Corona and cluster value problems in infinite-dimensional spaces
Sofía Ortega Castillo
Department of Mathematics Texas A& M University
April 13, 2014 Informal Analysis Seminar, Universality Weekend Department of Mathematical Sciences at Kent State University
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 1 / 22 Outline
Introduction: Main Concepts Our Cluster Value Theorems Open Cluster Value Problems and Related Questions References
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 2 / 22 Outline
Introduction: Main Concepts Our Cluster Value Theorems Open Cluster Value Problems and Related Questions References
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 2 / 22 Outline
Introduction: Main Concepts Our Cluster Value Theorems Open Cluster Value Problems and Related Questions References
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 2 / 22 Outline
Introduction: Main Concepts Our Cluster Value Theorems Open Cluster Value Problems and Related Questions References
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 2 / 22 Outline
Introduction: Main Concepts Our Cluster Value Theorems Open Cluster Value Problems and Related Questions References
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 2 / 22 Introduction: Main Concepts
Given a Banach space X with B its open unit ball, we are interested in studying certain Banach algebras of bounded analytic functions on B, that contain X∗ (the continuous linear functionals on X) and 1. H(B) will denote any such algebra.
Remark ∗ ∗ ∗ ∗ ∀x ∈ X , x : B → C acts linearly and continuously, thus each x is analytic and bounded.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 3 / 22 Introduction: Main Concepts
Given a Banach space X with B its open unit ball, we are interested in studying certain Banach algebras of bounded analytic functions on B, that contain X∗ (the continuous linear functionals on X) and 1. H(B) will denote any such algebra.
Remark ∗ ∗ ∗ ∗ ∀x ∈ X , x : B → C acts linearly and continuously, thus each x is analytic and bounded.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 3 / 22 Introduction: Main Concepts
Examples of Banach algebras H(B): H∞(B): all bounded analytic functions on B. Two generalizations of the disk algebra:
Au(B): bounded and uniformly continuous analytic functions on B. A(B): uniform limits on B of polynomials in the functions in X∗.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 4 / 22 Introduction: Main Concepts
Examples of Banach algebras H(B): H∞(B): all bounded analytic functions on B. Two generalizations of the disk algebra:
Au(B): bounded and uniformly continuous analytic functions on B. A(B): uniform limits on B of polynomials in the functions in X∗.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 4 / 22 Introduction: Main Concepts
Examples of Banach algebras H(B): H∞(B): all bounded analytic functions on B. Two generalizations of the disk algebra:
Au(B): bounded and uniformly continuous analytic functions on B. A(B): uniform limits on B of polynomials in the functions in X∗.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 4 / 22 Introduction: Main Concepts
Examples of Banach algebras H(B): H∞(B): all bounded analytic functions on B. Two generalizations of the disk algebra:
Au(B): bounded and uniformly continuous analytic functions on B. A(B): uniform limits on B of polynomials in the functions in X∗.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 4 / 22 Introduction: Main Concepts
Examples of Banach algebras H(B): H∞(B): all bounded analytic functions on B. Two generalizations of the disk algebra:
Au(B): bounded and uniformly continuous analytic functions on B. A(B): uniform limits on B of polynomials in the functions in X∗.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 4 / 22 Introduction: Main Concepts
One of the most important topics in the study of Banach algebras H(B) is the study of its set of characters, the nonzero algebra homomorphisms from H(B) to C, called the spectrum of H(B), and denoted by MH(B).
The study of the spectrum is simplified by fibering it over B¯∗∗ (the ∗∗ ∗∗ closed unit ball of X ) via the surjective mapping π : MH(B) → B¯ given by π(τ) = τ|X∗ .
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 5 / 22 Introduction: Main Concepts
One of the most important topics in the study of Banach algebras H(B) is the study of its set of characters, the nonzero algebra homomorphisms from H(B) to C, called the spectrum of H(B), and denoted by MH(B).
The study of the spectrum is simplified by fibering it over B¯∗∗ (the ∗∗ ∗∗ closed unit ball of X ) via the surjective mapping π : MH(B) → B¯ given by π(τ) = τ|X∗ .
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 5 / 22 Introduction: Main Concepts
One of the most important topics in the study of Banach algebras H(B) is the study of its set of characters, the nonzero algebra homomorphisms from H(B) to C, called the spectrum of H(B), and denoted by MH(B).
The study of the spectrum is simplified by fibering it over B¯∗∗ (the ∗∗ ∗∗ closed unit ball of X ) via the surjective mapping π : MH(B) → B¯ given by π(τ) = τ|X∗ .
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 5 / 22 Introduction: Main Concepts
One of the most important topics in the study of Banach algebras H(B) is the study of its set of characters, the nonzero algebra homomorphisms from H(B) to C, called the spectrum of H(B), and denoted by MH(B).
The study of the spectrum is simplified by fibering it over B¯∗∗ (the ∗∗ ∗∗ closed unit ball of X ) via the surjective mapping π : MH(B) → B¯ given by π(τ) = τ|X∗ .
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 5 / 22 Introduction: Main Concepts
Given f ∈ H(B), the Gelfand Transform of f is the continuous map
ˆ f : MH(B) → C given by τ 7→ τ(f ).
Note: The Gelfand Transform is a generalization of the Fourier Transform for L1(R) under convolution.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 6 / 22 Introduction: Main Concepts
Given f ∈ H(B), the Gelfand Transform of f is the continuous map
ˆ f : MH(B) → C given by τ 7→ τ(f ).
Note: The Gelfand Transform is a generalization of the Fourier Transform for L1(R) under convolution.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 6 / 22 Introduction: Main Concepts
Given f ∈ H(B), the Gelfand Transform of f is the continuous map
ˆ f : MH(B) → C given by τ 7→ τ(f ).
Note: The Gelfand Transform is a generalization of the Fourier Transform for L1(R) under convolution.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 6 / 22 Introduction: Main Concepts
Big open problems in the study of algebras H(B):
∗ Corona problem: Is B dense in MH(B) (in the w topology)?
Remark
B ⊂ MH(B) via δ : B → MH(B) such that x 7→ δx, where δx : H(B) → C is defined by f 7→ f (x).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 7 / 22 Introduction: Main Concepts
Big open problems in the study of algebras H(B):
∗ Corona problem: Is B dense in MH(B) (in the w topology)?
Remark
B ⊂ MH(B) via δ : B → MH(B) such that x 7→ δx, where δx : H(B) → C is defined by f 7→ f (x).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 7 / 22 Introduction: Main Concepts
Big open problems in the study of algebras H(B):
∗ Corona problem: Is B dense in MH(B) (in the w topology)?
Remark
B ⊂ MH(B) via δ : B → MH(B) such that x 7→ δx, where δx : H(B) → C is defined by f 7→ f (x).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 7 / 22 Introduction: Main Concepts
Results related to Corona problem: Carleson, ’62: Corona theorem for the unit disk ∆ ⊂ C. Gamelin, ’70; Garnett and Jones, ’85: Corona theorems for other planar domains. Sibony, ’87: Pseudoconvex counterexample in C3 to Corona problem. Sibony, ’93: Pseudoconvex and strictly pseudoconvex (except at one point) counterexample in C2 to Corona problem. n Corona problem open for the unit ball and polydisk in C for n ≥ 2.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 8 / 22 Introduction: Main Concepts
Results related to Corona problem: Carleson, ’62: Corona theorem for the unit disk ∆ ⊂ C. Gamelin, ’70; Garnett and Jones, ’85: Corona theorems for other planar domains. Sibony, ’87: Pseudoconvex counterexample in C3 to Corona problem. Sibony, ’93: Pseudoconvex and strictly pseudoconvex (except at one point) counterexample in C2 to Corona problem. n Corona problem open for the unit ball and polydisk in C for n ≥ 2.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 8 / 22 Introduction: Main Concepts
Results related to Corona problem: Carleson, ’62: Corona theorem for the unit disk ∆ ⊂ C. Gamelin, ’70; Garnett and Jones, ’85: Corona theorems for other planar domains. Sibony, ’87: Pseudoconvex counterexample in C3 to Corona problem. Sibony, ’93: Pseudoconvex and strictly pseudoconvex (except at one point) counterexample in C2 to Corona problem. n Corona problem open for the unit ball and polydisk in C for n ≥ 2.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 8 / 22 Introduction: Main Concepts
Results related to Corona problem: Carleson, ’62: Corona theorem for the unit disk ∆ ⊂ C. Gamelin, ’70; Garnett and Jones, ’85: Corona theorems for other planar domains. Sibony, ’87: Pseudoconvex counterexample in C3 to Corona problem. Sibony, ’93: Pseudoconvex and strictly pseudoconvex (except at one point) counterexample in C2 to Corona problem. n Corona problem open for the unit ball and polydisk in C for n ≥ 2.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 8 / 22 Introduction: Main Concepts
Results related to Corona problem: Carleson, ’62: Corona theorem for the unit disk ∆ ⊂ C. Gamelin, ’70; Garnett and Jones, ’85: Corona theorems for other planar domains. Sibony, ’87: Pseudoconvex counterexample in C3 to Corona problem. Sibony, ’93: Pseudoconvex and strictly pseudoconvex (except at one point) counterexample in C2 to Corona problem. n Corona problem open for the unit ball and polydisk in C for n ≥ 2.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 8 / 22 Introduction: Main Concepts
Results related to Corona problem: Carleson, ’62: Corona theorem for the unit disk ∆ ⊂ C. Gamelin, ’70; Garnett and Jones, ’85: Corona theorems for other planar domains. Sibony, ’87: Pseudoconvex counterexample in C3 to Corona problem. Sibony, ’93: Pseudoconvex and strictly pseudoconvex (except at one point) counterexample in C2 to Corona problem. n Corona problem open for the unit ball and polydisk in C for n ≥ 2.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 8 / 22 Introduction: Main Concepts
Another set of big open problems in the study of algebras H(B): Cluster value problems.
The cluster value theorem for H(B) asserts that, for a given x∗∗ ∈ B¯∗∗, the sets of cluster values
∗∗ w∗ ∗∗ ClB(f , x ) := {λ : f (xα) → λ, xα −→ x }
coincides with the Gelfand transform of f evaluated on the fiber −1 ∗∗ Mx∗∗ (B) := π (x ),
ˆf (Mx∗∗ (B)) = {τ(f ): τ ∈ Mx∗∗ },
for all f ∈ H(B).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 9 / 22 Introduction: Main Concepts
Another set of big open problems in the study of algebras H(B): Cluster value problems.
The cluster value theorem for H(B) asserts that, for a given x∗∗ ∈ B¯∗∗, the sets of cluster values
∗∗ w∗ ∗∗ ClB(f , x ) := {λ : f (xα) → λ, xα −→ x }
coincides with the Gelfand transform of f evaluated on the fiber −1 ∗∗ Mx∗∗ (B) := π (x ),
ˆf (Mx∗∗ (B)) = {τ(f ): τ ∈ Mx∗∗ },
for all f ∈ H(B).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 9 / 22 Introduction: Main Concepts
Another set of big open problems in the study of algebras H(B): Cluster value problems.
The cluster value theorem for H(B) asserts that, for a given x∗∗ ∈ B¯∗∗, the sets of cluster values
∗∗ w∗ ∗∗ ClB(f , x ) := {λ : f (xα) → λ, xα −→ x }
coincides with the Gelfand transform of f evaluated on the fiber −1 ∗∗ Mx∗∗ (B) := π (x ),
ˆf (Mx∗∗ (B)) = {τ(f ): τ ∈ Mx∗∗ },
for all f ∈ H(B).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 9 / 22 Introduction: Main Concepts
Another set of big open problems in the study of algebras H(B): Cluster value problems.
The cluster value theorem for H(B) asserts that, for a given x∗∗ ∈ B¯∗∗, the sets of cluster values
∗∗ w∗ ∗∗ ClB(f , x ) := {λ : f (xα) → λ, xα −→ x }
coincides with the Gelfand transform of f evaluated on the fiber −1 ∗∗ Mx∗∗ (B) := π (x ),
ˆf (Mx∗∗ (B)) = {τ(f ): τ ∈ Mx∗∗ },
for all f ∈ H(B).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 9 / 22 Introduction: Main Concepts
Another set of big open problems in the study of algebras H(B): Cluster value problems.
The cluster value theorem for H(B) asserts that, for a given x∗∗ ∈ B¯∗∗, the sets of cluster values
∗∗ w∗ ∗∗ ClB(f , x ) := {λ : f (xα) → λ, xα −→ x }
coincides with the Gelfand transform of f evaluated on the fiber −1 ∗∗ Mx∗∗ (B) := π (x ),
ˆf (Mx∗∗ (B)) = {τ(f ): τ ∈ Mx∗∗ },
for all f ∈ H(B).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 9 / 22 Introduction: Main Concepts
Results related to cluster value problems:
Kakutani, ’55: Considers first cluster value problems for domains in C. I. J. Schark, ’61: Cluster value theorem for H(∆), ∆ unit disk of C. n n n Gamelin, ’73: Cluster value theorem for H(∆ ), ∆ polydisk in C .
McDonald, ’79: Cluster value theorem for H(BCn ), and actually for n H(U) for U any strongly pseudoconvex domain in C with smooth boundary. The proof if this last result uses a solution to a ∂¯ problem in strongly pseudoconvex domains (Kerzman, ’71).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 10 / 22 Introduction: Main Concepts
Results related to cluster value problems:
Kakutani, ’55: Considers first cluster value problems for domains in C. I. J. Schark, ’61: Cluster value theorem for H(∆), ∆ unit disk of C. n n n Gamelin, ’73: Cluster value theorem for H(∆ ), ∆ polydisk in C .
McDonald, ’79: Cluster value theorem for H(BCn ), and actually for n H(U) for U any strongly pseudoconvex domain in C with smooth boundary. The proof if this last result uses a solution to a ∂¯ problem in strongly pseudoconvex domains (Kerzman, ’71).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 10 / 22 Introduction: Main Concepts
Results related to cluster value problems:
Kakutani, ’55: Considers first cluster value problems for domains in C. I. J. Schark, ’61: Cluster value theorem for H(∆), ∆ unit disk of C. n n n Gamelin, ’73: Cluster value theorem for H(∆ ), ∆ polydisk in C .
McDonald, ’79: Cluster value theorem for H(BCn ), and actually for n H(U) for U any strongly pseudoconvex domain in C with smooth boundary. The proof if this last result uses a solution to a ∂¯ problem in strongly pseudoconvex domains (Kerzman, ’71).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 10 / 22 Introduction: Main Concepts
Results related to cluster value problems:
Kakutani, ’55: Considers first cluster value problems for domains in C. I. J. Schark, ’61: Cluster value theorem for H(∆), ∆ unit disk of C. n n n Gamelin, ’73: Cluster value theorem for H(∆ ), ∆ polydisk in C .
McDonald, ’79: Cluster value theorem for H(BCn ), and actually for n H(U) for U any strongly pseudoconvex domain in C with smooth boundary. The proof if this last result uses a solution to a ∂¯ problem in strongly pseudoconvex domains (Kerzman, ’71).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 10 / 22 Introduction: Main Concepts
Results related to cluster value problems:
Kakutani, ’55: Considers first cluster value problems for domains in C. I. J. Schark, ’61: Cluster value theorem for H(∆), ∆ unit disk of C. n n n Gamelin, ’73: Cluster value theorem for H(∆ ), ∆ polydisk in C .
McDonald, ’79: Cluster value theorem for H(BCn ), and actually for n H(U) for U any strongly pseudoconvex domain in C with smooth boundary. The proof if this last result uses a solution to a ∂¯ problem in strongly pseudoconvex domains (Kerzman, ’71).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 10 / 22 Introduction: Main Concepts
Results related to cluster value problems:
Kakutani, ’55: Considers first cluster value problems for domains in C. I. J. Schark, ’61: Cluster value theorem for H(∆), ∆ unit disk of C. n n n Gamelin, ’73: Cluster value theorem for H(∆ ), ∆ polydisk in C .
McDonald, ’79: Cluster value theorem for H(BCn ), and actually for n H(U) for U any strongly pseudoconvex domain in C with smooth boundary. The proof if this last result uses a solution to a ∂¯ problem in strongly pseudoconvex domains (Kerzman, ’71).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 10 / 22 Introduction: Main Concepts
What happens when B is the unit ball of an infinite-dimensional Banach space? Let us first overview the basic theory of analytic functions on arbitrary Banach spaces.
Given U an open subset of a Banach space X, f : U → C is analytic if for every x ∈ U there exists r > 0 and continuous polynomials on X, m ∞ m (P f (x))m=0, where P f (x) is m-homogeneous, such that, if ky − xk < r P∞ m then f (y) = m=0 P f (x)(y − x), and the convergence is uniform on B(x, r) (and rcf (x) is the supremum of such r).
An m-homogeneous polynomial Lˆ on X, for m ∈ N, is the restriction to m the diagonal of a m-linear mapping L : X → C, i.e. Lˆ(x) = L(x, ··· , x) (and it is a constant function for m = 0).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 11 / 22 Introduction: Main Concepts
What happens when B is the unit ball of an infinite-dimensional Banach space? Let us first overview the basic theory of analytic functions on arbitrary Banach spaces.
Given U an open subset of a Banach space X, f : U → C is analytic if for every x ∈ U there exists r > 0 and continuous polynomials on X, m ∞ m (P f (x))m=0, where P f (x) is m-homogeneous, such that, if ky − xk < r P∞ m then f (y) = m=0 P f (x)(y − x), and the convergence is uniform on B(x, r) (and rcf (x) is the supremum of such r).
An m-homogeneous polynomial Lˆ on X, for m ∈ N, is the restriction to m the diagonal of a m-linear mapping L : X → C, i.e. Lˆ(x) = L(x, ··· , x) (and it is a constant function for m = 0).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 11 / 22 Introduction: Main Concepts
What happens when B is the unit ball of an infinite-dimensional Banach space? Let us first overview the basic theory of analytic functions on arbitrary Banach spaces.
Given U an open subset of a Banach space X, f : U → C is analytic if for every x ∈ U there exists r > 0 and continuous polynomials on X, m ∞ m (P f (x))m=0, where P f (x) is m-homogeneous, such that, if ky − xk < r P∞ m then f (y) = m=0 P f (x)(y − x), and the convergence is uniform on B(x, r) (and rcf (x) is the supremum of such r).
An m-homogeneous polynomial Lˆ on X, for m ∈ N, is the restriction to m the diagonal of a m-linear mapping L : X → C, i.e. Lˆ(x) = L(x, ··· , x) (and it is a constant function for m = 0).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 11 / 22 Introduction: Main Concepts
What happens when B is the unit ball of an infinite-dimensional Banach space? Let us first overview the basic theory of analytic functions on arbitrary Banach spaces.
Given U an open subset of a Banach space X, f : U → C is analytic if for every x ∈ U there exists r > 0 and continuous polynomials on X, m ∞ m (P f (x))m=0, where P f (x) is m-homogeneous, such that, if ky − xk < r P∞ m then f (y) = m=0 P f (x)(y − x), and the convergence is uniform on B(x, r) (and rcf (x) is the supremum of such r).
An m-homogeneous polynomial Lˆ on X, for m ∈ N, is the restriction to m the diagonal of a m-linear mapping L : X → C, i.e. Lˆ(x) = L(x, ··· , x) (and it is a constant function for m = 0).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 11 / 22 Introduction: Main Concepts
What happens when B is the unit ball of an infinite-dimensional Banach space? Let us first overview the basic theory of analytic functions on arbitrary Banach spaces.
Given U an open subset of a Banach space X, f : U → C is analytic if for every x ∈ U there exists r > 0 and continuous polynomials on X, m ∞ m (P f (x))m=0, where P f (x) is m-homogeneous, such that, if ky − xk < r P∞ m then f (y) = m=0 P f (x)(y − x), and the convergence is uniform on B(x, r) (and rcf (x) is the supremum of such r).
An m-homogeneous polynomial Lˆ on X, for m ∈ N, is the restriction to m the diagonal of a m-linear mapping L : X → C, i.e. Lˆ(x) = L(x, ··· , x) (and it is a constant function for m = 0).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 11 / 22 Introduction: Main Concepts
What happens when B is the unit ball of an infinite-dimensional Banach space? Let us first overview the basic theory of analytic functions on arbitrary Banach spaces.
Given U an open subset of a Banach space X, f : U → C is analytic if for every x ∈ U there exists r > 0 and continuous polynomials on X, m ∞ m (P f (x))m=0, where P f (x) is m-homogeneous, such that, if ky − xk < r P∞ m then f (y) = m=0 P f (x)(y − x), and the convergence is uniform on B(x, r) (and rcf (x) is the supremum of such r).
An m-homogeneous polynomial Lˆ on X, for m ∈ N, is the restriction to m the diagonal of a m-linear mapping L : X → C, i.e. Lˆ(x) = L(x, ··· , x) (and it is a constant function for m = 0).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 11 / 22 Introduction: Main Concepts
What happens when B is the unit ball of an infinite-dimensional Banach space? Let us first overview the basic theory of analytic functions on arbitrary Banach spaces.
Given U an open subset of a Banach space X, f : U → C is analytic if for every x ∈ U there exists r > 0 and continuous polynomials on X, m ∞ m (P f (x))m=0, where P f (x) is m-homogeneous, such that, if ky − xk < r P∞ m then f (y) = m=0 P f (x)(y − x), and the convergence is uniform on B(x, r) (and rcf (x) is the supremum of such r).
An m-homogeneous polynomial Lˆ on X, for m ∈ N, is the restriction to m the diagonal of a m-linear mapping L : X → C, i.e. Lˆ(x) = L(x, ··· , x) (and it is a constant function for m = 0).
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 11 / 22 Introduction: Main Concepts
Theorem Given f : U ⊂ X → Y, TFAE: f is analytic, f is continuous and analytic on each complex line, i.e.
λ 7→ f (a + λb) is analytic for all a ∈ U and b 6= 0 ∈ E,
on {ζ ∈ C : a + ζb ∈ U}, f is Fréchet C-differentiable. Example ∗ ∗ ∗ ∗ w P∞ ∗ m If xm ∈ X and xm −→ 0, then m=0(xm) is analytic on X.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 12 / 22 Introduction: Main Concepts
Theorem Given f : U ⊂ X → Y, TFAE: f is analytic, f is continuous and analytic on each complex line, i.e.
λ 7→ f (a + λb) is analytic for all a ∈ U and b 6= 0 ∈ E,
on {ζ ∈ C : a + ζb ∈ U}, f is Fréchet C-differentiable. Example ∗ ∗ ∗ ∗ w P∞ ∗ m If xm ∈ X and xm −→ 0, then m=0(xm) is analytic on X.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 12 / 22 Introduction: Main Concepts
Theorem Given f : U ⊂ X → Y, TFAE: f is analytic, f is continuous and analytic on each complex line, i.e.
λ 7→ f (a + λb) is analytic for all a ∈ U and b 6= 0 ∈ E,
on {ζ ∈ C : a + ζb ∈ U}, f is Fréchet C-differentiable. Example ∗ ∗ ∗ ∗ w P∞ ∗ m If xm ∈ X and xm −→ 0, then m=0(xm) is analytic on X.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 12 / 22 Introduction: Main Concepts
Theorem Given f : U ⊂ X → Y, TFAE: f is analytic, f is continuous and analytic on each complex line, i.e.
λ 7→ f (a + λb) is analytic for all a ∈ U and b 6= 0 ∈ E,
on {ζ ∈ C : a + ζb ∈ U}, f is Fréchet C-differentiable. Example ∗ ∗ ∗ ∗ w P∞ ∗ m If xm ∈ X and xm −→ 0, then m=0(xm) is analytic on X.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 12 / 22 Introduction: Main Concepts
Theorem Given f : U ⊂ X → Y, TFAE: f is analytic, f is continuous and analytic on each complex line, i.e.
λ 7→ f (a + λb) is analytic for all a ∈ U and b 6= 0 ∈ E,
on {ζ ∈ C : a + ζb ∈ U}, f is Fréchet C-differentiable. Example ∗ ∗ ∗ ∗ w P∞ ∗ m If xm ∈ X and xm −→ 0, then m=0(xm) is analytic on X.
S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 12 / 22 Introduction: Main Concepts
Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem.
More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Properties of analytic functions that extend to ∞-dimensional setting: Open Mapping Principle, Maximum Principle, Liouville’s Theorem. More properties of analytic functions that extend: Given f : U ⊂ X → Y analytic, a ∈ U, b ∈ X, and r small enough, m 1 R f (a+ζb) P f (a)(b) = 2πi |ζ|=r ζm+1 dζ if λ ∈ r∆ and m ∈ N0, m −m ||P f (a)(b)|| ≤ r sup|ζ|=r ||f (a + ζb)||, ||x−a|| ||f (x) − f (a)|| ≤ 2 sup||z−a|| S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 13 / 22 Introduction: Main Concepts Back to Corona and cluster value problems: When B is the unit ball of an infinite-dimensional Banach space, there are no positive solutions to the Corona problem. Aron, Carando, Gamelin, Lasalle, Maestre, ’12 [1]: Cluster value theorem for x = 0 and Au(B), when X has a shrinking 1-unconditional basis. Example The previous condition is satisfied by `p for 1 < p < ∞ and c0, but not by `1, `∞ nor Lp(0, 1) for 1 ≤ p 6= 2 ≤ ∞. S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 14 / 22 Introduction: Main Concepts Back to Corona and cluster value problems: When B is the unit ball of an infinite-dimensional Banach space, there are no positive solutions to the Corona problem. Aron, Carando, Gamelin, Lasalle, Maestre, ’12 [1]: Cluster value theorem for x = 0 and Au(B), when X has a shrinking 1-unconditional basis. Example The previous condition is satisfied by `p for 1 < p < ∞ and c0, but not by `1, `∞ nor Lp(0, 1) for 1 ≤ p 6= 2 ≤ ∞. S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 14 / 22 Introduction: Main Concepts Back to Corona and cluster value problems: When B is the unit ball of an infinite-dimensional Banach space, there are no positive solutions to the Corona problem. Aron, Carando, Gamelin, Lasalle, Maestre, ’12 [1]: Cluster value theorem for x = 0 and Au(B), when X has a shrinking 1-unconditional basis. Example The previous condition is satisfied by `p for 1 < p < ∞ and c0, but not by `1, `∞ nor Lp(0, 1) for 1 ≤ p 6= 2 ≤ ∞. S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 14 / 22 Introduction: Main Concepts Back to Corona and cluster value problems: When B is the unit ball of an infinite-dimensional Banach space, there are no positive solutions to the Corona problem. Aron, Carando, Gamelin, Lasalle, Maestre, ’12 [1]: Cluster value theorem for x = 0 and Au(B), when X has a shrinking 1-unconditional basis. Example The previous condition is satisfied by `p for 1 < p < ∞ and c0, but not by `1, `∞ nor Lp(0, 1) for 1 ≤ p 6= 2 ≤ ∞. S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 14 / 22 Introduction: Main Concepts Back to Corona and cluster value problems: When B is the unit ball of an infinite-dimensional Banach space, there are no positive solutions to the Corona problem. Aron, Carando, Gamelin, Lasalle, Maestre, ’12 [1]: Cluster value theorem for x = 0 and Au(B), when X has a shrinking 1-unconditional basis. Example The previous condition is satisfied by `p for 1 < p < ∞ and c0, but not by `1, `∞ nor Lp(0, 1) for 1 ≤ p 6= 2 ≤ ∞. S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 14 / 22 Introduction: Main Concepts Cluster value theorem for Au(B) when X is a Hilbert space. ∞ Cluster value theorem for H (B) when X = c0. Remark Hilbert space and c0 are infinite-dimensional analogues of the unit ball and the polydisk of Euclidean space. More known cluster value theorems: Farmer, ’98: There is a cluster value theorem for each point in ∂B and Au(B), when B is the unit ball of a uniformly convex Banach space, like `p and Lp for 1 < p < ∞; Acosta and Lourenzo, ’07: There is a cluster value theorem for Au(B`1 ) at each point in ∂B`1 . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 15 / 22 Introduction: Main Concepts Cluster value theorem for Au(B) when X is a Hilbert space. ∞ Cluster value theorem for H (B) when X = c0. Remark Hilbert space and c0 are infinite-dimensional analogues of the unit ball and the polydisk of Euclidean space. More known cluster value theorems: Farmer, ’98: There is a cluster value theorem for each point in ∂B and Au(B), when B is the unit ball of a uniformly convex Banach space, like `p and Lp for 1 < p < ∞; Acosta and Lourenzo, ’07: There is a cluster value theorem for Au(B`1 ) at each point in ∂B`1 . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 15 / 22 Introduction: Main Concepts Cluster value theorem for Au(B) when X is a Hilbert space. ∞ Cluster value theorem for H (B) when X = c0. Remark Hilbert space and c0 are infinite-dimensional analogues of the unit ball and the polydisk of Euclidean space. More known cluster value theorems: Farmer, ’98: There is a cluster value theorem for each point in ∂B and Au(B), when B is the unit ball of a uniformly convex Banach space, like `p and Lp for 1 < p < ∞; Acosta and Lourenzo, ’07: There is a cluster value theorem for Au(B`1 ) at each point in ∂B`1 . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 15 / 22 Introduction: Main Concepts Cluster value theorem for Au(B) when X is a Hilbert space. ∞ Cluster value theorem for H (B) when X = c0. Remark Hilbert space and c0 are infinite-dimensional analogues of the unit ball and the polydisk of Euclidean space. More known cluster value theorems: Farmer, ’98: There is a cluster value theorem for each point in ∂B and Au(B), when B is the unit ball of a uniformly convex Banach space, like `p and Lp for 1 < p < ∞; Acosta and Lourenzo, ’07: There is a cluster value theorem for Au(B`1 ) at each point in ∂B`1 . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 15 / 22 Introduction: Main Concepts Cluster value theorem for Au(B) when X is a Hilbert space. ∞ Cluster value theorem for H (B) when X = c0. Remark Hilbert space and c0 are infinite-dimensional analogues of the unit ball and the polydisk of Euclidean space. More known cluster value theorems: Farmer, ’98: There is a cluster value theorem for each point in ∂B and Au(B), when B is the unit ball of a uniformly convex Banach space, like `p and Lp for 1 < p < ∞; Acosta and Lourenzo, ’07: There is a cluster value theorem for Au(B`1 ) at each point in ∂B`1 . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 15 / 22 Introduction: Main Concepts Cluster value theorem for Au(B) when X is a Hilbert space. ∞ Cluster value theorem for H (B) when X = c0. Remark Hilbert space and c0 are infinite-dimensional analogues of the unit ball and the polydisk of Euclidean space. More known cluster value theorems: Farmer, ’98: There is a cluster value theorem for each point in ∂B and Au(B), when B is the unit ball of a uniformly convex Banach space, like `p and Lp for 1 < p < ∞; Acosta and Lourenzo, ’07: There is a cluster value theorem for Au(B`1 ) at each point in ∂B`1 . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 15 / 22 Our Cluster Value Theorems Generalizing the ideas and techniques of Aron, Carando, Gamelin, Lasalle and Maestre, we proved: Theorem (Johnson, O., ’13) Suppose that for each finite-dimensional subspace E of X∗ and > 0 ∗ there exists a finite rank operator S on X so that k(I − S )|Ek < and kI − Sk = 1. Then the cluster value theorem holds for Au(B) at 0. Remark If X is a Banach space with a shrinking reverse monotone Finite Dimensional Decomposition, we have a cluster value theorem for Au(B) at 0. S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 16 / 22 Our Cluster Value Theorems Generalizing the ideas and techniques of Aron, Carando, Gamelin, Lasalle and Maestre, we proved: Theorem (Johnson, O., ’13) Suppose that for each finite-dimensional subspace E of X∗ and > 0 ∗ there exists a finite rank operator S on X so that k(I − S )|Ek < and kI − Sk = 1. Then the cluster value theorem holds for Au(B) at 0. Remark If X is a Banach space with a shrinking reverse monotone Finite Dimensional Decomposition, we have a cluster value theorem for Au(B) at 0. S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 16 / 22 Our Cluster Value Theorems Generalizing the ideas and techniques of Aron, Carando, Gamelin, Lasalle and Maestre, we proved: Theorem (Johnson, O., ’13) Suppose that for each finite-dimensional subspace E of X∗ and > 0 ∗ there exists a finite rank operator S on X so that k(I − S )|Ek < and kI − Sk = 1. Then the cluster value theorem holds for Au(B) at 0. Remark If X is a Banach space with a shrinking reverse monotone Finite Dimensional Decomposition, we have a cluster value theorem for Au(B) at 0. S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 16 / 22 Our Cluster Value Theorems We also proved the following relationship with the help of Aron and Maestre: Lemma (Aron, Maestre) If Y is a closed finite-codimensional subspace of X and f ∈ Au(B), then ClB(f , 0) = ClBY (f |Y , 0), where BY is the unit ball of Y. Since c is one-codimensional in c0 and c0 satisfies a cluster value theorem, the previous result suggests that c satisfies a cluster value thorem for Au(B). It turns out that Au(Bc) = A(Bc) because c is isomorphic to c0, so c indeed satisfies a cluster value theorem for Au(Bc). Moreover, Au(B) = A(B) when X = C(K) for any K compact, Hausdorff and dispersed, so X satisfies a cluster value theorem for Au(B). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 17 / 22 Our Cluster Value Theorems We also proved the following relationship with the help of Aron and Maestre: Lemma (Aron, Maestre) If Y is a closed finite-codimensional subspace of X and f ∈ Au(B), then ClB(f , 0) = ClBY (f |Y , 0), where BY is the unit ball of Y. Since c is one-codimensional in c0 and c0 satisfies a cluster value theorem, the previous result suggests that c satisfies a cluster value thorem for Au(B). It turns out that Au(Bc) = A(Bc) because c is isomorphic to c0, so c indeed satisfies a cluster value theorem for Au(Bc). Moreover, Au(B) = A(B) when X = C(K) for any K compact, Hausdorff and dispersed, so X satisfies a cluster value theorem for Au(B). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 17 / 22 Our Cluster Value Theorems We also proved the following relationship with the help of Aron and Maestre: Lemma (Aron, Maestre) If Y is a closed finite-codimensional subspace of X and f ∈ Au(B), then ClB(f , 0) = ClBY (f |Y , 0), where BY is the unit ball of Y. Since c is one-codimensional in c0 and c0 satisfies a cluster value theorem, the previous result suggests that c satisfies a cluster value thorem for Au(B). It turns out that Au(Bc) = A(Bc) because c is isomorphic to c0, so c indeed satisfies a cluster value theorem for Au(Bc). Moreover, Au(B) = A(B) when X = C(K) for any K compact, Hausdorff and dispersed, so X satisfies a cluster value theorem for Au(B). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 17 / 22 Our Cluster Value Theorems We also proved the following relationship with the help of Aron and Maestre: Lemma (Aron, Maestre) If Y is a closed finite-codimensional subspace of X and f ∈ Au(B), then ClB(f , 0) = ClBY (f |Y , 0), where BY is the unit ball of Y. Since c is one-codimensional in c0 and c0 satisfies a cluster value theorem, the previous result suggests that c satisfies a cluster value thorem for Au(B). It turns out that Au(Bc) = A(Bc) because c is isomorphic to c0, so c indeed satisfies a cluster value theorem for Au(Bc). Moreover, Au(B) = A(B) when X = C(K) for any K compact, Hausdorff and dispersed, so X satisfies a cluster value theorem for Au(B). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 17 / 22 Our Cluster Value Theorems We also proved the following relationship with the help of Aron and Maestre: Lemma (Aron, Maestre) If Y is a closed finite-codimensional subspace of X and f ∈ Au(B), then ClB(f , 0) = ClBY (f |Y , 0), where BY is the unit ball of Y. Since c is one-codimensional in c0 and c0 satisfies a cluster value theorem, the previous result suggests that c satisfies a cluster value thorem for Au(B). It turns out that Au(Bc) = A(Bc) because c is isomorphic to c0, so c indeed satisfies a cluster value theorem for Au(Bc). Moreover, Au(B) = A(B) when X = C(K) for any K compact, Hausdorff and dispersed, so X satisfies a cluster value theorem for Au(B). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 17 / 22 Our Cluster Value Theorems Following the proof by Aron, Carando, Gamelin, Lasalle and Maestre ∞ that c0 satisfies a cluster value theorem for H (B), and using that ∗ C(K) = `1(K) when K is compact, Hausdorff and dispersed, we obtain a cluster value theorem for H∞(B) when X = C(K), and K is compact, Hausdorff and dispersed. ∗∗ ∗∗ We consider the following in [2]: Given f0 ∈ B , the cluster value ∞ ∗∗ ∞ problem for H (B) over Au(B) at f0 asks whether for all ψ ∈ H (B) and τ ∈ M ∗∗ (B), can we find a net (f ) ⊂ B such that ψ(f ) → τ(ψ) f0 α α ∗∗ and fα converges to f0 in the polynomial-star topology (that we denote ∗∗ by τ(ψ) ∈ ClB(ψ, f0 ))? Theorem ∞ The cluster value theorem of H (B) over Au(B) at 0 is equivalent to the ∞ cluster value theorem of H (B) over Au(B) at any f0 ∈ B, for X = C(K). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 18 / 22 Our Cluster Value Theorems Following the proof by Aron, Carando, Gamelin, Lasalle and Maestre ∞ that c0 satisfies a cluster value theorem for H (B), and using that ∗ C(K) = `1(K) when K is compact, Hausdorff and dispersed, we obtain a cluster value theorem for H∞(B) when X = C(K), and K is compact, Hausdorff and dispersed. ∗∗ ∗∗ We consider the following in [2]: Given f0 ∈ B , the cluster value ∞ ∗∗ ∞ problem for H (B) over Au(B) at f0 asks whether for all ψ ∈ H (B) and τ ∈ M ∗∗ (B), can we find a net (f ) ⊂ B such that ψ(f ) → τ(ψ) f0 α α ∗∗ and fα converges to f0 in the polynomial-star topology (that we denote ∗∗ by τ(ψ) ∈ ClB(ψ, f0 ))? Theorem ∞ The cluster value theorem of H (B) over Au(B) at 0 is equivalent to the ∞ cluster value theorem of H (B) over Au(B) at any f0 ∈ B, for X = C(K). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 18 / 22 Our Cluster Value Theorems Following the proof by Aron, Carando, Gamelin, Lasalle and Maestre ∞ that c0 satisfies a cluster value theorem for H (B), and using that ∗ C(K) = `1(K) when K is compact, Hausdorff and dispersed, we obtain a cluster value theorem for H∞(B) when X = C(K), and K is compact, Hausdorff and dispersed. ∗∗ ∗∗ We consider the following in [2]: Given f0 ∈ B , the cluster value ∞ ∗∗ ∞ problem for H (B) over Au(B) at f0 asks whether for all ψ ∈ H (B) and τ ∈ M ∗∗ (B), can we find a net (f ) ⊂ B such that ψ(f ) → τ(ψ) f0 α α ∗∗ and fα converges to f0 in the polynomial-star topology (that we denote ∗∗ by τ(ψ) ∈ ClB(ψ, f0 ))? Theorem ∞ The cluster value theorem of H (B) over Au(B) at 0 is equivalent to the ∞ cluster value theorem of H (B) over Au(B) at any f0 ∈ B, for X = C(K). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 18 / 22 Our Cluster Value Theorems Following the proof by Aron, Carando, Gamelin, Lasalle and Maestre ∞ that c0 satisfies a cluster value theorem for H (B), and using that ∗ C(K) = `1(K) when K is compact, Hausdorff and dispersed, we obtain a cluster value theorem for H∞(B) when X = C(K), and K is compact, Hausdorff and dispersed. ∗∗ ∗∗ We consider the following in [2]: Given f0 ∈ B , the cluster value ∞ ∗∗ ∞ problem for H (B) over Au(B) at f0 asks whether for all ψ ∈ H (B) and τ ∈ M ∗∗ (B), can we find a net (f ) ⊂ B such that ψ(f ) → τ(ψ) f0 α α ∗∗ and fα converges to f0 in the polynomial-star topology (that we denote ∗∗ by τ(ψ) ∈ ClB(ψ, f0 ))? Theorem ∞ The cluster value theorem of H (B) over Au(B) at 0 is equivalent to the ∞ cluster value theorem of H (B) over Au(B) at any f0 ∈ B, for X = C(K). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 18 / 22 Our Cluster Value Theorems In [3] we prove that for any separable Banach space Y, a cluster value ∞ problem for H(BY ) (H = H or H = Au) can be reduced to a cluster value problem for H(BX) for some Banach space X that is an `1-sum of a sequence of finite-dimensional spaces. In particular, if H(B`1 ) satisfies the cluster value theorem, then so does H(BL1 ). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 19 / 22 Our Cluster Value Theorems In [3] we prove that for any separable Banach space Y, a cluster value ∞ problem for H(BY ) (H = H or H = Au) can be reduced to a cluster value problem for H(BX) for some Banach space X that is an `1-sum of a sequence of finite-dimensional spaces. In particular, if H(B`1 ) satisfies the cluster value theorem, then so does H(BL1 ). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 19 / 22 Open Cluster Value Problems and Related Questions ∞ Does the cluster value theorem hold for H (B) or Au(B), when B = B`1 or B = BX and X is uniformly convex? Remark ¯ Lempert, ’99: There is a solution to the ∂ problem in B`1 . Is the previous solution weakly continuous? Is there a solution to the ∂¯ problem in BX for X uniformly convex? If so, is it weakly continuous? Remark Kerzman, ’71: There is a solution to the ∂¯ problem in strongly n pseudoconvex domains in C . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 20 / 22 Open Cluster Value Problems and Related Questions ∞ Does the cluster value theorem hold for H (B) or Au(B), when B = B`1 or B = BX and X is uniformly convex? Remark ¯ Lempert, ’99: There is a solution to the ∂ problem in B`1 . Is the previous solution weakly continuous? Is there a solution to the ∂¯ problem in BX for X uniformly convex? If so, is it weakly continuous? Remark Kerzman, ’71: There is a solution to the ∂¯ problem in strongly n pseudoconvex domains in C . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 20 / 22 Open Cluster Value Problems and Related Questions ∞ Does the cluster value theorem hold for H (B) or Au(B), when B = B`1 or B = BX and X is uniformly convex? Remark ¯ Lempert, ’99: There is a solution to the ∂ problem in B`1 . Is the previous solution weakly continuous? Is there a solution to the ∂¯ problem in BX for X uniformly convex? If so, is it weakly continuous? Remark Kerzman, ’71: There is a solution to the ∂¯ problem in strongly n pseudoconvex domains in C . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 20 / 22 Open Cluster Value Problems and Related Questions ∞ Does the cluster value theorem hold for H (B) or Au(B), when B = B`1 or B = BX and X is uniformly convex? Remark ¯ Lempert, ’99: There is a solution to the ∂ problem in B`1 . Is the previous solution weakly continuous? Is there a solution to the ∂¯ problem in BX for X uniformly convex? If so, is it weakly continuous? Remark Kerzman, ’71: There is a solution to the ∂¯ problem in strongly n pseudoconvex domains in C . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 20 / 22 Open Cluster Value Problems and Related Questions ∞ Does the cluster value theorem hold for H (B) or Au(B), when B = B`1 or B = BX and X is uniformly convex? Remark ¯ Lempert, ’99: There is a solution to the ∂ problem in B`1 . Is the previous solution weakly continuous? Is there a solution to the ∂¯ problem in BX for X uniformly convex? If so, is it weakly continuous? Remark Kerzman, ’71: There is a solution to the ∂¯ problem in strongly n pseudoconvex domains in C . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 20 / 22 Open Cluster Value Problems and Related Questions ∞ Does the cluster value theorem hold for H (B) or Au(B), when B = B`1 or B = BX and X is uniformly convex? Remark ¯ Lempert, ’99: There is a solution to the ∂ problem in B`1 . Is the previous solution weakly continuous? Is there a solution to the ∂¯ problem in BX for X uniformly convex? If so, is it weakly continuous? Remark Kerzman, ’71: There is a solution to the ∂¯ problem in strongly n pseudoconvex domains in C . S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 20 / 22 Open Cluster Value Problems and Related Questions Is there a solution to the ∂¯ problem in strongly pseudoconvex domains in any Banach space? ∞ Does the cluster value theorem for H (B n ) hold? `1 Remark Lempert, ’99: There is a solution to the ∂¯ problem in B n . `1 Is the solution to the ∂¯ problem in B n uniformly continuous? `1 S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 21 / 22 Open Cluster Value Problems and Related Questions Is there a solution to the ∂¯ problem in strongly pseudoconvex domains in any Banach space? ∞ Does the cluster value theorem for H (B n ) hold? `1 Remark Lempert, ’99: There is a solution to the ∂¯ problem in B n . `1 Is the solution to the ∂¯ problem in B n uniformly continuous? `1 S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 21 / 22 Open Cluster Value Problems and Related Questions Is there a solution to the ∂¯ problem in strongly pseudoconvex domains in any Banach space? ∞ Does the cluster value theorem for H (B n ) hold? `1 Remark Lempert, ’99: There is a solution to the ∂¯ problem in B n . `1 Is the solution to the ∂¯ problem in B n uniformly continuous? `1 S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 21 / 22 Open Cluster Value Problems and Related Questions Is there a solution to the ∂¯ problem in strongly pseudoconvex domains in any Banach space? ∞ Does the cluster value theorem for H (B n ) hold? `1 Remark Lempert, ’99: There is a solution to the ∂¯ problem in B n . `1 Is the solution to the ∂¯ problem in B n uniformly continuous? `1 S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 21 / 22 References R. M. Aron, D. Carando, T. W. Gamelin, S. Lasalle, M. Maestre, Cluster Values of Analytic Functions on a Banach space, Math. Ann. 353 (2012), pp. 293-303. W. B. Johnson, S. Ortega Castillo, The cluster value problem in spaces of continuous functions, to appear in Proceedings of the AMS (2013). W. B. Johnson, S. Ortega Castillo, The cluster value problem for Banach spaces, submitted to Illinois Journal of Mathematics (2013). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 22 / 22 References R. M. Aron, D. Carando, T. W. Gamelin, S. Lasalle, M. Maestre, Cluster Values of Analytic Functions on a Banach space, Math. Ann. 353 (2012), pp. 293-303. W. B. Johnson, S. Ortega Castillo, The cluster value problem in spaces of continuous functions, to appear in Proceedings of the AMS (2013). W. B. Johnson, S. Ortega Castillo, The cluster value problem for Banach spaces, submitted to Illinois Journal of Mathematics (2013). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 22 / 22 References R. M. Aron, D. Carando, T. W. Gamelin, S. Lasalle, M. Maestre, Cluster Values of Analytic Functions on a Banach space, Math. Ann. 353 (2012), pp. 293-303. W. B. Johnson, S. Ortega Castillo, The cluster value problem in spaces of continuous functions, to appear in Proceedings of the AMS (2013). W. B. Johnson, S. Ortega Castillo, The cluster value problem for Banach spaces, submitted to Illinois Journal of Mathematics (2013). S. Ortega Castillo (TAMU) Corona and cluster value p. in ∞-dim. spaces 04/13/2014 22 / 22