂ M(H∞(D))
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
M(Au(D)) = {δx : x ∈ D}
Corona Theorem (Carlesson, 1962)
If we denote by D the open unit disk of C
M(H∞(D)) = {δx : x ∈ D}w∗.
But!βN \ N ⊂ M(H∞(D))
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
M(Au(D)) = {δx : x ∈ D}
Corona Theorem (Carlesson, 1962)
If we denote by D the open unit disk of C
M(H∞(D)) = {δx : x ∈ D}w∗.
But!βN \ N ⊂ M(H∞(D))
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
From now on X is an infinite dimensional complex Banach space!!!
{δx : x ∈ BX } ⊂ M(H∞(BX ))
{δx : x ∈ B̄X } ⊂ M(Au(BX ))
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Bull. Soc. math. France,
106, 1978, p. 3-24.
A HAHN-BANACH EXTENSION THEOREM
FOR ANALYTIC MAPPINGS
BY
RICHARD M. ARON 0 and PAUL D. BERNER (2)
[Dublin]
RESUME. — Soient E un sous-espace vectoriel ferme d'un espace de Banach G,
U un ouvert de E, et F un espace de Banach. On considere Ie probleme du prolongement
des applications analytiques de U a valeur dans F a un ouvert de G, et on trouve des
conditions necessaires et suffisantes pour 1'existence de tels prolongements. Ces conditions
entrainent 1'existence d'une application lineaire continue de prolongement de E ' a G'
ce qui, a tour de role, se rapporte au theoreme de Hahn-Banach vectoriel.
ABSTRACT. — Let E be a closed subspace of a Banach space G, let U be an open subset
of E, and let F be another Banach space. The problem of extending analytic F-valued
mappings defined on U to an open subset of G is discussed, and necessary and sufficient
conditions are found for such extensions to exist. These conditions involve the exis-
tence of a continuous linear extension mapping of E ' to G\ which in turn is related tothe Hahn-Banach theorem for linear transformations.
We consider the problem of extending an analytic mapping defined on
an open subset U of a closed subspace E of a Banach space G to an analytic
mapping defined on an open neighbourhood of U in G. Our general
approach is to obtain extensions to the whole space G of polynomials defined
on E, and then to use local Taylor series representations to extend analytic
functions locally. It is necessary to show that the local extensions are
"coherent in the overlaps". This can be done when one can define a linear
and continuous extension mapping taking polynomials defined on E to their
extensions defined on G, which in turn is closely related to the vector-valued
Hahn-Banach property as studied by NACHBIN, LINDENSTRAUSS, and others.
The general question of extending analytic mappings on topological vector
spaces was raised by DINEEN in [4]. He and other authors (HIRSCHOWITZ,
(1) The research of this author was supported in part by the Universite de Nancy
(France).(2) The research of this author was supported by a post-doctoral fellowship from
An Roinn Oideachais, Department of Education (Ireland).
BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem: Davie-Gamelin (after Aron-Berner)
{δ̃z : z ∈ BX∗∗ } ⊂ M(H∞(BX )).
{δ̃z : z ∈ B̄X∗∗ } ⊂ M(Au(BX )).
δ̃z < {δx : x ∈ BX }
if z ∈ BX∗∗ \ BX .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem: Davie-Gamelin (after Aron-Berner)
{δ̃z : z ∈ BX∗∗ } ⊂ M(H∞(BX )).
{δ̃z : z ∈ B̄X∗∗ } ⊂ M(Au(BX )).
δ̃z < {δx : x ∈ BX }
if z ∈ BX∗∗ \ BX .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem: Davie-Gamelin (after Aron-Berner)
{δ̃z : z ∈ BX∗∗ } ⊂ M(H∞(BX )).
{δ̃z : z ∈ B̄X∗∗ } ⊂ M(Au(BX )).
δ̃z < {δx : x ∈ BX }
if z ∈ BX∗∗ \ BX .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Definition
Given U an open subset of a Banach space X , then Hb (U) will be the set all allFréchet differentiable functions such that are bounded on the bounded subsets ofU that have a positive distance to the boundary of U.
Hb (U) is a Fréchet algebra when endowed with the topology of uniformconvergence on the sequence
Ur ={x ∈ X : ||x || 6 r and dist (x,X\U) >
1r
}.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
If we denote by A either Au(BX ) or H∞(BX ) or Hb (U), since X ∗ ⊂ A , we can define
π : A → X ∗∗
byπ(φ)(x∗) := φ(x∗).
For each z ∈ X ∗∗, We will call the setMz(A ) the fiber ofM(A ) at z, where
Mz(A ) ={φ ∈ M(A ) : π(φ) = z
}.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
If we denote by A either Au(BX ) or H∞(BX ) or Hb (U), since X ∗ ⊂ A , we can define
π : A → X ∗∗
byπ(φ)(x∗) := φ(x∗).
For each z ∈ X ∗∗, We will call the setMz(A ) the fiber ofM(A ) at z, where
Mz(A ) ={φ ∈ M(A ) : π(φ) = z
}.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
If we denote by A either Au(BX ) or H∞(BX ) or Hb (U), since X ∗ ⊂ A , we can define
π : A → X ∗∗
byπ(φ)(x∗) := φ(x∗).
For each z ∈ X ∗∗, We will call the setMz(A ) the fiber ofM(A ) at z, where
Mz(A ) ={φ ∈ M(A ) : π(φ) = z
}.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
If we denote by A either Au(BX ) or H∞(BX ) or Hb (U), since X ∗ ⊂ A , we can define
π : A → X ∗∗
byπ(φ)(x∗) := φ(x∗).
For each z ∈ X ∗∗, We will call the setMz(A ) the fiber ofM(A ) at z, where
Mz(A ) ={φ ∈ M(A ) : π(φ) = z
}.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem. R. Aron, P. Galindo, D. García, , M.M. TAMS 1996
Let X be a symmetrically regular Banach space and U an open subset of X , thenMb (U) has a Riemann analytic manifold structure.
Definition
A Banach X is (symmetrically) regular Banach space if every (symmetric)operator T : X → X ∗ is weakly-compact.
T : X → X ∗ is symmetric ifT (x)(y) = T (y)(x),
for every x and y in X .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem. R. Aron, P. Galindo, D. García, , M.M. TAMS 1996
Let X be a symmetrically regular Banach space and U an open subset of X , thenMb (U) has a Riemann analytic manifold structure.
Definition
A Banach X is (symmetrically) regular Banach space if every (symmetric)operator T : X → X ∗ is weakly-compact.
T : X → X ∗ is symmetric ifT (x)(y) = T (y)(x),
for every x and y in X .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem. R. Aron, P. Galindo, D. García, , M.M. TAMS 1996
Let X be a symmetrically regular Banach space and U an open subset of X , thenMb (U) has a Riemann analytic manifold structure.
Definition
A Banach X is (symmetrically) regular Banach space if every (symmetric)operator T : X → X ∗ is weakly-compact.
T : X → X ∗ is symmetric ifT (x)(y) = T (y)(x),
for every x and y in X .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
For each φ ∈ Mb (U), there is a bounded subsetUr = {x ∈ X : ||x || 6 r and dist (x,X\U) > 1
r } such that |φ(f )| 6 ||f ||Ur for allf ∈ Hb (U).
Given φ in Mb (U) and w ∈ X ∗∗ with ‖w‖ < 1r ,
φw : Hb (U)→ C
by
φw (f ) =∞∑
n=0
φ(P̃n(w)
),
where∑∞
n=0 Pn(x)(·) is the Taylor series expansion of f at x ∈ U.Defined Vφ,ε = {φw : ‖w‖ < ε}, then the familyV := {Vφ,ε : φ ∈ Mb (U) and ε > 0}is a basic neighborhood system for a Hausdorff topology on Mb (U).Moreover,
π : Mb (U)→ X ∗∗
is a local homeomorphism over X ∗∗ and Mb (U) has a Riemann analytic structureover X ∗∗.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
For each φ ∈ Mb (U), there is a bounded subsetUr = {x ∈ X : ||x || 6 r and dist (x,X\U) > 1
r } such that |φ(f )| 6 ||f ||Ur for allf ∈ Hb (U).Given φ in Mb (U) and w ∈ X ∗∗ with ‖w‖ < 1
r ,
φw : Hb (U)→ C
by
φw (f ) =∞∑
n=0
φ(P̃n(w)
),
where∑∞
n=0 Pn(x)(·) is the Taylor series expansion of f at x ∈ U.Defined Vφ,ε = {φw : ‖w‖ < ε}, then the familyV := {Vφ,ε : φ ∈ Mb (U) and ε > 0}is a basic neighborhood system for a Hausdorff topology on Mb (U).Moreover,
π : Mb (U)→ X ∗∗
is a local homeomorphism over X ∗∗ and Mb (U) has a Riemann analytic structureover X ∗∗.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
For each φ ∈ Mb (U), there is a bounded subsetUr = {x ∈ X : ||x || 6 r and dist (x,X\U) > 1
r } such that |φ(f )| 6 ||f ||Ur for allf ∈ Hb (U).Given φ in Mb (U) and w ∈ X ∗∗ with ‖w‖ < 1
r ,
φw : Hb (U)→ C
by
φw (f ) =∞∑
n=0
φ(P̃n(w)
),
where∑∞
n=0 Pn(x)(·) is the Taylor series expansion of f at x ∈ U.
Defined Vφ,ε = {φw : ‖w‖ < ε}, then the familyV := {Vφ,ε : φ ∈ Mb (U) and ε > 0}is a basic neighborhood system for a Hausdorff topology on Mb (U).Moreover,
π : Mb (U)→ X ∗∗
is a local homeomorphism over X ∗∗ and Mb (U) has a Riemann analytic structureover X ∗∗.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
For each φ ∈ Mb (U), there is a bounded subsetUr = {x ∈ X : ||x || 6 r and dist (x,X\U) > 1
r } such that |φ(f )| 6 ||f ||Ur for allf ∈ Hb (U).Given φ in Mb (U) and w ∈ X ∗∗ with ‖w‖ < 1
r ,
φw : Hb (U)→ C
by
φw (f ) =∞∑
n=0
φ(P̃n(w)
),
where∑∞
n=0 Pn(x)(·) is the Taylor series expansion of f at x ∈ U.Defined Vφ,ε = {φw : ‖w‖ < ε}, then the familyV := {Vφ,ε : φ ∈ Mb (U) and ε > 0}is a basic neighborhood system for a Hausdorff topology on Mb (U).
Moreover,π : Mb (U)→ X ∗∗
is a local homeomorphism over X ∗∗ and Mb (U) has a Riemann analytic structureover X ∗∗.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
For each φ ∈ Mb (U), there is a bounded subsetUr = {x ∈ X : ||x || 6 r and dist (x,X\U) > 1
r } such that |φ(f )| 6 ||f ||Ur for allf ∈ Hb (U).Given φ in Mb (U) and w ∈ X ∗∗ with ‖w‖ < 1
r ,
φw : Hb (U)→ C
by
φw (f ) =∞∑
n=0
φ(P̃n(w)
),
where∑∞
n=0 Pn(x)(·) is the Taylor series expansion of f at x ∈ U.Defined Vφ,ε = {φw : ‖w‖ < ε}, then the familyV := {Vφ,ε : φ ∈ Mb (U) and ε > 0}is a basic neighborhood system for a Hausdorff topology on Mb (U).Moreover,
π : Mb (U)→ X ∗∗
is a local homeomorphism over X ∗∗ and Mb (U) has a Riemann analytic structureover X ∗∗.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
If we denote by D the open unit disk of C
Mx (H∞(D)) = {δx },
for all x ∈ D
But for all x ∈ T
βN \ N ⊂ Mx (H∞(D)),
Aron, Cole and Gamelin 1991.
Suppose that X is an infinite dimensional Banach space. Then the fiberMz(H∞(B)) contains a copy of β(N) \ N, for every z ∈ B∗∗.
Aron, Cole and Gamelin 1991
M(Au(Bc0 )) = {δ̃z : z ∈ B̄`∞ }.
Aron, Cole and Gamelin 1991.
Actually βN \ N ⊂ Mx (Au(B`p )), for any 1 < p < ∞, for every x ∈ B`p .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
If we denote by D the open unit disk of C
Mx (H∞(D)) = {δx },
for all x ∈ D
But for all x ∈ T
βN \ N ⊂ Mx (H∞(D)),
Aron, Cole and Gamelin 1991.
Suppose that X is an infinite dimensional Banach space. Then the fiberMz(H∞(B)) contains a copy of β(N) \ N, for every z ∈ B∗∗.
Aron, Cole and Gamelin 1991
M(Au(Bc0 )) = {δ̃z : z ∈ B̄`∞ }.
Aron, Cole and Gamelin 1991.
Actually βN \ N ⊂ Mx (Au(B`p )), for any 1 < p < ∞, for every x ∈ B`p .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
If we denote by D the open unit disk of C
Mx (H∞(D)) = {δx },
for all x ∈ D
But for all x ∈ T
βN \ N ⊂ Mx (H∞(D)),
Aron, Cole and Gamelin 1991.
Suppose that X is an infinite dimensional Banach space. Then the fiberMz(H∞(B)) contains a copy of β(N) \ N, for every z ∈ B∗∗.
Aron, Cole and Gamelin 1991
M(Au(Bc0 )) = {δ̃z : z ∈ B̄`∞ }.
Aron, Cole and Gamelin 1991.
Actually βN \ N ⊂ Mx (Au(B`p )), for any 1 < p < ∞, for every x ∈ B`p .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
If we denote by D the open unit disk of C
Mx (H∞(D)) = {δx },
for all x ∈ D
But for all x ∈ T
βN \ N ⊂ Mx (H∞(D)),
Aron, Cole and Gamelin 1991.
Suppose that X is an infinite dimensional Banach space. Then the fiberMz(H∞(B)) contains a copy of β(N) \ N, for every z ∈ B∗∗.
Aron, Cole and Gamelin 1991
M(Au(Bc0 )) = {δ̃z : z ∈ B̄`∞ }.
Aron, Cole and Gamelin 1991.
Actually βN \ N ⊂ Mx (Au(B`p )), for any 1 < p < ∞, for every x ∈ B`p .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
If we denote by D the open unit disk of C
Mx (H∞(D)) = {δx },
for all x ∈ D
But for all x ∈ T
βN \ N ⊂ Mx (H∞(D)),
Aron, Cole and Gamelin 1991.
Suppose that X is an infinite dimensional Banach space. Then the fiberMz(H∞(B)) contains a copy of β(N) \ N, for every z ∈ B∗∗.
Aron, Cole and Gamelin 1991
M(Au(Bc0 )) = {δ̃z : z ∈ B̄`∞ }.
Aron, Cole and Gamelin 1991.
Actually βN \ N ⊂ Mx (Au(B`p )), for any 1 < p < ∞, for every x ∈ B`p .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem. R. Aron, D. Carando, Ted. Gamelin, S.Lassalle, M.M. 2012
For every f ∈ H∞(Bc0 ) and z ∈ B`∞ ,
{ϕ(f ) : ϕ ∈ Mz(H∞(Bc0 )) ∩ {δx : x ∈ Bc0 }w∗} = {ϕ(f ) : ϕ ∈ Mz(H∞(Bc0 ))}.
Theorem. The weak Theorem holds for A (B`2 ).
For f ∈ Au(B`2 ), we have
f̂ (Mz(Au(B`2 )) ∩ {δx : x ∈ B`2 }w∗
) = f̂ (Mz(Au(B`2 ))),
for every z ∈ B̄`2 .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem. R. Aron, D. Carando, Ted. Gamelin, S.Lassalle, M.M. 2012
For every f ∈ H∞(Bc0 ) and z ∈ B`∞ ,
{ϕ(f ) : ϕ ∈ Mz(H∞(Bc0 )) ∩ {δx : x ∈ Bc0 }w∗} = {ϕ(f ) : ϕ ∈ Mz(H∞(Bc0 ))}.
Theorem. The weak Theorem holds for A (B`2 ).
For f ∈ Au(B`2 ), we have
f̂ (Mz(Au(B`2 )) ∩ {δx : x ∈ B`2 }w∗
) = f̂ (Mz(Au(B`2 ))),
for every z ∈ B̄`2 .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem, R. Aron, D. Carando, S. Lassalle M.M. 2016
If x0 ∈ `1 and ‖x0‖ = 1 thenMx0 (Au(B`1 )) = {δx0 }.
For all x0 ∈ B`1 , then βN is embedded inMx0 (Au(B`1 )).
For every z ∈ B`∗∗1we have Card
(Mz(Au(B`1 ))
)> c.
There exist z ∈ S`∗∗1such that Card
(Mz(Au(B`1 ))
)> c,
where c stands for the cardinal of the continuum
.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem, R. Aron, D. Carando, S. Lassalle M.M. 2016
If x0 ∈ `1 and ‖x0‖ = 1 thenMx0 (Au(B`1 )) = {δx0 }.
For all x0 ∈ B`1 , then βN is embedded inMx0 (Au(B`1 )).
For every z ∈ B`∗∗1we have Card
(Mz(Au(B`1 ))
)> c.
There exist z ∈ S`∗∗1such that Card
(Mz(Au(B`1 ))
)> c,
where c stands for the cardinal of the continuum
.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem, R. Aron, D. Carando, S. Lassalle M.M. 2016
If x0 ∈ `1 and ‖x0‖ = 1 thenMx0 (Au(B`1 )) = {δx0 }.
For all x0 ∈ B`1 , then βN is embedded inMx0 (Au(B`1 )).
For every z ∈ B`∗∗1we have Card
(Mz(Au(B`1 ))
)> c.
There exist z ∈ S`∗∗1such that Card
(Mz(Au(B`1 ))
)> c,
where c stands for the cardinal of the continuum
.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem, R. Aron, D. Carando, S. Lassalle M.M. 2016
If x0 ∈ `1 and ‖x0‖ = 1 thenMx0 (Au(B`1 )) = {δx0 }.
For all x0 ∈ B`1 , then βN is embedded inMx0 (Au(B`1 )).
For every z ∈ B`∗∗1we have Card
(Mz(Au(B`1 ))
)> c.
There exist z ∈ S`∗∗1such that Card
(Mz(Au(B`1 ))
)> c,
where c stands for the cardinal of the continuum
.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem, R. Aron, D. Carando, S. Lassalle M.M. 2016
If x0 ∈ `1 and ‖x0‖ = 1 thenMx0 (Au(B`1 )) = {δx0 }.
For all x0 ∈ B`1 , then βN is embedded inMx0 (Au(B`1 )).
For every z ∈ B`∗∗1we have Card
(Mz(Au(B`1 ))
)> c.
There exist z ∈ S`∗∗1such that Card
(Mz(Au(B`1 ))
)> c,
where c stands for the cardinal of the continuum.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Example R. Aron, D. Carando, T. Gamelin, S. Lassalle M.M. 2012
Let {aj | j ∈ N} be a dense sequence in D, and let f : B`2 → C be given byf (x) =
∑∞n=1 anx2
n . Let (nj) be an arbitrary subsequence of natural numbers, and letϕ ∈ {δenj
| j ∈ N} be an accumulation point inM(Au(B`2 )). Then π(ϕ) = 0 for everysuch ϕ. In addition, for any λ0 ∈ D, if (nj) is chosen so that anj → λ0, then thecorresponding ϕ satisfies ϕ(f ) = λ0. Hence
D ⊂ f̂ (M0((Au(B`2 ))).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Injecting analytic structures
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem I. J. Schark 1961
The complex disk D can be analytically and homeomorphically injected intoM1(H∞(D)).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem: Cole, Gamelin and Johnson 1992
If X = `2, then the unit ball of a nonseparable Hilbert space injects into the fiberM0(H∞(B)) via an analytic map which is uniformly bicontinuous from the metric ofthe unit ball of the Hilbert space to the Gleason metric of its image inM(H∞(B)).
Gleason metric inM(H∞(B)) is defined as
ρ(ϕ, ψ) = sup{|f̂ (ϕ) − f̂ (ψ)| : f ∈ H∞(B), ‖f‖ ≤ 1}
Theorem: Cole, Gamelin and Johnson 1992
Suppose that X has a normalized basis (ej) that is shrinking, with associatedfunctionals (e∗j ) satisfying that there exists a positive integer N > 1 such that
∞∑j=1
|e∗j (x)|N < ∞
for all x =∑∞
j=1 e∗j (x)xj in X . Then there is an analytic injection of the countableinfinite dimensional polydisk DN into the fiberM0(H∞(B)).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem: Cole, Gamelin and Johnson 1992
If X = `2, then the unit ball of a nonseparable Hilbert space injects into the fiberM0(H∞(B)) via an analytic map which is uniformly bicontinuous from the metric ofthe unit ball of the Hilbert space to the Gleason metric of its image inM(H∞(B)).
Gleason metric inM(H∞(B)) is defined as
ρ(ϕ, ψ) = sup{|f̂ (ϕ) − f̂ (ψ)| : f ∈ H∞(B), ‖f‖ ≤ 1}
Theorem: Cole, Gamelin and Johnson 1992
Suppose that X has a normalized basis (ej) that is shrinking, with associatedfunctionals (e∗j ) satisfying that there exists a positive integer N > 1 such that
∞∑j=1
|e∗j (x)|N < ∞
for all x =∑∞
j=1 e∗j (x)xj in X . Then there is an analytic injection of the countableinfinite dimensional polydisk DN into the fiberM0(H∞(B)).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem: Cole, Gamelin and Johnson 1992
If X = `2, then the unit ball of a nonseparable Hilbert space injects into the fiberM0(H∞(B)) via an analytic map which is uniformly bicontinuous from the metric ofthe unit ball of the Hilbert space to the Gleason metric of its image inM(H∞(B)).
Gleason metric inM(H∞(B)) is defined as
ρ(ϕ, ψ) = sup{|f̂ (ϕ) − f̂ (ψ)| : f ∈ H∞(B), ‖f‖ ≤ 1}
Theorem: Cole, Gamelin and Johnson 1992
Suppose that X has a normalized basis (ej) that is shrinking, with associatedfunctionals (e∗j ) satisfying that there exists a positive integer N > 1 such that
∞∑j=1
|e∗j (x)|N < ∞
for all x =∑∞
j=1 e∗j (x)xj in X . Then there is an analytic injection of the countableinfinite dimensional polydisk DN into the fiberM0(H∞(B)).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Theorem: Cole, Gamelin and Johnson, 1992
Let X be an infinite dimensional Banach space. Suppose that (zk ) is a sequencein B∗∗ which converges weak-star to 0, such that the distance from zk to the linearspan of z1, . . . , zk−1 tends to 1 as k → ∞ (e.g. c0 and `p for 1 < p < ∞). Then,passing to a subsequence, we can find a sequence of analytic disks λ→ zk (λ)(λ ∈ D, k > 1) in B∗∗ with zk (0) = zk , such that for each λ ∈ D, (zk (λ)) is aninterpolating sequence for H∞(B). Furthermore, the correspondence(k , λ)→ zk (λ) extends to an embedding
Ψ : β(N) × D→M(H∞(B))
such thatΨ((β(N) \ N) × D) ⊂ M0(H∞(B))
and f̂ ◦Ψ is analytic on each slice {p} × D for all f ∈ H∞(B) and p ∈ β(N).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Remark
Let U be an open subset of a Banach space and X and φ ∈ Mb (U). There existsr > 0 such that
F : BX∗∗ (0,1r
)→ Mb (U),
F(w) = φw : Hb (U)→ C,
is an analytic injection.
Where r > 0 satisfies that|φ(f )| 6 sup
x∈Ur
|f (x)|,
for all f ∈ Hb (U), where Ur = {x ∈ X : ‖x‖ 6 r and dist (x,X\U) > 1r }.
Recall
φw (f ) =∞∑
n=0
φ(P̃n(w)
),
where∑∞
n=0 Pn(x) is the Taylor series expansion of f about x ∈ U.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Remark
Let U be an open subset of a Banach space and X and φ ∈ Mb (U). There existsr > 0 such that
F : BX∗∗ (0,1r
)→ Mb (U),
F(w) = φw : Hb (U)→ C,
is an analytic injection.
Where r > 0 satisfies that|φ(f )| 6 sup
x∈Ur
|f (x)|,
for all f ∈ Hb (U), where Ur = {x ∈ X : ‖x‖ 6 r and dist (x,X\U) > 1r }.
Recall
φw (f ) =∞∑
n=0
φ(P̃n(w)
),
where∑∞
n=0 Pn(x) is the Taylor series expansion of f about x ∈ U.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Remark
Let U be an open subset of a Banach space and X and φ ∈ Mb (U). There existsr > 0 such that
F : BX∗∗ (0,1r
)→ Mb (U),
F(w) = φw : Hb (U)→ C,
is an analytic injection.
Where r > 0 satisfies that|φ(f )| 6 sup
x∈Ur
|f (x)|,
for all f ∈ Hb (U), where Ur = {x ∈ X : ‖x‖ 6 r and dist (x,X\U) > 1r }.
Recall
φw (f ) =∞∑
n=0
φ(P̃n(w)
),
where∑∞
n=0 Pn(x) is the Taylor series expansion of f about x ∈ U.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Aron, Cole and Gamelin 1991
M(Hu(Bc0 )) = {δ̃z : z ∈ `∞}.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Proposition, R. Aron, J. Falcó, D. García and M.M, 2016
Let X , {0} be a Banach space and x0 ∈ SX . Then the complex disk D can beanalytically injected intoMx0 (H∞(BX )).
Theorem, R. Aron, J. Falcó, D. García and M.M, 2016
Let z0 = (z1, z2, ..., zn, ...) be a point of the distinguished boundary Tℵ0 of B`∞ (i.e.|zj | = 1 for all j ∈ N). Then there exists an injection Ψ : B`∞ →Mz0 (H∞(Bc0 ))which is biholomorphic onto its image.
R. Aron, J. Falcó, D. García and M.M, 2016
Let K be an infinite scattered compact Hausdorff set. We have that B`∞ iscontinuously injected in the fiberM0(H∞(BC(K ))).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Proposition, R. Aron, J. Falcó, D. García and M.M, 2016
Let X , {0} be a Banach space and x0 ∈ SX . Then the complex disk D can beanalytically injected intoMx0 (H∞(BX )).
Theorem, R. Aron, J. Falcó, D. García and M.M, 2016
Let z0 = (z1, z2, ..., zn, ...) be a point of the distinguished boundary Tℵ0 of B`∞ (i.e.|zj | = 1 for all j ∈ N). Then there exists an injection Ψ : B`∞ →Mz0 (H∞(Bc0 ))which is biholomorphic onto its image.
R. Aron, J. Falcó, D. García and M.M, 2016
Let K be an infinite scattered compact Hausdorff set. We have that B`∞ iscontinuously injected in the fiberM0(H∞(BC(K ))).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Proposition, R. Aron, J. Falcó, D. García and M.M, 2016
Let X , {0} be a Banach space and x0 ∈ SX . Then the complex disk D can beanalytically injected intoMx0 (H∞(BX )).
Theorem, R. Aron, J. Falcó, D. García and M.M, 2016
Let z0 = (z1, z2, ..., zn, ...) be a point of the distinguished boundary Tℵ0 of B`∞ (i.e.|zj | = 1 for all j ∈ N). Then there exists an injection Ψ : B`∞ →Mz0 (H∞(Bc0 ))which is biholomorphic onto its image.
R. Aron, J. Falcó, D. García and M.M, 2016
Let K be an infinite scattered compact Hausdorff set. We have that B`∞ iscontinuously injected in the fiberM0(H∞(BC(K ))).
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Proposition, R. Aron, J. Falcó, D. García and M.M, 2016
There exists a continuous injection B`2 intoM0(Au(B`2 )) that restricted to B`2 isanalytic.
Proposition, R. Aron, J. Falcó, D. García and M.M, 2016
Let X be a Banach space such that there exists a polynomial P satisfying thatP |BX is not weakly continuous at some point of BX . Then the complex disk D canbe analytically injected inMz(Au(BX )) for every z ∈ BX∗∗ .
Remark
If every polynomial is weakly continuous at each point of BX and X ∗ has theapproximation property, then
Mz(Au(BX )) = {δ̃z}
for every z ∈ BX∗∗ .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Proposition, R. Aron, J. Falcó, D. García and M.M, 2016
There exists a continuous injection B`2 intoM0(Au(B`2 )) that restricted to B`2 isanalytic.
Proposition, R. Aron, J. Falcó, D. García and M.M, 2016
Let X be a Banach space such that there exists a polynomial P satisfying thatP |BX is not weakly continuous at some point of BX . Then the complex disk D canbe analytically injected inMz(Au(BX )) for every z ∈ BX∗∗ .
Remark
If every polynomial is weakly continuous at each point of BX and X ∗ has theapproximation property, then
Mz(Au(BX )) = {δ̃z}
for every z ∈ BX∗∗ .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
Proposition, R. Aron, J. Falcó, D. García and M.M, 2016
There exists a continuous injection B`2 intoM0(Au(B`2 )) that restricted to B`2 isanalytic.
Proposition, R. Aron, J. Falcó, D. García and M.M, 2016
Let X be a Banach space such that there exists a polynomial P satisfying thatP |BX is not weakly continuous at some point of BX . Then the complex disk D canbe analytically injected inMz(Au(BX )) for every z ∈ BX∗∗ .
Remark
If every polynomial is weakly continuous at each point of BX and X ∗ has theapproximation property, then
Mz(Au(BX )) = {δ̃z}
for every z ∈ BX∗∗ .
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
R. M. Aron, P. D. Berner, A Hahn-Banach extension theorem for analyticmappings. Bull. Soc. Math. France 106 (1978), no. 1, 3–24.
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R. M. Aron, J. Falcó, D. García and M. Maestre, Embedding analytic disks infibers of the spectra, preprint.
Manuel Maestre Analytic structures in maximal ideal spaces
Introduction. The algebras of holomorphic functions in CN .Infinite dimensional setting. Size of the fibres
Injecting analytic structures
D. Carando, G. García, M. Maestre, Homomorphisms and compositionoperators on algebras of analytic functions of bounded type, Advances inMathematics 197 (2005), 607-629.
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A. M. Davie and T. W. Gamelin, A theorem on polynomial-starapproximation, Proc. Amer. Math. Soc. 106 (1989), no. 2, 351–356.
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Manuel Maestre Analytic structures in maximal ideal spaces