The Maximal Ideal Space of Subalgebras of the Disk Algebra

1975 ◽  
Vol 18 (1) ◽  
pp. 61-65 ◽  
Author(s):  
Bruce Lund
Keyword(s):  
Unit Disk ◽  
Maximal Ideal ◽  
Function Algebra ◽  
Continuous Functions ◽  
Maximal Ideal Space ◽  
Open Unit ◽  
Ideal Space ◽  
Disk Algebra ◽  
Closed Subalgebra ◽  
Uniformly Closed

Let X be a compact Hausdorff space and C(X) the complexvalued continuous functions on X. We say A is a function algebra on X if A is a point separating, uniformly closed subalgebra of C(X) containing the constant functions. Equipped with the sup-norm ‖f‖ = sup{|f(x)|: x ∊ X} for f ∊ A, A is a Banach algebra. Let MA denote the maximal ideal space.Let D be the closed unit disk in C and let U be the open unit disk. We call A(D)={f ∊ C(D):f is analytic on U} the disk algebra. Let T be the unit circle and set C1(T) = {f ∊ C(T): f'(t) ∊ C(T)}.

The Maximal Ideal Space of H∞ + C on the ball in Cn

1979 ◽  
Vol 31 (1) ◽  
pp. 79-86 ◽  
Author(s):  
Gerard Mcdonald
Keyword(s):  
Maximal Ideal ◽  
Banach Algebras ◽  
Holomorphic Functions ◽  
Maximal Ideal Space ◽  
Open Unit ◽  
Ideal Space ◽  
Open Unit Ball ◽  
Closed Subalgebra ◽  
Image Position ◽  
Bounded Holomorphic

Let S denote the unit sphere in Cn, B the (open) unit ball in Cn and H∞(B) the collection of all bounded holomorphic functions on B. For f ∈ H∞(B) the limitsexist for almost every ζ in S, and the map ƒ → ƒ* defines an isometric isomorphism from H∞(B) onto a closed subalgebra of L∞(S), denoted H∞(S). (The only measure on S we will refer to in this paper is the Lebesgue measure, dσ, generated by Euclidean surface area.) Rudin has shown in [4] that the spaces H∞(B) + C(B) and H∞(S) + C(S) are Banach algebras in the sup norm. In this paper we will show that the maximal ideal space of H∞(B) + C(B), Σ (H∞(B) + C(B)), is naturally homeomorphic to Σ (H∞(B)) and that Z (H∞(S) + C(S)) is naturally homeomorphic to Σ (H∞(S))\B.

A Filter Description of the Homomorphisms of H∞

1965 ◽  
Vol 17 ◽  
pp. 734-757 ◽  
Author(s):  
A. Kerr-Lawson
Keyword(s):  
Unit Disk ◽  
Maximal Ideal ◽  
Continuous Mapping ◽  
Open Unit ◽  
Supremum Norm ◽  
Ideal Space ◽  
Open Unit Disk ◽  
Classical Theorem ◽  
Bounded Analytic Functions ◽  
The Hyperbolic Metric

The algebra of bounded analytic functions on the open unit disk D, usually written H∞ i is a commutative Banach algebra under the supremum norm. Since its compact maximal ideal space M (space of complex homomorphisms) is an extension space of the unit disk, there must be a continuous mapping form βD, the Stone-Čech compactification of D, onto M. R. C. Buck has remarked (4), that this mapping fails to be one-one, in the light of a classical theorem of Pick. If the points of βD are represented by filters of subsets of D, we can identify those filters which are sufficiently close in terms of the hyperbolic metric on D in an attempt to get a one-one correspondence between filters and points of M.

Homeomorphic Analytic Maps into the Maximal Ideal Space of H∞

1999 ◽  
Vol 51 (1) ◽  
pp. 147-163 ◽  
Author(s):  
Daniel Suárez
Keyword(s):  
Maximal Ideal ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Gleason Part ◽  
Interpolating Sequences ◽  
Closed Subalgebra ◽  
Analytic Maps ◽  
The Ideal

AbstractLet m be a point of the maximal ideal space of H∞ with nontrivial Gleason part P(m). If Lm : D → P(m) is the Hoffman map, we show that H∞ ° Lm is a closed subalgebra of H∞. We characterize the points m for which Lm is a homeomorphism in terms of interpolating sequences, and we show that in this case H∞ ° Lm coincides with H∞. Also, if Im is the ideal of functions in H∞ that identically vanish on P(m), we estimate the distance of any f ϵ H∞ to Im.

scholarly journals Toeplitz operators and algebras of bounded analytic functions on the disk

1991 ◽  
Vol 33 (2) ◽  
pp. 181-185
Author(s):  
R. C. Smith
Keyword(s):  
Maximal Ideal ◽  
Orthogonal Projection ◽  
Analytic Functions ◽  
Toeplitz Operators ◽  
Multiplication Operator ◽  
Ideal Space ◽  
Bounded Analytic Functions ◽  
Disk Algebra ◽  
Closed Subalgebra ◽  
Image Position

Here and throughout, A is a closed subalgebra of H∞ that contains the disk algebra and M(A) denotes the maximal ideal space of A. Because A contains the function fo(z) = z, we can define the fiber Mλ(A) of M(A) for λ ε ∂D (the unit circle) in the usual way; i.e., Mλ(A) = {φ ∈ M(A): fo(φ) = λ}. The Bergman space of the unit disk D is the L2(D, dx dy)-closure of A. Let be the orthogonal projection. For f ∈ L∞(D, dx dy), define the multiplication operator Mf: L2(D, dx dy)→ L2, (D, dx dy) byand define the Toeplitz operator by

Ideals and Subalgebras of a Function Algebra

1974 ◽  
Vol 26 (02) ◽  
pp. 405-411 ◽  
Author(s):  
Bruce Lund
Keyword(s):  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Ideal Space ◽  
Shilov Boundary ◽  
Closed Point ◽  
Continuous Complex ◽  
Complex Valued ◽  
Uniformly Closed

Let X be a compact Hausdorff space and C(X) the set of all continuous complex-valued functions on X. A function algebra A on X is a uniformly closed, point separating subalgebra of C(X) which contains the constants. Equipped with the sup-norm, A becomes a Banach algebra. We let MA denote the maximal ideal space and SA the Shilov boundary.

Real linear isometries between function algebras. II

2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Osamu Hatori ◽  
Takeshi Miura
Keyword(s):  
Maximal Ideal ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Hausdorff Spaces ◽  
Shilov Boundary ◽  
Locally Compact ◽  
Choquet Boundary ◽  
Function Algebras ◽  
Real Linear ◽  
Uniformly Closed

AbstractWe describe the general form of isometries between uniformly closed function algebras on locally compact Hausdorff spaces in a continuation of the study by Miura. We can actually obtain the form on the Shilov boundary, rather than just on the Choquet boundary. We also give an example showing that the form cannot be extended to the whole maximal ideal space.

scholarly journals Maximal ideal space of function algebras

1997 ◽  
Vol 63 (1) ◽  
pp. 78-90
Author(s):  
Jorge Bustamante González ◽  
Raul Escobedo Conde
Keyword(s):  
Real Function ◽  
Normed Space ◽  
Maximal Ideal ◽  
Function Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Uniform Spaces ◽  
Differentiable Functions ◽  
Function Algebras ◽  
Real Function Algebra

AbstractWe present a representation theory for the maximal ideal space of a real function algebra, endowed with the Gelfand topology, using the theory of uniform spaces. Application are given to algebras of differentiable functions in a normęd space, improving and generalizing some known results.

Algebras of Bounded Analytic Functions containing the Disk Algebra

1986 ◽  
Vol 38 (1) ◽  
pp. 87-108 ◽  
Author(s):  
Keiji Izuchi ◽  
Yuko Izuchi
Keyword(s):  
Analytic Functions ◽  
Toeplitz Operators ◽  
Continuous Functions ◽  
Open Unit ◽  
Bounded Analytic Functions ◽  
Disk Algebra ◽  
Closed Subalgebra ◽  
Backward Shift ◽  
Image Position ◽  
Douglas Algebras

Let D be the open unit disk and let ∂D be its boundary. We denote by C the algebra of continuous functions on ∂d, and by L∞ the algebra of essentially bounded measurable functions with respect to the normalized Lebesgue measure m on ∂D. Let H∞ be the algebra of bounded analytic functions on D. Identifying with their boundary functions, we regard H∞ as a closed subalgebra of L∞. Let A = H∞ Pi C, which is called the disk algebra. The algebras A and H∞ have been studied extensively [5, 6, 7]. In these fifteen years, norm closed subalgebras between H∞ and L∞, called Douglas algebras, have received considerable attention in connection with Toeplitz operators [12]. A norm closed subalgebra between A and H∞ is called an analytic subalgebra. In [2], Dawson studied analytic subalgebras and he remarked that there are many different types of analytic subalgebras. One problem is to study which analytic subalgebras are backward shift invariant. Here, a subset E of H∞ is called backward shift invariant if

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