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Analytic structures in maximal ideal spaces Manuel Maestre Infinite Dimensional Analysis, October 28-30, 2016 Kent State University RICHARD’S PARTY

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Page 1: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

Analytic structures in maximal idealspaces

Manuel Maestre

Infinite Dimensional Analysis, October 28-30, 2016 Kent State University

RICHARD’S PARTY

Page 2: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 3: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 4: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 5: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 6: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 7: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 8: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 9: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 10: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 11: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 12: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 13: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 14: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 15: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 16: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 17: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 18: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 19: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 20: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 21: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 22: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 23: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 24: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 25: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 26: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 27: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 28: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 29: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 30: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 31: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 32: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 33: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 34: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 35: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 36: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 37: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 38: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 39: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 40: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 41: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 42: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 43: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 44: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 45: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 46: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 47: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 48: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 49: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 50: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 51: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 52: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 53: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 54: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 55: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 56: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 57: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 58: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 59: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 60: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 61: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 62: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 63: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 64: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 65: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 66: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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

Page 67: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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.

R. M. Aron, B. J. Cole, and T. W. Gamelin, Spectra of algebras of analyticfunctions on a Banach space, J. Reine Angew. Math. 415 (1991), 51–93.

R. M. Aron, B. Cole and T. Gamelin, Weak-star continuous analyticfunctions. Canad. J. Math. 47 (1995), 673–683.

R. M. Aron, D. Carando, T. Gamelin, S. Lasalle, M. Maestre, Cluster Valuesof Analytic Functions on a Banach Space Math. Annalen, 353 n.2, (2012)293–303.

Richard M. Aron, Daniel Carando, S. Lassalle and Manuel Maestre, Clustervalues of holomorphic functions of bounded type, Trans. Amer. Math. Soc.368, N. 4 (2016), 2355–2369.

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

Page 68: Analytic structures in maximal ideal spaceszvavitch/Infinite_Dimensiona_Analysis...X) the maximal ideal space ( spectrum) M(A) is the compact set of all non-null linear and multiplicative

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.

D. Carando, D. García, M. Maestre and P. Sevilla-Peris, A Riemann manifoldstructure on the spectra of algebras of weighted holomorphic functions,Topology 48 (2009), 54-65; doi:10.1016/j.top.2009.11.003.

B. J. Cole, T. W. Gamelin, W. B. Johnson, Analytic disks in fibers over theunit ball of a Banach space, Michigan Math. J. 39 (1992), 551–569.

A. M. Davie and T. W. Gamelin, A theorem on polynomial-starapproximation, Proc. Amer. Math. Soc. 106 (1989), no. 2, 351–356.

I. J. Schark, Maximal ideals in an algebra of bounded analytic functions. J.Math. Mech. 10 (1961), 735–746.

Manuel Maestre Analytic structures in maximal ideal spaces