Luiza A. Moraes Algebras of Lorch Analytic Mappings Defined In

Algebras of Lorch analytic mappings deﬁned in uniform algebras

Luiza A. Moraes Instituto de Matemática Universidade Federal do Rio de Janeiro

Conference on Non-Linear Functional Analysis Valencia 2017

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 1 / 29 All the results presented in this talk were obtained in collaboration with

Guilherme V.S.Mauro (Universidade Federal da Integração Latino-Americana)

These results are part of

Guilherme V.S.Mauro and L.A.M. , Algebras of Lorch analytic mappings on uniform algebras. , Pre-print. and

Guilherme V.S.Mauro, Espectros de Álgebras de Aplicações Analíticas no sentido de Lorch, Tese de Doutorado, UFRJ (2016)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 2 / 29 M(A) coincides with the set of the closed maximal ideals of A. As a consequence: \ R(A) = ϕ−1(0). ϕ∈M(A)

M(A) = the set of the non-null complex valued continuous homomorphisms deﬁned in A (= spectrum of A).

τG = topology induced in M(A) by the weak*-topology (Gelfand topology). M(A) is always considered endowed with τG. Well known: M(A) is a compact Hausdorff space whenever A is a (commutative) Banach algebra The radical R(A) of is the intersection of all maximal ideals in A. An algebra A is called semi-simple if R(A) = {0}.

Well known:

Deﬁnitions and Notation

A = complex commutative unital Fréchet algebra.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 3 / 29 M(A) coincides with the set of the closed maximal ideals of A. As a consequence: \ R(A) = ϕ−1(0). ϕ∈M(A)

Well known:

Deﬁnitions and Notation

A = complex commutative unital Fréchet algebra. M(A) = the set of the non-null complex valued continuous homomorphisms deﬁned in A (= spectrum of A).

Well known: M(A) is a compact Hausdorff space whenever A is a (commutative) Banach algebra The radical R(A) of is the intersection of all maximal ideals in A. An algebra A is called semi-simple if R(A) = {0}.

Well known:

Deﬁnitions and Notation

A = complex commutative unital Fréchet algebra. M(A) = the set of the non-null complex valued continuous homomorphisms deﬁned in A (= spectrum of A).

τG = topology induced in M(A) by the weak*-topology (Gelfand topology). M(A) is always considered endowed with τG.

The radical R(A) of is the intersection of all maximal ideals in A. An algebra A is called semi-simple if R(A) = {0}.

Well known:

Deﬁnitions and Notation

A = complex commutative unital Fréchet algebra. M(A) = the set of the non-null complex valued continuous homomorphisms deﬁned in A (= spectrum of A).

An algebra A is called semi-simple if R(A) = {0}.

Well known:

Deﬁnitions and Notation

A = complex commutative unital Fréchet algebra. M(A) = the set of the non-null complex valued continuous homomorphisms deﬁned in A (= spectrum of A).

Well known:

Deﬁnitions and Notation

A = complex commutative unital Fréchet algebra. M(A) = the set of the non-null complex valued continuous homomorphisms deﬁned in A (= spectrum of A).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 3 / 29 As a consequence: \ R(A) = ϕ−1(0). ϕ∈M(A)

Deﬁnitions and Notation

A = complex commutative unital Fréchet algebra. M(A) = the set of the non-null complex valued continuous homomorphisms deﬁned in A (= spectrum of A).

Well known: M(A) coincides with the set of the closed maximal ideals of A.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 3 / 29 Deﬁnitions and Notation

A = complex commutative unital Fréchet algebra. M(A) = the set of the non-null complex valued continuous homomorphisms deﬁned in A (= spectrum of A).

Well known: M(A) coincides with the set of the closed maximal ideals of A. As a consequence: \ R(A) = ϕ−1(0). ϕ∈M(A)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 3 / 29 B ⊂ U is U-bounded if B is bounded and dist(B, ∂U) > 0.

Hb(U, E) denotes the space of holomorphic mappings from U into E which are bounded on the U-bounded subsets of U endowed with the topology τb of uniform convergence on the U-bounded subsets of U.

Well known: Hb(U, E) endowed with the usual pointwise operations is an algebra.

U = open connected subset of E.

Br (z0) = {z ∈ E : kz − z0k < r} for all z0 ∈ E and r > 0.

B1(0) will be denoted by BE .

dU (a) = dist (a, ∂U) > 0 for every a ∈ U.

Deﬁnitions and Notation

From now E = commutative complex Banach algebra with a unit e s.th. kek = 1.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 4 / 29 B ⊂ U is U-bounded if B is bounded and dist(B, ∂U) > 0.

Hb(U, E) denotes the space of holomorphic mappings from U into E which are bounded on the U-bounded subsets of U endowed with the topology τb of uniform convergence on the U-bounded subsets of U.

Well known: Hb(U, E) endowed with the usual pointwise operations is an algebra.

Br (z0) = {z ∈ E : kz − z0k < r} for all z0 ∈ E and r > 0.

B1(0) will be denoted by BE .

dU (a) = dist (a, ∂U) > 0 for every a ∈ U.

Deﬁnitions and Notation

From now E = commutative complex Banach algebra with a unit e s.th. kek = 1. U = open connected subset of E.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 4 / 29 B ⊂ U is U-bounded if B is bounded and dist(B, ∂U) > 0.

Hb(U, E) denotes the space of holomorphic mappings from U into E which are bounded on the U-bounded subsets of U endowed with the topology τb of uniform convergence on the U-bounded subsets of U.

Well known: Hb(U, E) endowed with the usual pointwise operations is an algebra.

B1(0) will be denoted by BE .

dU (a) = dist (a, ∂U) > 0 for every a ∈ U.

Deﬁnitions and Notation

From now E = commutative complex Banach algebra with a unit e s.th. kek = 1. U = open connected subset of E.

Br (z0) = {z ∈ E : kz − z0k < r} for all z0 ∈ E and r > 0.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 4 / 29 B ⊂ U is U-bounded if B is bounded and dist(B, ∂U) > 0.

Hb(U, E) denotes the space of holomorphic mappings from U into E which are bounded on the U-bounded subsets of U endowed with the topology τb of uniform convergence on the U-bounded subsets of U.

Well known: Hb(U, E) endowed with the usual pointwise operations is an algebra.

dU (a) = dist (a, ∂U) > 0 for every a ∈ U.

Deﬁnitions and Notation

From now E = commutative complex Banach algebra with a unit e s.th. kek = 1. U = open connected subset of E.

Br (z0) = {z ∈ E : kz − z0k < r} for all z0 ∈ E and r > 0.

B1(0) will be denoted by BE .

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 4 / 29 B ⊂ U is U-bounded if B is bounded and dist(B, ∂U) > 0.

Hb(U, E) denotes the space of holomorphic mappings from U into E which are bounded on the U-bounded subsets of U endowed with the topology τb of uniform convergence on the U-bounded subsets of U.

Well known: Hb(U, E) endowed with the usual pointwise operations is an algebra.

Deﬁnitions and Notation

From now E = commutative complex Banach algebra with a unit e s.th. kek = 1. U = open connected subset of E.

Br (z0) = {z ∈ E : kz − z0k < r} for all z0 ∈ E and r > 0.

B1(0) will be denoted by BE .

dU (a) = dist (a, ∂U) > 0 for every a ∈ U.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 4 / 29 Hb(U, E) denotes the space of holomorphic mappings from U into E which are bounded on the U-bounded subsets of U endowed with the topology τb of uniform convergence on the U-bounded subsets of U.

Well known: Hb(U, E) endowed with the usual pointwise operations is an algebra.

Deﬁnitions and Notation

From now E = commutative complex Banach algebra with a unit e s.th. kek = 1. U = open connected subset of E.

Br (z0) = {z ∈ E : kz − z0k < r} for all z0 ∈ E and r > 0.

B1(0) will be denoted by BE .

dU (a) = dist (a, ∂U) > 0 for every a ∈ U.

B ⊂ U is U-bounded if B is bounded and dist(B, ∂U) > 0.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 4 / 29 Well known: Hb(U, E) endowed with the usual pointwise operations is an algebra.

Deﬁnitions and Notation

From now E = commutative complex Banach algebra with a unit e s.th. kek = 1. U = open connected subset of E.

Br (z0) = {z ∈ E : kz − z0k < r} for all z0 ∈ E and r > 0.

B1(0) will be denoted by BE .

dU (a) = dist (a, ∂U) > 0 for every a ∈ U.

B ⊂ U is U-bounded if B is bounded and dist(B, ∂U) > 0.

Hb(U, E) denotes the space of holomorphic mappings from U into E which are bounded on the U-bounded subsets of U endowed with the topology τb of uniform convergence on the U-bounded subsets of U.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 4 / 29 Deﬁnitions and Notation

From now E = commutative complex Banach algebra with a unit e s.th. kek = 1. U = open connected subset of E.

Br (z0) = {z ∈ E : kz − z0k < r} for all z0 ∈ E and r > 0.

B1(0) will be denoted by BE .

dU (a) = dist (a, ∂U) > 0 for every a ∈ U.

B ⊂ U is U-bounded if B is bounded and dist(B, ∂U) > 0.

Hb(U, E) denotes the space of holomorphic mappings from U into E which are bounded on the U-bounded subsets of U endowed with the topology τb of uniform convergence on the U-bounded subsets of U.

Well known: Hb(U, E) endowed with the usual pointwise operations is an algebra.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 4 / 29 Easy to check: f is holomorphic in U in the standard sense in holomorphy whenever f is Lorch analytic in U .

The converse is not true. Classical example: The linear mapping 2 2 f : C → C deﬁned by f (z1, z2) = (z2, z1) is not Lorch analytic.

We say that f is Lorch analytic in U if f has a Lorch-derivative at each point of U.

The Lorch analytic Mappings

Deﬁnition If U is an open subset of commutative complex Banach algebra E, we say 0 that f : U → E has a Lorch-derivative f (z0) ∈ E at z0 ∈ U if for each ε > 0 there exists δ > 0 such that for all h ∈ E satisfying khk < δ we have

0 kf (z0 + h) − f (z0) − h f (z0)k < εkhk.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 5 / 29 Easy to check: f is holomorphic in U in the standard sense in holomorphy whenever f is Lorch analytic in U .

The converse is not true. Classical example: The linear mapping 2 2 f : C → C deﬁned by f (z1, z2) = (z2, z1) is not Lorch analytic.

The Lorch analytic Mappings

0 kf (z0 + h) − f (z0) − h f (z0)k < εkhk.

We say that f is Lorch analytic in U if f has a Lorch-derivative at each point of U.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 5 / 29 The converse is not true. Classical example: The linear mapping 2 2 f : C → C deﬁned by f (z1, z2) = (z2, z1) is not Lorch analytic.

The Lorch analytic Mappings

0 kf (z0 + h) − f (z0) − h f (z0)k < εkhk.

We say that f is Lorch analytic in U if f has a Lorch-derivative at each point of U.

Easy to check: f is holomorphic in U in the standard sense in holomorphy whenever f is Lorch analytic in U .

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 5 / 29 The Lorch analytic Mappings

0 kf (z0 + h) − f (z0) − h f (z0)k < εkhk.

We say that f is Lorch analytic in U if f has a Lorch-derivative at each point of U.

Easy to check: f is holomorphic in U in the standard sense in holomorphy whenever f is Lorch analytic in U .

The converse is not true. Classical example: The linear mapping 2 2 f : C → C deﬁned by f (z1, z2) = (z2, z1) is not Lorch analytic.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 5 / 29 HL(U) denotes the vector space of all Lorch analytic mappings in an open subset U of E.

Proposition

Let U1 and U2 be connected open subsets of E. If f ∈ HL(U1) is such that f (U1) ⊂ U2 and g ∈ HL(U2), then g ◦ f ∈ HL(U1) and

(g ◦ f )0(z) = g0(f (z))f 0(z)

for all z ∈ U1.

Proposition

A mapping f : U → E is Lorch analytic in U if and only if given any z0 ∈ U there exists r > 0 and there exist (unique) elements an ∈ E, such that P∞ n Br (z0) ⊂ U and f (z) = n=0 an(z − z0) , for all z ∈ Br (z0).

The Lorch analytic mappings

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 6 / 29 Proposition

Let U1 and U2 be connected open subsets of E. If f ∈ HL(U1) is such that f (U1) ⊂ U2 and g ∈ HL(U2), then g ◦ f ∈ HL(U1) and

(g ◦ f )0(z) = g0(f (z))f 0(z)

for all z ∈ U1.

Proposition

The Lorch analytic mappings

HL(U) denotes the vector space of all Lorch analytic mappings in an open subset U of E.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 6 / 29 Proposition

The Lorch analytic mappings

HL(U) denotes the vector space of all Lorch analytic mappings in an open subset U of E.

Proposition

Let U1 and U2 be connected open subsets of E. If f ∈ HL(U1) is such that f (U1) ⊂ U2 and g ∈ HL(U2), then g ◦ f ∈ HL(U1) and

(g ◦ f )0(z) = g0(f (z))f 0(z)

for all z ∈ U1.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 6 / 29 The Lorch analytic mappings

HL(U) denotes the vector space of all Lorch analytic mappings in an open subset U of E.

Proposition

Let U1 and U2 be connected open subsets of E. If f ∈ HL(U1) is such that f (U1) ⊂ U2 and g ∈ HL(U2), then g ◦ f ∈ HL(U1) and

(g ◦ f )0(z) = g0(f (z))f 0(z)

for all z ∈ U1.

Proposition

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 6 / 29 Since HL(E) ⊂ Hb(E, E), we have that f ∈ HL(E) if and only if there 1 exists (an)n ⊂ E satisfying lim kank n = 0 and such that n→∞

∞ X n f (z) = anz , for all z ∈ E. n=0

With the usual operations, HL(E) and HL(Br (z0)) are closed subalgebras of Hb(E, E) and Hb(Br (z0), E) respectively, and so (HL(E), τb) and (Hb(Br (z0), E), τb) are commutative Fréchet algebras (with identity).

For arbitrary U we may have HL(U) 6⊂ Hb(U, E) and in this case we can not consider HL(U) endowed with the topology τb.

We are going to introduce a convenient topology in HL(U).

The spaces HL(E) and HL(U)

Known:

HL(U) $ Hb(U, E) in cases U = E and U = Br (z0).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 7 / 29 With the usual operations, HL(E) and HL(Br (z0)) are closed subalgebras of Hb(E, E) and Hb(Br (z0), E) respectively, and so (HL(E), τb) and (Hb(Br (z0), E), τb) are commutative Fréchet algebras (with identity).

For arbitrary U we may have HL(U) 6⊂ Hb(U, E) and in this case we can not consider HL(U) endowed with the topology τb.

We are going to introduce a convenient topology in HL(U).

The spaces HL(E) and HL(U)

Known:

HL(U) $ Hb(U, E) in cases U = E and U = Br (z0). Since HL(E) ⊂ Hb(E, E), we have that f ∈ HL(E) if and only if there 1 exists (an)n ⊂ E satisfying lim kank n = 0 and such that n→∞

∞ X n f (z) = anz , for all z ∈ E. n=0

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 7 / 29 For arbitrary U we may have HL(U) 6⊂ Hb(U, E) and in this case we can not consider HL(U) endowed with the topology τb.

We are going to introduce a convenient topology in HL(U).

The spaces HL(E) and HL(U)

Known:

HL(U) $ Hb(U, E) in cases U = E and U = Br (z0). Since HL(E) ⊂ Hb(E, E), we have that f ∈ HL(E) if and only if there 1 exists (an)n ⊂ E satisfying lim kank n = 0 and such that n→∞

∞ X n f (z) = anz , for all z ∈ E. n=0

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 7 / 29 We are going to introduce a convenient topology in HL(U).

The spaces HL(E) and HL(U)

Known:

HL(U) $ Hb(U, E) in cases U = E and U = Br (z0). Since HL(E) ⊂ Hb(E, E), we have that f ∈ HL(E) if and only if there 1 exists (an)n ⊂ E satisfying lim kank n = 0 and such that n→∞

∞ X n f (z) = anz , for all z ∈ E. n=0

For arbitrary U we may have HL(U) 6⊂ Hb(U, E) and in this case we can not consider HL(U) endowed with the topology τb.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 7 / 29 We are going to introduce a convenient topology in HL(U).

The spaces HL(E) and HL(U)

Known:

HL(U) $ Hb(U, E) in cases U = E and U = Br (z0). Since HL(E) ⊂ Hb(E, E), we have that f ∈ HL(E) if and only if there 1 exists (an)n ⊂ E satisfying lim kank n = 0 and such that n→∞

∞ X n f (z) = anz , for all z ∈ E. n=0

For arbitrary U we may have HL(U) 6⊂ Hb(U, E) and in this case we can not consider HL(U) endowed with the topology τb.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 7 / 29 The spaces HL(E) and HL(U)

Known:

HL(U) $ Hb(U, E) in cases U = E and U = Br (z0). Since HL(E) ⊂ Hb(E, E), we have that f ∈ HL(E) if and only if there 1 exists (an)n ⊂ E satisfying lim kank n = 0 and such that n→∞

∞ X n f (z) = anz , for all z ∈ E. n=0

For arbitrary U we may have HL(U) 6⊂ Hb(U, E) and in this case we can not consider HL(U) endowed with the topology τb.

We are going to introduce a convenient topology in HL(U).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 7 / 29 Hb(U; F) ⊂ Hd (U; F) ⊂ H(U; F)

HL(U) ⊂ Hd (U; E).

If U = Br (z0), where z0 ∈ E, r > 0, HL(Br (z0)) & Hb(Br (z0), E) = Hd (Br (z0), E) and τd = τb in HL(Br (z0)).

In particular HL(BE ) & Hb(BE , E) = Hd (BE , E) and τb = τd .

For f : U → F (where F = complex Banach space) and Br (z) ⊂ U, let

kf kBr (z) =: sup{kf (w)k ; w ∈ Br (z)}.

Hd (U, F) =: {f ∈ H(U; F): kf kBr (z) < ∞ ∀z ∈ U and ∀ 0 < r < dU (z)}

Let τd = topology generated in Hd (U, F) by the family of semi-norms

{k · kBr (z) : 0 < r < dU (z) and z ∈ U}.

The space HL(U, E)

It is known that if f ∈ HL(U) and w0 ∈ U, then rbf (w0) = dU (w0) where rbf (w0) = radius of boundedness of f at w0.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 8 / 29 Hb(U; F) ⊂ Hd (U; F) ⊂ H(U; F)

HL(U) ⊂ Hd (U; E).

If U = Br (z0), where z0 ∈ E, r > 0, HL(Br (z0)) & Hb(Br (z0), E) = Hd (Br (z0), E) and τd = τb in HL(Br (z0)).

In particular HL(BE ) & Hb(BE , E) = Hd (BE , E) and τb = τd .

For f : U → F (where F = complex Banach space) and Br (z) ⊂ U, let

kf kBr (z) =: sup{kf (w)k ; w ∈ Br (z)}.

Hd (U, F) =: {f ∈ H(U; F): kf kBr (z) < ∞ ∀z ∈ U and ∀ 0 < r < dU (z)}

Let τd = topology generated in Hd (U, F) by the family of semi-norms

{k · kBr (z) : 0 < r < dU (z) and z ∈ U}.

The space HL(U, E)

It is known that if f ∈ HL(U) and w0 ∈ U, then rbf (w0) = dU (w0) where rbf (w0) = radius of boundedness of f at w0.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 8 / 29 Hb(U; F) ⊂ Hd (U; F) ⊂ H(U; F)

HL(U) ⊂ Hd (U; E).

If U = Br (z0), where z0 ∈ E, r > 0, HL(Br (z0)) & Hb(Br (z0), E) = Hd (Br (z0), E) and τd = τb in HL(Br (z0)).

In particular HL(BE ) & Hb(BE , E) = Hd (BE , E) and τb = τd .

Let τd = topology generated in Hd (U, F) by the family of semi-norms

{k · kBr (z) : 0 < r < dU (z) and z ∈ U}.

The space HL(U, E)

It is known that if f ∈ HL(U) and w0 ∈ U, then rbf (w0) = dU (w0) where rbf (w0) = radius of boundedness of f at w0.

For f : U → F (where F = complex Banach space) and Br (z) ⊂ U, let

kf kBr (z) =: sup{kf (w)k ; w ∈ Br (z)}.

Hd (U, F) =: {f ∈ H(U; F): kf kBr (z) < ∞ ∀z ∈ U and ∀ 0 < r < dU (z)}

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 8 / 29 If U = Br (z0), where z0 ∈ E, r > 0, HL(Br (z0)) & Hb(Br (z0), E) = Hd (Br (z0), E) and τd = τb in HL(Br (z0)).

In particular HL(BE ) & Hb(BE , E) = Hd (BE , E) and τb = τd .

HL(U) ⊂ Hd (U; E).

Let τd = topology generated in Hd (U, F) by the family of semi-norms

{k · kBr (z) : 0 < r < dU (z) and z ∈ U}.

The space HL(U, E)

It is known that if f ∈ HL(U) and w0 ∈ U, then rbf (w0) = dU (w0) where rbf (w0) = radius of boundedness of f at w0.

For f : U → F (where F = complex Banach space) and Br (z) ⊂ U, let

kf kBr (z) =: sup{kf (w)k ; w ∈ Br (z)}.

Hd (U, F) =: {f ∈ H(U; F): kf kBr (z) < ∞ ∀z ∈ U and ∀ 0 < r < dU (z)}

Hb(U; F) ⊂ Hd (U; F) ⊂ H(U; F)

In particular HL(BE ) & Hb(BE , E) = Hd (BE , E) and τb = τd .

Let τd = topology generated in Hd (U, F) by the family of semi-norms

{k · kBr (z) : 0 < r < dU (z) and z ∈ U}.

The space HL(U, E)

It is known that if f ∈ HL(U) and w0 ∈ U, then rbf (w0) = dU (w0) where rbf (w0) = radius of boundedness of f at w0.

For f : U → F (where F = complex Banach space) and Br (z) ⊂ U, let

kf kBr (z) =: sup{kf (w)k ; w ∈ Br (z)}.

Hd (U, F) =: {f ∈ H(U; F): kf kBr (z) < ∞ ∀z ∈ U and ∀ 0 < r < dU (z)}

Hb(U; F) ⊂ Hd (U; F) ⊂ H(U; F)

HL(U) ⊂ Hd (U; E).

In particular HL(BE ) & Hb(BE , E) = Hd (BE , E) and τb = τd .

Let τd = topology generated in Hd (U, F) by the family of semi-norms

{k · kBr (z) : 0 < r < dU (z) and z ∈ U}.

The space HL(U, E)

It is known that if f ∈ HL(U) and w0 ∈ U, then rbf (w0) = dU (w0) where rbf (w0) = radius of boundedness of f at w0.

For f : U → F (where F = complex Banach space) and Br (z) ⊂ U, let

kf kBr (z) =: sup{kf (w)k ; w ∈ Br (z)}.

Hd (U, F) =: {f ∈ H(U; F): kf kBr (z) < ∞ ∀z ∈ U and ∀ 0 < r < dU (z)}

Hb(U; F) ⊂ Hd (U; F) ⊂ H(U; F)

HL(U) ⊂ Hd (U; E).

In particular HL(BE ) & Hb(BE , E) = Hd (BE , E) and τb = τd .

The space HL(U, E)

It is known that if f ∈ HL(U) and w0 ∈ U, then rbf (w0) = dU (w0) where rbf (w0) = radius of boundedness of f at w0.

For f : U → F (where F = complex Banach space) and Br (z) ⊂ U, let

kf kBr (z) =: sup{kf (w)k ; w ∈ Br (z)}.

Hd (U, F) =: {f ∈ H(U; F): kf kBr (z) < ∞ ∀z ∈ U and ∀ 0 < r < dU (z)}

Hb(U; F) ⊂ Hd (U; F) ⊂ H(U; F)

HL(U) ⊂ Hd (U; E).

Let τd = topology generated in Hd (U, F) by the family of semi-norms

{k · kBr (z) : 0 < r < dU (z) and z ∈ U}.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 8 / 29 In particular HL(BE ) & Hb(BE , E) = Hd (BE , E) and τb = τd .

The space HL(U, E)

It is known that if f ∈ HL(U) and w0 ∈ U, then rbf (w0) = dU (w0) where rbf (w0) = radius of boundedness of f at w0.

For f : U → F (where F = complex Banach space) and Br (z) ⊂ U, let

kf kBr (z) =: sup{kf (w)k ; w ∈ Br (z)}.

Hd (U, F) =: {f ∈ H(U; F): kf kBr (z) < ∞ ∀z ∈ U and ∀ 0 < r < dU (z)}

Hb(U; F) ⊂ Hd (U; F) ⊂ H(U; F)

HL(U) ⊂ Hd (U; E).

Let τd = topology generated in Hd (U, F) by the family of semi-norms

{k · kBr (z) : 0 < r < dU (z) and z ∈ U}.

If U = Br (z0), where z0 ∈ E, r > 0, HL(Br (z0)) & Hb(Br (z0), E) = Hd (Br (z0), E) and τd = τb in HL(Br (z0)).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 8 / 29 The space HL(U, E)

It is known that if f ∈ HL(U) and w0 ∈ U, then rbf (w0) = dU (w0) where rbf (w0) = radius of boundedness of f at w0.

For f : U → F (where F = complex Banach space) and Br (z) ⊂ U, let

kf kBr (z) =: sup{kf (w)k ; w ∈ Br (z)}.

Hd (U, F) =: {f ∈ H(U; F): kf kBr (z) < ∞ ∀z ∈ U and ∀ 0 < r < dU (z)}

Hb(U; F) ⊂ Hd (U; F) ⊂ H(U; F)

HL(U) ⊂ Hd (U; E).

Let τd = topology generated in Hd (U, F) by the family of semi-norms

{k · kBr (z) : 0 < r < dU (z) and z ∈ U}.

If U = Br (z0), where z0 ∈ E, r > 0, HL(Br (z0)) & Hb(Br (z0), E) = Hd (Br (z0), E) and τd = τb in HL(Br (z0)).

In particular HL(BE ) & Hb(BE , E) = Hd (BE , E) and τb = τd .

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 8 / 29 From now HL(U) denotes (HL(U), τd ).

Proposition

HL(U) is a closed subspace of (Hd (U; E), τd ) whenever E is separable. Consequently, (HL(U), τd ) is a Fréchet space whenever E is separable.

The spectrum of the Fréchet algebra (with identity) HL(U) has been described in cases U = E and by L.A.M. and A.F. Pereira in case U = E and by G.V.S. Mauro, L.A.M. and A.F. Pereira in case U = Br (z0). In both cases τd = τb.

The space HL(U, E)

E separable ⇒ (Hd (U, F), τd ) is a Fréchet space (Dineen and Venkova, for F = C)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 9 / 29 From now HL(U) denotes (HL(U), τd ).

Proposition

HL(U) is a closed subspace of (Hd (U; E), τd ) whenever E is separable. Consequently, (HL(U), τd ) is a Fréchet space whenever E is separable.

The space HL(U, E)

E separable ⇒ (Hd (U, F), τd ) is a Fréchet space (Dineen and Venkova, for F = C)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 9 / 29 From now HL(U) denotes (HL(U), τd ).

The space HL(U, E)

E separable ⇒ (Hd (U, F), τd ) is a Fréchet space (Dineen and Venkova, for F = C)

Proposition

HL(U) is a closed subspace of (Hd (U; E), τd ) whenever E is separable. Consequently, (HL(U), τd ) is a Fréchet space whenever E is separable.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 9 / 29 The space HL(U, E)

E separable ⇒ (Hd (U, F), τd ) is a Fréchet space (Dineen and Venkova, for F = C)

Proposition

HL(U) is a closed subspace of (Hd (U; E), τd ) whenever E is separable. Consequently, (HL(U), τd ) is a Fréchet space whenever E is separable.

From now HL(U) denotes (HL(U), τd ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 9 / 29 Let EΩ = {x ∈ E : σ(x) ⊂ Ω} (where σ(x) denotes the spectrum of x).

It is well known that EΩ is open in E. The symbolic calculus associates to each analytic function f :Ω −→ C the ˜ mapping f : EΩ −→ E deﬁned by Z ˜f (x) = f (λ)(λe − x)−1dλ. Γ Where Γ is any contour surrounding σ(x) in Ω.

We are going to present a description of the spectrum of (HL(EΩ), τd ) in case Ω (C is a simply connected domain and E is a uniform algebra. By using this characterization we can show that in this case the algebra (HL(EΩ), τd ) is semi-simple.

The symbolic calculus (or functional calculus)

Take any non empty open subset Ω of C.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 10 / 29 It is well known that EΩ is open in E. The symbolic calculus associates to each analytic function f :Ω −→ C the ˜ mapping f : EΩ −→ E deﬁned by Z ˜f (x) = f (λ)(λe − x)−1dλ. Γ Where Γ is any contour surrounding σ(x) in Ω.

The symbolic calculus (or functional calculus)

Take any non empty open subset Ω of C.

Let EΩ = {x ∈ E : σ(x) ⊂ Ω} (where σ(x) denotes the spectrum of x).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 10 / 29 The symbolic calculus associates to each analytic function f :Ω −→ C the ˜ mapping f : EΩ −→ E deﬁned by Z ˜f (x) = f (λ)(λe − x)−1dλ. Γ Where Γ is any contour surrounding σ(x) in Ω.

The symbolic calculus (or functional calculus)

Take any non empty open subset Ω of C.

Let EΩ = {x ∈ E : σ(x) ⊂ Ω} (where σ(x) denotes the spectrum of x).

It is well known that EΩ is open in E.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 10 / 29 We are going to present a description of the spectrum of (HL(EΩ), τd ) in case Ω (C is a simply connected domain and E is a uniform algebra. By using this characterization we can show that in this case the algebra (HL(EΩ), τd ) is semi-simple.

The symbolic calculus (or functional calculus)

Take any non empty open subset Ω of C.

Let EΩ = {x ∈ E : σ(x) ⊂ Ω} (where σ(x) denotes the spectrum of x).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 10 / 29 The symbolic calculus (or functional calculus)

Take any non empty open subset Ω of C.

Let EΩ = {x ∈ E : σ(x) ⊂ Ω} (where σ(x) denotes the spectrum of x).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 10 / 29 n Given any a ∈ E, and n ∈ N Pa,n(z) = az ∀z ∈ E.

Easy to check: φ(EΩ) = Ω for every φ ∈ M(E). In particular, φ(Br (z0)) = ∆r (φ(z0)).

In particular we will write Pe,1(z) = z and Pa,0(z) = a ∀z ∈ E. Using this notation we frequently write

∞ X n f = Pan,0(Pe,1 − Pz0,0) in U n=0 instead of ∞ X n f (z) = an(z − z0) , for all z ∈ U. n=0

Notation and Deﬁnitions As usual, {λ ∈ C; |λ − λ0| < r} will be denoted by ∆r (λ0) and ∆1(0) will be denoted by ∆.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 11 / 29 n Given any a ∈ E, and n ∈ N Pa,n(z) = az ∀z ∈ E.

In particular, φ(Br (z0)) = ∆r (φ(z0)).

In particular we will write Pe,1(z) = z and Pa,0(z) = a ∀z ∈ E. Using this notation we frequently write

∞ X n f = Pan,0(Pe,1 − Pz0,0) in U n=0 instead of ∞ X n f (z) = an(z − z0) , for all z ∈ U. n=0

Notation and Deﬁnitions As usual, {λ ∈ C; |λ − λ0| < r} will be denoted by ∆r (λ0) and ∆1(0) will be denoted by ∆. Easy to check: φ(EΩ) = Ω for every φ ∈ M(E).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 11 / 29 n Given any a ∈ E, and n ∈ N Pa,n(z) = az ∀z ∈ E.

In particular we will write Pe,1(z) = z and Pa,0(z) = a ∀z ∈ E. Using this notation we frequently write

∞ X n f = Pan,0(Pe,1 − Pz0,0) in U n=0 instead of ∞ X n f (z) = an(z − z0) , for all z ∈ U. n=0

Notation and Deﬁnitions As usual, {λ ∈ C; |λ − λ0| < r} will be denoted by ∆r (λ0) and ∆1(0) will be denoted by ∆. Easy to check: φ(EΩ) = Ω for every φ ∈ M(E). In particular, φ(Br (z0)) = ∆r (φ(z0)).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 11 / 29 Using this notation we frequently write

∞ X n f = Pan,0(Pe,1 − Pz0,0) in U n=0 instead of ∞ X n f (z) = an(z − z0) , for all z ∈ U. n=0

n Given any a ∈ E, and n ∈ N Pa,n(z) = az ∀z ∈ E.

In particular we will write Pe,1(z) = z and Pa,0(z) = a ∀z ∈ E.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 11 / 29 Using this notation we frequently write

∞ X n f = Pan,0(Pe,1 − Pz0,0) in U n=0 instead of ∞ X n f (z) = an(z − z0) , for all z ∈ U. n=0

n Given any a ∈ E, and n ∈ N Pa,n(z) = az ∀z ∈ E.

In particular we will write Pe,1(z) = z and Pa,0(z) = a ∀z ∈ E.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 11 / 29 Notation and Deﬁnitions As usual, {λ ∈ C; |λ − λ0| < r} will be denoted by ∆r (λ0) and ∆1(0) will be denoted by ∆. Easy to check: φ(EΩ) = Ω for every φ ∈ M(E). In particular, φ(Br (z0)) = ∆r (φ(z0)).

n Given any a ∈ E, and n ∈ N Pa,n(z) = az ∀z ∈ E.

In particular we will write Pe,1(z) = z and Pa,0(z) = a ∀z ∈ E. Using this notation we frequently write

∞ X n f = Pan,0(Pe,1 − Pz0,0) in U n=0 instead of ∞ X n f (z) = an(z − z0) , for all z ∈ U. n=0

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 11 / 29 ψ0 ∈ M(E)

ψ(Pe,1) ∈ ∆r (ψ0(z0)) for every ψ ∈ M(HL(Br (z0))).

It is easy to check

Deﬁnition (Glickfeld)

Let f ∈ HL(U) and φ ∈ M (E). If there is a (necessarily unique) g ∈ H(φ(U)) so that g ◦ φ = φ ◦ f on U, we say that g is the quotient function of f with respect to φ, and we write g = fφ.

U = Br (z0) where z0 ∈ E and r > 0 ⇒ fφ exists for all φ ∈ M (E). (easy)

U = star-shaped ⇒ fφ exists for all φ ∈ M (E).(Glickfeld)

U = simply connected ; fφ exists for all φ ∈ M (E).(Glickfeld)

Notation and Deﬁnitions

Given any ψ ∈ M(HL(U)), deﬁne ψ0(a) = ψ(Pa,0) for all a ∈ U.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 12 / 29 ψ(Pe,1) ∈ ∆r (ψ0(z0)) for every ψ ∈ M(HL(Br (z0))).

Deﬁnition (Glickfeld)

U = Br (z0) where z0 ∈ E and r > 0 ⇒ fφ exists for all φ ∈ M (E). (easy)

U = star-shaped ⇒ fφ exists for all φ ∈ M (E).(Glickfeld)

U = simply connected ; fφ exists for all φ ∈ M (E).(Glickfeld)

Notation and Deﬁnitions

Given any ψ ∈ M(HL(U)), deﬁne ψ0(a) = ψ(Pa,0) for all a ∈ U. It is easy to check

ψ0 ∈ M(E)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 12 / 29 Deﬁnition (Glickfeld)

U = Br (z0) where z0 ∈ E and r > 0 ⇒ fφ exists for all φ ∈ M (E). (easy)

U = star-shaped ⇒ fφ exists for all φ ∈ M (E).(Glickfeld)

U = simply connected ; fφ exists for all φ ∈ M (E).(Glickfeld)

Notation and Deﬁnitions

Given any ψ ∈ M(HL(U)), deﬁne ψ0(a) = ψ(Pa,0) for all a ∈ U. It is easy to check

ψ0 ∈ M(E)

ψ(Pe,1) ∈ ∆r (ψ0(z0)) for every ψ ∈ M(HL(Br (z0))).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 12 / 29 Deﬁnition (Glickfeld)

U = Br (z0) where z0 ∈ E and r > 0 ⇒ fφ exists for all φ ∈ M (E). (easy)

U = star-shaped ⇒ fφ exists for all φ ∈ M (E).(Glickfeld)

U = simply connected ; fφ exists for all φ ∈ M (E).(Glickfeld)

Notation and Deﬁnitions

Given any ψ ∈ M(HL(U)), deﬁne ψ0(a) = ψ(Pa,0) for all a ∈ U. It is easy to check

ψ0 ∈ M(E)

ψ(Pe,1) ∈ ∆r (ψ0(z0)) for every ψ ∈ M(HL(Br (z0))).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 12 / 29 U = Br (z0) where z0 ∈ E and r > 0 ⇒ fφ exists for all φ ∈ M (E). (easy)

U = star-shaped ⇒ fφ exists for all φ ∈ M (E).(Glickfeld)

U = simply connected ; fφ exists for all φ ∈ M (E).(Glickfeld)

Notation and Deﬁnitions

Given any ψ ∈ M(HL(U)), deﬁne ψ0(a) = ψ(Pa,0) for all a ∈ U. It is easy to check

ψ0 ∈ M(E)

ψ(Pe,1) ∈ ∆r (ψ0(z0)) for every ψ ∈ M(HL(Br (z0))).

Deﬁnition (Glickfeld)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 12 / 29 U = star-shaped ⇒ fφ exists for all φ ∈ M (E).(Glickfeld)

U = simply connected ; fφ exists for all φ ∈ M (E).(Glickfeld)

Notation and Deﬁnitions

Given any ψ ∈ M(HL(U)), deﬁne ψ0(a) = ψ(Pa,0) for all a ∈ U. It is easy to check

ψ0 ∈ M(E)

ψ(Pe,1) ∈ ∆r (ψ0(z0)) for every ψ ∈ M(HL(Br (z0))).

Deﬁnition (Glickfeld)

U = Br (z0) where z0 ∈ E and r > 0 ⇒ fφ exists for all φ ∈ M (E). (easy)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 12 / 29 U = simply connected ; fφ exists for all φ ∈ M (E).(Glickfeld)

Notation and Deﬁnitions

Given any ψ ∈ M(HL(U)), deﬁne ψ0(a) = ψ(Pa,0) for all a ∈ U. It is easy to check

ψ0 ∈ M(E)

ψ(Pe,1) ∈ ∆r (ψ0(z0)) for every ψ ∈ M(HL(Br (z0))).

Deﬁnition (Glickfeld)

U = Br (z0) where z0 ∈ E and r > 0 ⇒ fφ exists for all φ ∈ M (E). (easy)

U = star-shaped ⇒ fφ exists for all φ ∈ M (E).(Glickfeld)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 12 / 29 Notation and Deﬁnitions

Given any ψ ∈ M(HL(U)), deﬁne ψ0(a) = ψ(Pa,0) for all a ∈ U. It is easy to check

ψ0 ∈ M(E)

ψ(Pe,1) ∈ ∆r (ψ0(z0)) for every ψ ∈ M(HL(Br (z0))).

Deﬁnition (Glickfeld)

U = Br (z0) where z0 ∈ E and r > 0 ⇒ fφ exists for all φ ∈ M (E). (easy)

U = star-shaped ⇒ fφ exists for all φ ∈ M (E).(Glickfeld)

U = simply connected ; fφ exists for all φ ∈ M (E).(Glickfeld)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 12 / 29 (He(EΩ), τd ) is a Fréchet algebra whenever E is separable.

Proposition ˜ Given any f ∈ H(Ω) and φ ∈ M(E) the quotient function efφ of f with respect to φ exists and efφ = f .

This guarantees the existence of Fφ for every F ∈ He(Ω) ⊂ HL(EΩ). We will show that this is true for all the elements of HL(EΩ).

The algebras He(Ω) and (HL(EΩ), τd )

Let He(Ω) = {˜f : f ∈ H(Ω)}. It is not difﬁcult to show that

He(Ω) is a closed subalgebra of (HL(EΩ), τd ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 13 / 29 Proposition ˜ Given any f ∈ H(Ω) and φ ∈ M(E) the quotient function efφ of f with respect to φ exists and efφ = f .

This guarantees the existence of Fφ for every F ∈ He(Ω) ⊂ HL(EΩ). We will show that this is true for all the elements of HL(EΩ).

The algebras He(Ω) and (HL(EΩ), τd )

Let He(Ω) = {˜f : f ∈ H(Ω)}. It is not difﬁcult to show that

He(Ω) is a closed subalgebra of (HL(EΩ), τd ).

(He(EΩ), τd ) is a Fréchet algebra whenever E is separable.

This guarantees the existence of Fφ for every F ∈ He(Ω) ⊂ HL(EΩ). We will show that this is true for all the elements of HL(EΩ).

The algebras He(Ω) and (HL(EΩ), τd )

Let He(Ω) = {˜f : f ∈ H(Ω)}. It is not difﬁcult to show that

He(Ω) is a closed subalgebra of (HL(EΩ), τd ).

(He(EΩ), τd ) is a Fréchet algebra whenever E is separable.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 13 / 29 This guarantees the existence of Fφ for every F ∈ He(Ω) ⊂ HL(EΩ). We will show that this is true for all the elements of HL(EΩ).

The algebras He(Ω) and (HL(EΩ), τd )

Let He(Ω) = {˜f : f ∈ H(Ω)}. It is not difﬁcult to show that

He(Ω) is a closed subalgebra of (HL(EΩ), τd ).

(He(EΩ), τd ) is a Fréchet algebra whenever E is separable.

Proposition ˜ Given any f ∈ H(Ω) and φ ∈ M(E) the quotient function efφ of f with respect to φ exists and efφ = f .

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 13 / 29 The algebras He(Ω) and (HL(EΩ), τd )

Let He(Ω) = {˜f : f ∈ H(Ω)}. It is not difﬁcult to show that

He(Ω) is a closed subalgebra of (HL(EΩ), τd ).

(He(EΩ), τd ) is a Fréchet algebra whenever E is separable.

Proposition ˜ Given any f ∈ H(Ω) and φ ∈ M(E) the quotient function efφ of f with respect to φ exists and efφ = f .

This guarantees the existence of Fφ for every F ∈ He(Ω) ⊂ HL(EΩ). We will show that this is true for all the elements of HL(EΩ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 13 / 29 It seems natural to try to prove a Riemann Mapping Theorem in case of Banach algebras E in the context of the Lorch analytic mappings.

Deﬁnition Let U, V be two domains in E. We say that an homeomorphism f : U → V is −1 an L-homeomorphism between U and V if f ∈ HL(U) and f ∈ HL(V ). We say that U and V are L-homeomorphic if there exists an L-homeomorphism between U and V .

A ﬁrst attempt to show a Riemann Mapping Theorem in the context of the Lorch analytic mappings in E would be to show that the open unit ball BE is L-homeomorphic to every bounded homeomorphic image of itself. But Glick- feld showed that this cannot be proved for general Banach algebras by pre- senting an example of a simply connected domain D contained in the algebra E = C([0, 1]) such that D é homeomorphic to the ball BE but D is not L-homeomorphic to BE .

The algebras He(Ω) and (HL(EΩ), τd )

If Ω (C is a simply-connected domain, by the Riemann Mapping Theorem there exists a one-to-one analytic mapping h from Ω onto ∆ s.th. h−1 is analytic.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 14 / 29 Deﬁnition Let U, V be two domains in E. We say that an homeomorphism f : U → V is −1 an L-homeomorphism between U and V if f ∈ HL(U) and f ∈ HL(V ). We say that U and V are L-homeomorphic if there exists an L-homeomorphism between U and V .

The algebras He(Ω) and (HL(EΩ), τd )

If Ω (C is a simply-connected domain, by the Riemann Mapping Theorem there exists a one-to-one analytic mapping h from Ω onto ∆ s.th. h−1 is analytic. It seems natural to try to prove a Riemann Mapping Theorem in case of Banach algebras E in the context of the Lorch analytic mappings.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 14 / 29 A ﬁrst attempt to show a Riemann Mapping Theorem in the context of the Lorch analytic mappings in E would be to show that the open unit ball BE is L-homeomorphic to every bounded homeomorphic image of itself. But Glick- feld showed that this cannot be proved for general Banach algebras by pre- senting an example of a simply connected domain D contained in the algebra E = C([0, 1]) such that D é homeomorphic to the ball BE but D is not L-homeomorphic to BE .

The algebras He(Ω) and (HL(EΩ), τd )

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 14 / 29 The algebras He(Ω) and (HL(EΩ), τd )

L. Moraes (UFRJ) Algebras of Lorch analytic mappings defined in uniform algebras Valencia, 20/10/2017 14 / 29 Recall that a simple closed curve in a Banach algebra E is, by definition, a continuous mapping Γ: ∂∆ → E such that, for each φ ∈ M(E), φ ◦ Γ is a homeomorphism between ∂∆ and φ ◦ Γ(∂∆) ⊂ C. The image of each φ ◦ Γ is always a Jordan curve in C. By definition, the interior of the curve Γ is the set

int(Γ) = {z ∈ E ; φ(z) ∈ int(φ ◦ Γ) for all φ ∈ M(E)} ,

where int(φ ◦ Γ) is the interior of the Jordan curve φ ◦ Γ in the usual sense. Warren considered the case when E is a semi-simple Banach algebra and showed that in this case BE is L-homeomorphic to int(Γ) by a mapping g : ∗ BE → int(Γ) whenever Γ satisﬁes the following condition: for all T ∈ E , T ◦ Γ has a continuous extension to ∆ that is holomorphic in ∆.

The algebras He(Ω) and (HL(EΩ), τd )

Warren characterized which domains are L-homeomorphic to BE in case E = C(X), where X is a compact Hausdorff space.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 15 / 29 The image of each φ ◦ Γ is always a Jordan curve in C. By deﬁnition, the interior of the curve Γ is the set

int(Γ) = {z ∈ E ; φ(z) ∈ int(φ ◦ Γ) for all φ ∈ M(E)} ,

The algebras He(Ω) and (HL(EΩ), τd )

Warren characterized which domains are L-homeomorphic to BE in case E = C(X), where X is a compact Hausdorff space. Recall that a simple closed curve in a Banach algebra E is, by deﬁnition, a continuous mapping Γ: ∂∆ → E such that, for each φ ∈ M(E), φ ◦ Γ is a homeomorphism between ∂∆ and φ ◦ Γ(∂∆) ⊂ C.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 15 / 29 By deﬁnition, the interior of the curve Γ is the set

int(Γ) = {z ∈ E ; φ(z) ∈ int(φ ◦ Γ) for all φ ∈ M(E)} ,

The algebras He(Ω) and (HL(EΩ), τd )

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 15 / 29 Warren considered the case when E is a semi-simple Banach algebra and showed that in this case BE is L-homeomorphic to int(Γ) by a mapping g : ∗ BE → int(Γ) whenever Γ satisﬁes the following condition: for all T ∈ E , T ◦ Γ has a continuous extension to ∆ that is holomorphic in ∆.

The algebras He(Ω) and (HL(EΩ), τd )

int(Γ) = {z ∈ E ; φ(z) ∈ int(φ ◦ Γ) for all φ ∈ M(E)} ,

where int(φ ◦ Γ) is the interior of the Jordan curve φ ◦ Γ in the usual sense.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 15 / 29 The algebras He(Ω) and (HL(EΩ), τd )

int(Γ) = {z ∈ E ; φ(z) ∈ int(φ ◦ Γ) for all φ ∈ M(E)} ,

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 15 / 29 Theorem Let Ω (C be a simply connected domain and E be a Banach algebra. There is an L-homeomorphism between EΩ and E∆ given by he ∈ He(Ω) where h is a conformal mapping between Ω and ∆.

Now we are able to show:

Theorem Let Ω (C be a simply connected domain and let E be an unitary commutative semi-simple Banach algebra. Then BE and EΩ are L-homeomorphic.

The algebras He(Ω) and (HL(EΩ), τd )

If Ω (C is a simply-connected domain, by the Riemann Mapping Theorem there exists a one-to-one analytic mapping h from Ω onto ∆. We may take ˜ h ∈ He(EΩ) that corresponds to h via the functional calculus and get:

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 16 / 29 Now we are able to show:

Theorem Let Ω (C be a simply connected domain and let E be an unitary commutative semi-simple Banach algebra. Then BE and EΩ are L-homeomorphic.

The algebras He(Ω) and (HL(EΩ), τd )

Theorem Let Ω (C be a simply connected domain and E be a Banach algebra. There is an L-homeomorphism between EΩ and E∆ given by he ∈ He(Ω) where h is a conformal mapping between Ω and ∆.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 16 / 29 Theorem Let Ω (C be a simply connected domain and let E be an unitary commutative semi-simple Banach algebra. Then BE and EΩ are L-homeomorphic.

The algebras He(Ω) and (HL(EΩ), τd )

Now we are able to show:

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 16 / 29 The algebras He(Ω) and (HL(EΩ), τd )

Now we are able to show:

Theorem Let Ω (C be a simply connected domain and let E be an unitary commutative semi-simple Banach algebra. Then BE and EΩ are L-homeomorphic.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 16 / 29 The algebras He(Ω) and (HL(EΩ), τd )

Now we are able to show:

Theorem Let Ω (C be a simply connected domain and let E be an unitary commutative semi-simple Banach algebra. Then BE and EΩ are L-homeomorphic.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 16 / 29 Consider the simple closed curve Γ0 : ∂∆ −→ E given by Γ0(λ) = λe where e is the unit in E. It is easy to check that int(φ ◦ Γ0) = ∆ for all φ ∈ M(E) and hence int(Γ0) = E∆. Show that Warren’s result applies and get the existence of an L-homeomorphism g : BE → E∆. −1 By the previous theorem we have that hg : E∆ → EΩ is an L-homeomorphism between E∆ and EΩ. −1 Hence, hg ◦ g : BE → EΩ is an L-homeomorphism between BE and EΩ. This completes the proof.

Theorem 1 (G.V.S. Mauro, L.A.M. and A.F. Pereira) Let E be a commutative Banach algebra with a unit element e. The spectrum M(HL(BE )) is homeomorphic to M(E) × ∆ by the mapping

δ : M(E) × ∆ −→ M(HL(BE ))

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(BE ).

The algebras He(Ω) and (HL(EΩ), τd ) Idea of the proof:

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 17 / 29 Show that Warren’s result applies and get the existence of an L-homeomorphism g : BE → E∆. −1 By the previous theorem we have that hg : E∆ → EΩ is an L-homeomorphism between E∆ and EΩ. −1 Hence, hg ◦ g : BE → EΩ is an L-homeomorphism between BE and EΩ. This completes the proof.

Theorem 1 (G.V.S. Mauro, L.A.M. and A.F. Pereira) Let E be a commutative Banach algebra with a unit element e. The spectrum M(HL(BE )) is homeomorphic to M(E) × ∆ by the mapping

δ : M(E) × ∆ −→ M(HL(BE ))

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(BE ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 17 / 29 −1 By the previous theorem we have that hg : E∆ → EΩ is an L-homeomorphism between E∆ and EΩ. −1 Hence, hg ◦ g : BE → EΩ is an L-homeomorphism between BE and EΩ. This completes the proof.

Theorem 1 (G.V.S. Mauro, L.A.M. and A.F. Pereira) Let E be a commutative Banach algebra with a unit element e. The spectrum M(HL(BE )) is homeomorphic to M(E) × ∆ by the mapping

δ : M(E) × ∆ −→ M(HL(BE ))

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(BE ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 17 / 29 −1 Hence, hg ◦ g : BE → EΩ is an L-homeomorphism between BE and EΩ. This completes the proof.

Theorem 1 (G.V.S. Mauro, L.A.M. and A.F. Pereira) Let E be a commutative Banach algebra with a unit element e. The spectrum M(HL(BE )) is homeomorphic to M(E) × ∆ by the mapping

δ : M(E) × ∆ −→ M(HL(BE ))

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(BE ).

The algebras He(Ω) and (HL(EΩ), τd ) Idea of the proof: Consider the simple closed curve Γ0 : ∂∆ −→ E given by Γ0(λ) = λe where e is the unit in E. It is easy to check that int(φ ◦ Γ0) = ∆ for all φ ∈ M(E) and hence int(Γ0) = E∆. Show that Warren’s result applies and get the existence of an L-homeomorphism g : BE → E∆. −1 By the previous theorem we have that hg : E∆ → EΩ is an L-homeomorphism between E∆ and EΩ.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 17 / 29 Theorem 1 (G.V.S. Mauro, L.A.M. and A.F. Pereira) Let E be a commutative Banach algebra with a unit element e. The spectrum M(HL(BE )) is homeomorphic to M(E) × ∆ by the mapping

δ : M(E) × ∆ −→ M(HL(BE ))

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(BE ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 17 / 29 The algebras He(Ω) and (HL(EΩ), τd ) Idea of the proof: Consider the simple closed curve Γ0 : ∂∆ −→ E given by Γ0(λ) = λe where e is the unit in E. It is easy to check that int(φ ◦ Γ0) = ∆ for all φ ∈ M(E) and hence int(Γ0) = E∆. Show that Warren’s result applies and get the existence of an L-homeomorphism g : BE → E∆. −1 By the previous theorem we have that hg : E∆ → EΩ is an L-homeomorphism between E∆ and EΩ. −1 Hence, hg ◦ g : BE → EΩ is an L-homeomorphism between BE and EΩ. This completes the proof.

Theorem 1 (G.V.S. Mauro, L.A.M. and A.F. Pereira) Let E be a commutative Banach algebra with a unit element e. The spectrum M(HL(BE )) is homeomorphic to M(E) × ∆ by the mapping

δ : M(E) × ∆ −→ M(HL(BE ))

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(BE ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 17 / 29 Clearly our uniform algebras are always unitary commutative Banach algebras. Moreover, it is well known that every uniform algebra E is semi-simple and satisfy the equality kzk = kzˆk∞ for every z ∈ E where zˆ denotes the Gelfand transform of z. Consequently, E∆ = BE .

Our next goal is to describe the spectrum of the algebra (HL(EΩ), τd ) when E is a uniform algebra and Ω (C is a simply connected domain. We recall that (HL(EΩ), τd ) is a Fréchet algebra whenever E is separable. From now h denotes always a one-to-one analytic mapping h from Ω onto ∆.

−1 −1 −1 Note that he : EΩ → E∆, he : E∆ → EΩ and it is easy to show that he = hg.

Clearly E∆ = BE ⇒ he(z) ∈ BE for all z ∈ EΩ.

The spectrum of the algebra (HL(EΩ), τd )

Recall that a uniform algebra is a closed subalgebra of C(X), where X is a compact Hausdorff space, which contains the constants and separates the points of X.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 18 / 29 Our next goal is to describe the spectrum of the algebra (HL(EΩ), τd ) when E is a uniform algebra and Ω (C is a simply connected domain. We recall that (HL(EΩ), τd ) is a Fréchet algebra whenever E is separable. From now h denotes always a one-to-one analytic mapping h from Ω onto ∆.

−1 −1 −1 Note that he : EΩ → E∆, he : E∆ → EΩ and it is easy to show that he = hg.

Clearly E∆ = BE ⇒ he(z) ∈ BE for all z ∈ EΩ.

The spectrum of the algebra (HL(EΩ), τd )

Recall that a uniform algebra is a closed subalgebra of C(X), where X is a compact Hausdorff space, which contains the constants and separates the points of X. Clearly our uniform algebras are always unitary commutative Banach algebras. Moreover, it is well known that every uniform algebra E is semi-simple and satisfy the equality kzk = kzˆk∞ for every z ∈ E where zˆ denotes the Gelfand transform of z. Consequently, E∆ = BE .

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 18 / 29 From now h denotes always a one-to-one analytic mapping h from Ω onto ∆.

−1 −1 −1 Note that he : EΩ → E∆, he : E∆ → EΩ and it is easy to show that he = hg.

Clearly E∆ = BE ⇒ he(z) ∈ BE for all z ∈ EΩ.

The spectrum of the algebra (HL(EΩ), τd )

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 18 / 29 −1 −1 −1 Note that he : EΩ → E∆, he : E∆ → EΩ and it is easy to show that he = hg.

Clearly E∆ = BE ⇒ he(z) ∈ BE for all z ∈ EΩ.

The spectrum of the algebra (HL(EΩ), τd )

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 18 / 29 Clearly E∆ = BE ⇒ he(z) ∈ BE for all z ∈ EΩ.

The spectrum of the algebra (HL(EΩ), τd )

−1 −1 −1 Note that he : EΩ → E∆, he : E∆ → EΩ and it is easy to show that he = hg.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 18 / 29 The spectrum of the algebra (HL(EΩ), τd )

−1 −1 −1 Note that he : EΩ → E∆, he : E∆ → EΩ and it is easy to show that he = hg.

Clearly E∆ = BE ⇒ he(z) ∈ BE for all z ∈ EΩ.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 18 / 29 −1 Idea of the proof: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). Take

∞ −1 X n f ◦ hg(w) = anw for all w ∈ BE . n=0 ∞ −1 P n Clearly f (z) = (f ◦ hg)(he(z)) = an(he(z)) for every z ∈ EΩ. n=0

Since φ(EΩ) = Ω, by using the above equality show that fφ : φ(EΩ) → C deﬁned by ∞ X n fφ(λ) = φ(an)(h(λ)) for all λ ∈ Ω n=0

satisﬁes fφ ◦ φ = φ ◦ f . This completes the proof.

The spectrum of the algebra (HL(EΩ), τd )

Proposition Let E be a uniform algebra. If Ω (C is a simply connected domain and f ∈ HL(EΩ), then for every φ ∈ M(E) there exists a function fφ ∈ H(Ω) satisfying fφ(φ(z)) = φ(f (z)) for all z ∈ EΩ.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 19 / 29 −1 f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). Take

∞ −1 X n f ◦ hg(w) = anw for all w ∈ BE . n=0 ∞ −1 P n Clearly f (z) = (f ◦ hg)(he(z)) = an(he(z)) for every z ∈ EΩ. n=0

Since φ(EΩ) = Ω, by using the above equality show that fφ : φ(EΩ) → C deﬁned by ∞ X n fφ(λ) = φ(an)(h(λ)) for all λ ∈ Ω n=0

satisﬁes fφ ◦ φ = φ ◦ f . This completes the proof.

The spectrum of the algebra (HL(EΩ), τd )

Idea of the proof:

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 19 / 29 Take

∞ −1 X n f ◦ hg(w) = anw for all w ∈ BE . n=0 ∞ −1 P n Clearly f (z) = (f ◦ hg)(he(z)) = an(he(z)) for every z ∈ EΩ. n=0

Since φ(EΩ) = Ω, by using the above equality show that fφ : φ(EΩ) → C deﬁned by ∞ X n fφ(λ) = φ(an)(h(λ)) for all λ ∈ Ω n=0

satisﬁes fφ ◦ φ = φ ◦ f . This completes the proof.

The spectrum of the algebra (HL(EΩ), τd )

−1 Idea of the proof: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 19 / 29 ∞ −1 P n Clearly f (z) = (f ◦ hg)(he(z)) = an(he(z)) for every z ∈ EΩ. n=0

Since φ(EΩ) = Ω, by using the above equality show that fφ : φ(EΩ) → C deﬁned by ∞ X n fφ(λ) = φ(an)(h(λ)) for all λ ∈ Ω n=0

satisﬁes fφ ◦ φ = φ ◦ f . This completes the proof.

The spectrum of the algebra (HL(EΩ), τd )

−1 Idea of the proof: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). Take

∞ −1 X n f ◦ hg(w) = anw for all w ∈ BE . n=0

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 19 / 29 Since φ(EΩ) = Ω, by using the above equality show that fφ : φ(EΩ) → C deﬁned by ∞ X n fφ(λ) = φ(an)(h(λ)) for all λ ∈ Ω n=0

satisﬁes fφ ◦ φ = φ ◦ f . This completes the proof.

The spectrum of the algebra (HL(EΩ), τd )

−1 Idea of the proof: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). Take

∞ −1 X n f ◦ hg(w) = anw for all w ∈ BE . n=0 ∞ −1 P n Clearly f (z) = (f ◦ hg)(he(z)) = an(he(z)) for every z ∈ EΩ. n=0

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 19 / 29 The spectrum of the algebra (HL(EΩ), τd )

−1 Idea of the proof: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). Take

∞ −1 X n f ◦ hg(w) = anw for all w ∈ BE . n=0 ∞ −1 P n Clearly f (z) = (f ◦ hg)(he(z)) = an(he(z)) for every z ∈ EΩ. n=0

Since φ(EΩ) = Ω, by using the above equality show that fφ : φ(EΩ) → C deﬁned by ∞ X n fφ(λ) = φ(an)(h(λ)) for all λ ∈ Ω n=0

satisﬁes fφ ◦ φ = φ ◦ f . This completes the proof.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 19 / 29 We noted already that given if ψ ∈ M(HL(BE )) then it can be proved that ψ(Pe,1) ∈ ∆. Moreover we can get ψ0 ∈ M(E) by deﬁning ψ0(a) = ψ(Pa,0) for all a ∈ E.

Theorem 1 states that to each ψ ∈ M(HL(BE )) we can associate (φ, λ0) ∈ M(E) × ∆ such that ψ(f ) = δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(BE ). This can be proved by choosing φ = ψ0 ∈ M(E) and λ0 = ψ(Pe,1) ∈ ∆. A similar argument did not work when we tried to prove that δ is surjective in Theorem 2.

The spectrum of the algebra (HL(EΩ), τd )

Let τπ denote the restriction to M(E) × Ω of the product topology in M(E) × C. Theorem 2 If E is a uniform algebra and Ω (C is a simply connected domain, the mapping δ : M(E) × Ω −→ M(HL(EΩ))

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ) is a homeomorphism between (M(E) × Ω, τπ) and (M(HL(EΩ)), τG) .

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 20 / 29 Theorem 1 states that to each ψ ∈ M(HL(BE )) we can associate (φ, λ0) ∈ M(E) × ∆ such that ψ(f ) = δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(BE ). This can be proved by choosing φ = ψ0 ∈ M(E) and λ0 = ψ(Pe,1) ∈ ∆. A similar argument did not work when we tried to prove that δ is surjective in Theorem 2.

The spectrum of the algebra (HL(EΩ), τd )

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ) is a homeomorphism between (M(E) × Ω, τπ) and (M(HL(EΩ)), τG) .

We noted already that given if ψ ∈ M(HL(BE )) then it can be proved that ψ(Pe,1) ∈ ∆. Moreover we can get ψ0 ∈ M(E) by deﬁning ψ0(a) = ψ(Pa,0) for all a ∈ E.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 20 / 29 This can be proved by choosing φ = ψ0 ∈ M(E) and λ0 = ψ(Pe,1) ∈ ∆. A similar argument did not work when we tried to prove that δ is surjective in Theorem 2.

The spectrum of the algebra (HL(EΩ), τd )

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ) is a homeomorphism between (M(E) × Ω, τπ) and (M(HL(EΩ)), τG) .

We noted already that given if ψ ∈ M(HL(BE )) then it can be proved that ψ(Pe,1) ∈ ∆. Moreover we can get ψ0 ∈ M(E) by deﬁning ψ0(a) = ψ(Pa,0) for all a ∈ E.

Theorem 1 states that to each ψ ∈ M(HL(BE )) we can associate (φ, λ0) ∈ M(E) × ∆ such that ψ(f ) = δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(BE ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 20 / 29 A similar argument did not work when we tried to prove that δ is surjective in Theorem 2.

The spectrum of the algebra (HL(EΩ), τd )

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ) is a homeomorphism between (M(E) × Ω, τπ) and (M(HL(EΩ)), τG) .

We noted already that given if ψ ∈ M(HL(BE )) then it can be proved that ψ(Pe,1) ∈ ∆. Moreover we can get ψ0 ∈ M(E) by deﬁning ψ0(a) = ψ(Pa,0) for all a ∈ E.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 20 / 29 The spectrum of the algebra (HL(EΩ), τd )

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ) is a homeomorphism between (M(E) × Ω, τπ) and (M(HL(EΩ)), τG) .

We noted already that given if ψ ∈ M(HL(BE )) then it can be proved that ψ(Pe,1) ∈ ∆. Moreover we can get ψ0 ∈ M(E) by deﬁning ψ0(a) = ψ(Pa,0) for all a ∈ E.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 20 / 29 Lemma Let E be a uniform algebra, Ω (C be a simply connected domain and h be a conformal mapping between Ω and ∆. If z0 ∈ EΩ and

0 < r < dEΩ (z0) = inf kz0 − wk , then he(Br (z0)) is a BE -bounded subset of w∈EΩ BE .

The proof of this lemma uses strongly the equality kzk = kzˆk∞ for every z ∈ E where zˆ denotes the Gelfand transform of z. Let us show that δ is surjective.

Take ψ ∈ M(HL(EΩ)). We are going to get (φ, λ0) ∈ M(E) × Ω such that ψ(f ) = δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ).

The spectrum of the algebra (HL(EΩ), τd )

Next we are going to show that δ is surjective in Theorem 2. Our proof uses the following result:

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 21 / 29 The proof of this lemma uses strongly the equality kzk = kzˆk∞ for every z ∈ E where zˆ denotes the Gelfand transform of z. Let us show that δ is surjective.

Take ψ ∈ M(HL(EΩ)). We are going to get (φ, λ0) ∈ M(E) × Ω such that ψ(f ) = δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ).

The spectrum of the algebra (HL(EΩ), τd )

Next we are going to show that δ is surjective in Theorem 2. Our proof uses the following result:

Lemma Let E be a uniform algebra, Ω (C be a simply connected domain and h be a conformal mapping between Ω and ∆. If z0 ∈ EΩ and

0 < r < dEΩ (z0) = inf kz0 − wk , then he(Br (z0)) is a BE -bounded subset of w∈EΩ BE .

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 21 / 29 Let us show that δ is surjective.

Take ψ ∈ M(HL(EΩ)). We are going to get (φ, λ0) ∈ M(E) × Ω such that ψ(f ) = δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ).

The spectrum of the algebra (HL(EΩ), τd )

Next we are going to show that δ is surjective in Theorem 2. Our proof uses the following result:

Lemma Let E be a uniform algebra, Ω (C be a simply connected domain and h be a conformal mapping between Ω and ∆. If z0 ∈ EΩ and

0 < r < dEΩ (z0) = inf kz0 − wk , then he(Br (z0)) is a BE -bounded subset of w∈EΩ BE .

The proof of this lemma uses strongly the equality kzk = kzˆk∞ for every z ∈ E where zˆ denotes the Gelfand transform of z.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 21 / 29 Take ψ ∈ M(HL(EΩ)). We are going to get (φ, λ0) ∈ M(E) × Ω such that ψ(f ) = δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ).

The spectrum of the algebra (HL(EΩ), τd )

Next we are going to show that δ is surjective in Theorem 2. Our proof uses the following result:

Lemma Let E be a uniform algebra, Ω (C be a simply connected domain and h be a conformal mapping between Ω and ∆. If z0 ∈ EΩ and

0 < r < dEΩ (z0) = inf kz0 − wk , then he(Br (z0)) is a BE -bounded subset of w∈EΩ BE .

The proof of this lemma uses strongly the equality kzk = kzˆk∞ for every z ∈ E where zˆ denotes the Gelfand transform of z. Let us show that δ is surjective.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 21 / 29 The spectrum of the algebra (HL(EΩ), τd )

Next we are going to show that δ is surjective in Theorem 2. Our proof uses the following result:

Lemma Let E be a uniform algebra, Ω (C be a simply connected domain and h be a conformal mapping between Ω and ∆. If z0 ∈ EΩ and

0 < r < dEΩ (z0) = inf kz0 − wk , then he(Br (z0)) is a BE -bounded subset of w∈EΩ BE .

The proof of this lemma uses strongly the equality kzk = kzˆk∞ for every z ∈ E where zˆ denotes the Gelfand transform of z. Let us show that δ is surjective.

Take ψ ∈ M(HL(EΩ)). We are going to get (φ, λ0) ∈ M(E) × Ω such that ψ(f ) = δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 21 / 29 ψh ∈ M(HL(BE )). (The Lemma is used to show that ψh is continuous).

Now, deﬁne

ψh : HL(BE ) −→ C f 7−→ ψ(f ◦ he)

For every a ∈ E,

(ψh)0(a) = ψh(Pa,0) = ψ(Pa,0 ◦ he) = ψ(Pa,0) = ψ0(a).

−1 Now: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). And by Theorem 1, ψh = δ((ψh)0, µ0) where µ0 = ψh(Pe,1) ∈ ∆.

The spectrum of the algebra (HL(EΩ), τd )

Easy: f ∈ HL(BE ) ⇒ f ◦ he ∈ HL(EΩ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 22 / 29 ψh ∈ M(HL(BE )). (The Lemma is used to show that ψh is continuous).

For every a ∈ E,

(ψh)0(a) = ψh(Pa,0) = ψ(Pa,0 ◦ he) = ψ(Pa,0) = ψ0(a).

−1 Now: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). And by Theorem 1, ψh = δ((ψh)0, µ0) where µ0 = ψh(Pe,1) ∈ ∆.

The spectrum of the algebra (HL(EΩ), τd )

Easy: f ∈ HL(BE ) ⇒ f ◦ he ∈ HL(EΩ). Now, deﬁne

ψh : HL(BE ) −→ C f 7−→ ψ(f ◦ he)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 22 / 29 For every a ∈ E,

(ψh)0(a) = ψh(Pa,0) = ψ(Pa,0 ◦ he) = ψ(Pa,0) = ψ0(a).

−1 Now: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). And by Theorem 1, ψh = δ((ψh)0, µ0) where µ0 = ψh(Pe,1) ∈ ∆.

The spectrum of the algebra (HL(EΩ), τd )

Easy: f ∈ HL(BE ) ⇒ f ◦ he ∈ HL(EΩ). Now, deﬁne

ψh : HL(BE ) −→ C f 7−→ ψ(f ◦ he)

ψh ∈ M(HL(BE )). (The Lemma is used to show that ψh is continuous).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 22 / 29 For every a ∈ E,

(ψh)0(a) = ψh(Pa,0) = ψ(Pa,0 ◦ he) = ψ(Pa,0) = ψ0(a).

−1 Now: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). And by Theorem 1, ψh = δ((ψh)0, µ0) where µ0 = ψh(Pe,1) ∈ ∆.

The spectrum of the algebra (HL(EΩ), τd )

Easy: f ∈ HL(BE ) ⇒ f ◦ he ∈ HL(EΩ). Now, deﬁne

ψh : HL(BE ) −→ C f 7−→ ψ(f ◦ he)

ψh ∈ M(HL(BE )). (The Lemma is used to show that ψh is continuous).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 22 / 29 −1 Now: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). And by Theorem 1, ψh = δ((ψh)0, µ0) where µ0 = ψh(Pe,1) ∈ ∆.

The spectrum of the algebra (HL(EΩ), τd )

Easy: f ∈ HL(BE ) ⇒ f ◦ he ∈ HL(EΩ). Now, deﬁne

ψh : HL(BE ) −→ C f 7−→ ψ(f ◦ he)

ψh ∈ M(HL(BE )). (The Lemma is used to show that ψh is continuous).

For every a ∈ E,

(ψh)0(a) = ψh(Pa,0) = ψ(Pa,0 ◦ he) = ψ(Pa,0) = ψ0(a).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 22 / 29 And by Theorem 1, ψh = δ((ψh)0, µ0) where µ0 = ψh(Pe,1) ∈ ∆.

The spectrum of the algebra (HL(EΩ), τd )

Easy: f ∈ HL(BE ) ⇒ f ◦ he ∈ HL(EΩ). Now, deﬁne

ψh : HL(BE ) −→ C f 7−→ ψ(f ◦ he)

ψh ∈ M(HL(BE )). (The Lemma is used to show that ψh is continuous).

For every a ∈ E,

(ψh)0(a) = ψh(Pa,0) = ψ(Pa,0 ◦ he) = ψ(Pa,0) = ψ0(a).

−1 Now: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 22 / 29 The spectrum of the algebra (HL(EΩ), τd )

Easy: f ∈ HL(BE ) ⇒ f ◦ he ∈ HL(EΩ). Now, deﬁne

ψh : HL(BE ) −→ C f 7−→ ψ(f ◦ he)

ψh ∈ M(HL(BE )). (The Lemma is used to show that ψh is continuous).

For every a ∈ E,

(ψh)0(a) = ψh(Pa,0) = ψ(Pa,0 ◦ he) = ψ(Pa,0) = ψ0(a).

−1 Now: f ∈ HL(EΩ) ⇒ f ◦ hg ∈ HL(BE ). And by Theorem 1, ψh = δ((ψh)0, µ0) where µ0 = ψh(Pe,1) ∈ ∆.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 22 / 29 As h is a bijection between Ω and ∆, there exists λ0 ∈ Ω such that h(λ0) = µ0. So,

−1 −1 −1 ψ(f ) = (f ◦ hg)ψ0 (µ0) = (f ◦ hg)ψ0 (h(λ0)) = fψ0 ◦ (hg)ψ0 ◦ h(λ0) −1 = fψ0 ◦ h ◦ h(λ0) = fψ0 (λ0) = δ(ψ0, λ0)(f ).

Hence, ψ(f ) = fψ0 (λ0) for every f ∈ HL(EΩ). This shows that δ is surjective. As a consequence of Theorem 2 we get: Proposition Let E be a uniform algebra and Ω (C be a simply connected domain. Then, if E is a separable space, HL(EΩ) is a semi-simple algebra.

The spectrum of the algebra (HL(EΩ), τd )

Since (ψh)0 = ψ0, for every f ∈ HL(EΩ) we have

−1 −1 −1 ψ(f ) = ψ(f ◦ hg ◦ he) = ψh(f ◦ hg) = (f ◦ hg)ψ0 (µ0)

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 23 / 29 −1 −1 −1 ψ(f ) = (f ◦ hg)ψ0 (µ0) = (f ◦ hg)ψ0 (h(λ0)) = fψ0 ◦ (hg)ψ0 ◦ h(λ0) −1 = fψ0 ◦ h ◦ h(λ0) = fψ0 (λ0) = δ(ψ0, λ0)(f ).

The spectrum of the algebra (HL(EΩ), τd )

Since (ψh)0 = ψ0, for every f ∈ HL(EΩ) we have

−1 −1 −1 ψ(f ) = ψ(f ◦ hg ◦ he) = ψh(f ◦ hg) = (f ◦ hg)ψ0 (µ0)

As h is a bijection between Ω and ∆, there exists λ0 ∈ Ω such that h(λ0) = µ0. So,

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 23 / 29 As a consequence of Theorem 2 we get: Proposition Let E be a uniform algebra and Ω (C be a simply connected domain. Then, if E is a separable space, HL(EΩ) is a semi-simple algebra.

The spectrum of the algebra (HL(EΩ), τd )

Since (ψh)0 = ψ0, for every f ∈ HL(EΩ) we have

−1 −1 −1 ψ(f ) = ψ(f ◦ hg ◦ he) = ψh(f ◦ hg) = (f ◦ hg)ψ0 (µ0)

As h is a bijection between Ω and ∆, there exists λ0 ∈ Ω such that h(λ0) = µ0. So,

−1 −1 −1 ψ(f ) = (f ◦ hg)ψ0 (µ0) = (f ◦ hg)ψ0 (h(λ0)) = fψ0 ◦ (hg)ψ0 ◦ h(λ0) −1 = fψ0 ◦ h ◦ h(λ0) = fψ0 (λ0) = δ(ψ0, λ0)(f ).

Hence, ψ(f ) = fψ0 (λ0) for every f ∈ HL(EΩ). This shows that δ is surjective.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 23 / 29 The spectrum of the algebra (HL(EΩ), τd )

Since (ψh)0 = ψ0, for every f ∈ HL(EΩ) we have

−1 −1 −1 ψ(f ) = ψ(f ◦ hg ◦ he) = ψh(f ◦ hg) = (f ◦ hg)ψ0 (µ0)

As h is a bijection between Ω and ∆, there exists λ0 ∈ Ω such that h(λ0) = µ0. So,

−1 −1 −1 ψ(f ) = (f ◦ hg)ψ0 (µ0) = (f ◦ hg)ψ0 (h(λ0)) = fψ0 ◦ (hg)ψ0 ◦ h(λ0) −1 = fψ0 ◦ h ◦ h(λ0) = fψ0 (λ0) = δ(ψ0, λ0)(f ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 23 / 29 (HL(EΩ), τ0) is complete. Every separable C∗−álgebra A (not necessarily unitary) has an approximation of the identity, i.e., a sequence (en) ⊂ A such that n∈N lim kenz − zk = 0 for every z ∈ A and kenk ≤ 1 for every n ∈ . n→∞ N If A is a separable C∗−algebra and φ ∈ M(A), then ker(φ) is a separable C∗−algebra as well.

It is known:

Let M0(HL(EΩ)) denote the spectrum of (HL(EΩ), τ0) endowed with the Gelfand topology τG. Well known: If θ is a complex homomorphism deﬁned in H(Ω), then θ(f ) = f (λθ) for every f ∈ H(Ω) where λθ = θ(Id) ∈ Ω Lemma 1

If ψ is a complex homomorphism deﬁned in HL(EΩ), then ψ(ef ) = f (λ0) for every f ∈ H(Ω), where λ0 = ψ(Pe,1).

The algebra (HL(EΩ), τ0)

Let us consider the space locally convex space (HL(EΩ), τ0).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 24 / 29 Every separable C∗−álgebra A (not necessarily unitary) has an approximation of the identity, i.e., a sequence (en) ⊂ A such that n∈N lim kenz − zk = 0 for every z ∈ A and kenk ≤ 1 for every n ∈ . n→∞ N If A is a separable C∗−algebra and φ ∈ M(A), then ker(φ) is a separable C∗−algebra as well.

If ψ is a complex homomorphism deﬁned in HL(EΩ), then ψ(ef ) = f (λ0) for every f ∈ H(Ω), where λ0 = ψ(Pe,1).

The algebra (HL(EΩ), τ0)

Let us consider the space locally convex space (HL(EΩ), τ0). It is known:

(HL(EΩ), τ0) is complete.

If ψ is a complex homomorphism deﬁned in HL(EΩ), then ψ(ef ) = f (λ0) for every f ∈ H(Ω), where λ0 = ψ(Pe,1).

The algebra (HL(EΩ), τ0)

Let us consider the space locally convex space (HL(EΩ), τ0). It is known:

(HL(EΩ), τ0) is complete.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 24 / 29 If A is a separable C∗−algebra and φ ∈ M(A), then ker(φ) is a separable C∗−algebra as well.

If ψ is a complex homomorphism deﬁned in HL(EΩ), then ψ(ef ) = f (λ0) for every f ∈ H(Ω), where λ0 = ψ(Pe,1).

The algebra (HL(EΩ), τ0)

Let us consider the space locally convex space (HL(EΩ), τ0). It is known:

(HL(EΩ), τ0) is complete. Every separable C∗−álgebra A (not necessarily unitary) has an approximation of the identity, i.e., a sequence (en) ⊂ A such that n∈N lim kenz − zk = 0 for every z ∈ A and kenk ≤ 1 for every n ∈ . n→∞ N

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 24 / 29 Let M0(HL(EΩ)) denote the spectrum of (HL(EΩ), τ0) endowed with the Gelfand topology τG. Well known: If θ is a complex homomorphism deﬁned in H(Ω), then θ(f ) = f (λθ) for every f ∈ H(Ω) where λθ = θ(Id) ∈ Ω Lemma 1

If ψ is a complex homomorphism deﬁned in HL(EΩ), then ψ(ef ) = f (λ0) for every f ∈ H(Ω), where λ0 = ψ(Pe,1).

The algebra (HL(EΩ), τ0)

Let us consider the space locally convex space (HL(EΩ), τ0). It is known:

If ψ is a complex homomorphism deﬁned in HL(EΩ), then ψ(ef ) = f (λ0) for every f ∈ H(Ω), where λ0 = ψ(Pe,1).

The algebra (HL(EΩ), τ0)

Let us consider the space locally convex space (HL(EΩ), τ0). It is known:

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 24 / 29 Well known: If θ is a complex homomorphism deﬁned in H(Ω), then θ(f ) = f (λθ) for every f ∈ H(Ω) where λθ = θ(Id) ∈ Ω Lemma 1

If ψ is a complex homomorphism deﬁned in HL(EΩ), then ψ(ef ) = f (λ0) for every f ∈ H(Ω), where λ0 = ψ(Pe,1).

The algebra (HL(EΩ), τ0)

Let us consider the space locally convex space (HL(EΩ), τ0). It is known:

Let M0(HL(EΩ)) denote the spectrum of (HL(EΩ), τ0) endowed with the Gelfand topology τG.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 24 / 29 Lemma 1

If ψ is a complex homomorphism deﬁned in HL(EΩ), then ψ(ef ) = f (λ0) for every f ∈ H(Ω), where λ0 = ψ(Pe,1).

The algebra (HL(EΩ), τ0)

Let us consider the space locally convex space (HL(EΩ), τ0). It is known:

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 24 / 29 The algebra (HL(EΩ), τ0)

Let us consider the space locally convex space (HL(EΩ), τ0). It is known:

If ψ is a complex homomorphism deﬁned in HL(EΩ), then ψ(ef ) = f (λ0) for every f ∈ H(Ω), where λ0 = ψ(Pe,1).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 24 / 29 {Pen,1 ◦ f ; n ∈ N} ⊂ C(EΩ, E) is equicontinuous.

{(Pen,1 ◦ f )(z); n ∈ N} is a compact set for every z ∈ EΩ.

Idea of proof: ∗ ψ0 ∈ M(E) ⇒ ker(ψ0) is a separable C −álgebra ⇒ there exists an approximation of the identity (en) in ker(ψ ). n∈N 0 Show that:

τ0 Arzela-Ascoli Theorem ⇒ there exists g ∈ C(EΩ, E) such that Pen,1 ◦ f → g (go to a subsequence, if necessary) . In particular, g(z) = lim Pe ,1(f (z)) = lim enf (z) = f (z) n→∞ n n→∞

τ0 for all z ∈ EΩ, and hence Pen,1 ◦ f → f .

The algebra (HL(EΩ), τ0)

Lemma 2 ∗ Let E be a separable C −álgebra, f ∈ HL(EΩ) and ψ ∈ M0(HL(EΩ)). If f (z) ∈ ker(ψ0) for every z ∈ EΩ, then f ∈ ker(ψ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 25 / 29 {(Pen,1 ◦ f )(z); n ∈ N} is a compact set for every z ∈ EΩ.

τ0 for all z ∈ EΩ, and hence Pen,1 ◦ f → f .

The algebra (HL(EΩ), τ0)

Lemma 2 ∗ Let E be a separable C −álgebra, f ∈ HL(EΩ) and ψ ∈ M0(HL(EΩ)). If f (z) ∈ ker(ψ0) for every z ∈ EΩ, then f ∈ ker(ψ).

Idea of proof: ∗ ψ0 ∈ M(E) ⇒ ker(ψ0) is a separable C −álgebra ⇒ there exists an approximation of the identity (en) in ker(ψ ). n∈N 0 Show that:

{Pen,1 ◦ f ; n ∈ N} ⊂ C(EΩ, E) is equicontinuous.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 25 / 29 {(Pen,1 ◦ f )(z); n ∈ N} is a compact set for every z ∈ EΩ.

τ0 for all z ∈ EΩ, and hence Pen,1 ◦ f → f .

The algebra (HL(EΩ), τ0)

Lemma 2 ∗ Let E be a separable C −álgebra, f ∈ HL(EΩ) and ψ ∈ M0(HL(EΩ)). If f (z) ∈ ker(ψ0) for every z ∈ EΩ, then f ∈ ker(ψ).

Idea of proof: ∗ ψ0 ∈ M(E) ⇒ ker(ψ0) is a separable C −álgebra ⇒ there exists an approximation of the identity (en) in ker(ψ ). n∈N 0 Show that:

{Pen,1 ◦ f ; n ∈ N} ⊂ C(EΩ, E) is equicontinuous.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 25 / 29 τ0 Arzela-Ascoli Theorem ⇒ there exists g ∈ C(EΩ, E) such that Pen,1 ◦ f → g (go to a subsequence, if necessary) . In particular, g(z) = lim Pe ,1(f (z)) = lim enf (z) = f (z) n→∞ n n→∞

τ0 for all z ∈ EΩ, and hence Pen,1 ◦ f → f .

The algebra (HL(EΩ), τ0)

Lemma 2 ∗ Let E be a separable C −álgebra, f ∈ HL(EΩ) and ψ ∈ M0(HL(EΩ)). If f (z) ∈ ker(ψ0) for every z ∈ EΩ, then f ∈ ker(ψ).

Idea of proof: ∗ ψ0 ∈ M(E) ⇒ ker(ψ0) is a separable C −álgebra ⇒ there exists an approximation of the identity (en) in ker(ψ ). n∈N 0 Show that:

{Pen,1 ◦ f ; n ∈ N} ⊂ C(EΩ, E) is equicontinuous.

{(Pen,1 ◦ f )(z); n ∈ N} is a compact set for every z ∈ EΩ.

τ0 for all z ∈ EΩ, and hence Pen,1 ◦ f → f .

The algebra (HL(EΩ), τ0)

Lemma 2 ∗ Let E be a separable C −álgebra, f ∈ HL(EΩ) and ψ ∈ M0(HL(EΩ)). If f (z) ∈ ker(ψ0) for every z ∈ EΩ, then f ∈ ker(ψ).

Idea of proof: ∗ ψ0 ∈ M(E) ⇒ ker(ψ0) is a separable C −álgebra ⇒ there exists an approximation of the identity (en) in ker(ψ ). n∈N 0 Show that:

{Pen,1 ◦ f ; n ∈ N} ⊂ C(EΩ, E) is equicontinuous.

{(Pen,1 ◦ f )(z); n ∈ N} is a compact set for every z ∈ EΩ.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 25 / 29 In particular, g(z) = lim Pe ,1(f (z)) = lim enf (z) = f (z) n→∞ n n→∞

τ0 for all z ∈ EΩ, and hence Pen,1 ◦ f → f .

The algebra (HL(EΩ), τ0)

Lemma 2 ∗ Let E be a separable C −álgebra, f ∈ HL(EΩ) and ψ ∈ M0(HL(EΩ)). If f (z) ∈ ker(ψ0) for every z ∈ EΩ, then f ∈ ker(ψ).

Idea of proof: ∗ ψ0 ∈ M(E) ⇒ ker(ψ0) is a separable C −álgebra ⇒ there exists an approximation of the identity (en) in ker(ψ ). n∈N 0 Show that:

{Pen,1 ◦ f ; n ∈ N} ⊂ C(EΩ, E) is equicontinuous.

{(Pen,1 ◦ f )(z); n ∈ N} is a compact set for every z ∈ EΩ.

τ0 Arzela-Ascoli Theorem ⇒ there exists g ∈ C(EΩ, E) such that Pen,1 ◦ f → g (go to a subsequence, if necessary) .

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 25 / 29 The algebra (HL(EΩ), τ0)

Lemma 2 ∗ Let E be a separable C −álgebra, f ∈ HL(EΩ) and ψ ∈ M0(HL(EΩ)). If f (z) ∈ ker(ψ0) for every z ∈ EΩ, then f ∈ ker(ψ).

Idea of proof: ∗ ψ0 ∈ M(E) ⇒ ker(ψ0) is a separable C −álgebra ⇒ there exists an approximation of the identity (en) in ker(ψ ). n∈N 0 Show that:

{Pen,1 ◦ f ; n ∈ N} ⊂ C(EΩ, E) is equicontinuous.

{(Pen,1 ◦ f )(z); n ∈ N} is a compact set for every z ∈ EΩ.

τ0 for all z ∈ EΩ, and hence Pen,1 ◦ f → f .

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 25 / 29 Theorem If E is a separable commutative C∗−álgebra with a unit element e and Ω (C is a simply connected domain, the mapping

δ : M(E) × Ω −→ M0(HL(EΩ))

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ) is a homeomorphism between (M(E) × Ω, τπ) and (M0(HL(EΩ)), τG) .

We are going to show that δ is surjective.

The algebra (HL(EΩ), τ0)

Note that Pen,1 ◦ f = Pen,0 · f since Pen,1(f (z)) = enf (z) = Pen,0(z)f (z) for all z ∈ EΩ. τ0 Now, Pen,1 · f → f and ψ(Pen,0) = ψ0(en) = 0 for all n ∈ N implies

ψ(f ) = ψ( lim Pe ,0 · f ) = lim ψ(Pe ,0 · f ) = lim ψ0(en)ψ(f ) = 0. n→∞ n n→∞ n n→∞

Hence f ∈ ker(ψ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 26 / 29 We are going to show that δ is surjective.

The algebra (HL(EΩ), τ0)

Note that Pen,1 ◦ f = Pen,0 · f since Pen,1(f (z)) = enf (z) = Pen,0(z)f (z) for all z ∈ EΩ. τ0 Now, Pen,1 · f → f and ψ(Pen,0) = ψ0(en) = 0 for all n ∈ N implies

ψ(f ) = ψ( lim Pe ,0 · f ) = lim ψ(Pe ,0 · f ) = lim ψ0(en)ψ(f ) = 0. n→∞ n n→∞ n n→∞

Hence f ∈ ker(ψ).

Theorem If E is a separable commutative C∗−álgebra with a unit element e and Ω (C is a simply connected domain, the mapping

δ : M(E) × Ω −→ M0(HL(EΩ))

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ) is a homeomorphism between (M(E) × Ω, τπ) and (M0(HL(EΩ)), τG) .

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 26 / 29 The algebra (HL(EΩ), τ0)

Note that Pen,1 ◦ f = Pen,0 · f since Pen,1(f (z)) = enf (z) = Pen,0(z)f (z) for all z ∈ EΩ. τ0 Now, Pen,1 · f → f and ψ(Pen,0) = ψ0(en) = 0 for all n ∈ N implies

ψ(f ) = ψ( lim Pe ,0 · f ) = lim ψ(Pe ,0 · f ) = lim ψ0(en)ψ(f ) = 0. n→∞ n n→∞ n n→∞

Hence f ∈ ker(ψ).

Theorem If E is a separable commutative C∗−álgebra with a unit element e and Ω (C is a simply connected domain, the mapping

δ : M(E) × Ω −→ M0(HL(EΩ))

deﬁned by δ(φ, λ0)(f ) = fφ(λ0) for every f ∈ HL(EΩ) is a homeomorphism between (M(E) × Ω, τπ) and (M0(HL(EΩ)), τG) .

We are going to show that δ is surjective.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 26 / 29 Consider the mapping f − fψ0 . For every z ∈ EΩ,

ψ0(f (z) − fψ0 (z)) = fψ0 (ψ0(z)) − fψ0 (ψ0(z)) = 0. Hence, f − fψ0 (z) ∈ ker(ψ0) for all z ∈ EΩ.

By the Lemma2 , f − fψ0 ∈ ker(ψ) and, by Lemma 1, ψ(fψ0 ) = fψ0 (λ0). Conse- quently,

0 = ψ(f − fψ0 ) = ψ(f ) − ψ(fψ0 ) = ψ(f ) − fψ0 (λ0).

From this we get ψ(f ) = fψ0 (λ0)(f ) for every f ∈ HL(EΩ) and so ψ = δ(ψ0, λ0).

The algebra (HL(EΩ), τ0)

Given any ψ ∈ M0(HL(EΩ)), take ψ0 ∈ M(E) deﬁned as in the proof of Theo- rem 2 and λ0 = ψ(Pe,1) ∈ Ω.

f ∈ HL(EΩ) ⇒ fψ0 ∈ H(ψ0(EΩ)) = H(Ω) ⇒ fψ0 ∈ He(Ω) ⊂ HL(EΩ).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 27 / 29 Hence, f − fψ0 (z) ∈ ker(ψ0) for all z ∈ EΩ.

By the Lemma2 , f − fψ0 ∈ ker(ψ) and, by Lemma 1, ψ(fψ0 ) = fψ0 (λ0). Conse- quently,

0 = ψ(f − fψ0 ) = ψ(f ) − ψ(fψ0 ) = ψ(f ) − fψ0 (λ0).

From this we get ψ(f ) = fψ0 (λ0)(f ) for every f ∈ HL(EΩ) and so ψ = δ(ψ0, λ0).

The algebra (HL(EΩ), τ0)

Given any ψ ∈ M0(HL(EΩ)), take ψ0 ∈ M(E) deﬁned as in the proof of Theo- rem 2 and λ0 = ψ(Pe,1) ∈ Ω.

f ∈ HL(EΩ) ⇒ fψ0 ∈ H(ψ0(EΩ)) = H(Ω) ⇒ fψ0 ∈ He(Ω) ⊂ HL(EΩ).

Consider the mapping f − fψ0 . For every z ∈ EΩ,

ψ0(f (z) − fψ0 (z)) = fψ0 (ψ0(z)) − fψ0 (ψ0(z)) = 0.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 27 / 29 By the Lemma2 , f − fψ0 ∈ ker(ψ) and, by Lemma 1, ψ(fψ0 ) = fψ0 (λ0). Conse- quently,

0 = ψ(f − fψ0 ) = ψ(f ) − ψ(fψ0 ) = ψ(f ) − fψ0 (λ0).

From this we get ψ(f ) = fψ0 (λ0)(f ) for every f ∈ HL(EΩ) and so ψ = δ(ψ0, λ0).

The algebra (HL(EΩ), τ0)

Given any ψ ∈ M0(HL(EΩ)), take ψ0 ∈ M(E) deﬁned as in the proof of Theo- rem 2 and λ0 = ψ(Pe,1) ∈ Ω.

f ∈ HL(EΩ) ⇒ fψ0 ∈ H(ψ0(EΩ)) = H(Ω) ⇒ fψ0 ∈ He(Ω) ⊂ HL(EΩ).

Consider the mapping f − fψ0 . For every z ∈ EΩ,

ψ0(f (z) − fψ0 (z)) = fψ0 (ψ0(z)) − fψ0 (ψ0(z)) = 0. Hence, f − fψ0 (z) ∈ ker(ψ0) for all z ∈ EΩ.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 27 / 29 The algebra (HL(EΩ), τ0)

Given any ψ ∈ M0(HL(EΩ)), take ψ0 ∈ M(E) deﬁned as in the proof of Theo- rem 2 and λ0 = ψ(Pe,1) ∈ Ω.

f ∈ HL(EΩ) ⇒ fψ0 ∈ H(ψ0(EΩ)) = H(Ω) ⇒ fψ0 ∈ He(Ω) ⊂ HL(EΩ).

Consider the mapping f − fψ0 . For every z ∈ EΩ,

ψ0(f (z) − fψ0 (z)) = fψ0 (ψ0(z)) − fψ0 (ψ0(z)) = 0. Hence, f − fψ0 (z) ∈ ker(ψ0) for all z ∈ EΩ.

By the Lemma2 , f − fψ0 ∈ ker(ψ) and, by Lemma 1, ψ(fψ0 ) = fψ0 (λ0). Conse- quently,

0 = ψ(f − fψ0 ) = ψ(f ) − ψ(fψ0 ) = ψ(f ) − fψ0 (λ0).

From this we get ψ(f ) = fψ0 (λ0)(f ) for every f ∈ HL(EΩ) and so ψ = δ(ψ0, λ0).

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 27 / 29 Some References

1. B.W. Glickfeld, The Riemann sphere of a commutative Banach algebras, Trans. Amer. Math. Soc. 38 (1960), 414-425. 2. B.W. Glickfeld, Contributions to The Theory of Holomorphic Function in Commutative Banach Algebras, Thesis, Columbia University, New York, 1964. 3. B.W. Glickfeld, On the inverse function theorem in commutative Banach algebras, Illinois J. Math. 15 (1971) 212-221. 4. J. Dugundji, Topology, Allyn and Bacon, Boston, 1966. 5. E. Hille and R.S. Phillips, Functional Analysis and Semi-groups, American Mathematical Society, Colloquium Publications XXXI, Baltimore, 1957. 6. E.R. Lorch, The theory of analytic functions in normed abelian vector rings, Trans. Amer. Math. Soc. 54 (1943), 414–425. 7. D.H. Luecking and L.A. Rubel, Complex analysis. A Functional Analysis Approach, Springer-Verlag New York, 1984.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 28 / 29 Some References

8. G.V.S. Mauro, Espectros de Álgebras de Aplicações Analíticas no sentido de Lorch, Tese de Doutorado, UFRJ (2016) 9. G.V.S. Mauro, L.A. Moraes and A.F. Pereira, Topological and algebraic properties of spaces of Lorch analytic mappings, Math. Nachr. 289 (2015), 845 - 853. 10. G.V.S. Mauro and L.A. Moraes, Algebras of Lorch analytic mappings on uniform algebras. , Pre-print. 11. L.A. Moraes and A.F. Pereira, The spectra of algebras of Lorch analytic mappings, Topology 48 (2009), 91–99. 12. L.A. Moraes and A.F. Pereira, The algebra of Lorch analytic mappings with the Hadamard product, Publ. RIMS Kyoto Univ. 49 (2013) 111-122. 13. H.E. Warren, A Riemann mapping theorem for C(X), Proc. Amer. Math. Soc. 28 (1971), 147-154. 14. H.E. Warren, Analytic equivalence among simply connected domains in C(X), Trans. Amer. Math. Soc. 196 (1974), 265-288.

L. Moraes (UFRJ) Algebras of Lorch analytic mappings deﬁned in uniform algebras Valencia, 20/10/2017 29 / 29