scholarly journals A Note on Some Uniform Algebra Generated by Smooth Functions in the Plane

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Raymond Mortini ◽  
Rudolf Rupp
Keyword(s):  
Maximal Ideal ◽  
Stable Rank ◽  
Uniform Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Smooth Functions ◽  
Function Algebras ◽  
Classical Function ◽  
Complex Valued ◽  
Compact Planar

We determine, via classroom proofs, the maximal ideal space, the Bass stable rank as well as the topological and dense stable rank of the uniform closure of all complex-valued functions continuously differentiable on neighborhoods of a compact planar set and holomorphic in the interior of . In this spirit, we also give elementary approaches to the calculation of these stable ranks for some classical function algebras on .

Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs

2019 ◽  
pp. 1-18
Author(s):  
Alexander J. Izzo ◽  
Dimitris Papathanasiou
Keyword(s):  
Maximal Ideal ◽  
Uniform Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Regular Space ◽  
Compact Set ◽  
Gleason Part ◽  
Completely Regular ◽  
Analytic Discs ◽  
Compact Subspace

Abstract We strengthen, in various directions, the theorem of Garnett that every $\unicode[STIX]{x1D70E}$ -compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some $\mathbb{C}^{N}$ so that $\widehat{K}\backslash K$ contains a Gleason part homeomorphic to  $X$ , and $\widehat{K}$ contains no analytic discs.

Continuation of analytic structure in the maximal ideal space of a uniform algebra

1982 ◽  
Vol 92 (3) ◽  
pp. 437-449
Author(s):  
John R. Shackell
Keyword(s):  
Recent Result ◽  
Maximal Ideal ◽  
Uniform Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Analytic Structure ◽  
Classical Theorem ◽  
Shilov Boundary ◽  
One Dimensional

Let A be a uniform algebra with maximal ideal space M and Shilov boundary Σ; see (8), (18) or (21) for the basic definitions. If Σ is different from M, there is often analytic structure in M/Σ. However this is not always the case, as is shown by the classical example of Stolzenberg in (16). Hence much of the considerable amount of research on this topic has been devoted to finding conditions which ensure the presence of analytic structure in M/Σ One particularly fruitful line of development has been concerned with one-dimensional analytic structure; in particular we have in mind the classical theorem of Bishop (see (2), chapter 11) and the more recent result of Aupetit and Wermer(2).

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.

Analytic Structures for H∞ of Certain Domains in Cn

1978 ◽  
Vol 30 (4) ◽  
pp. 863-871
Author(s):  
Eric P. Kronstadt
Keyword(s):  
Bounded Domain ◽  
Maximal Ideal ◽  
Analytic Functions ◽  
Weak Topology ◽  
Uniform Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Bounded Analytic Functions ◽  
Gelfand Transform

Let Ω C Cn be a bounded domain; let H∞ (Ω) be the uniform algebra of bounded analytic functions on 12; and let ∑ (Ω) be the maximal ideal space of H∞ (Ω). In the weak-* topology of (H∞ (Ω))*, ∑ (Ω) is a compact Hausdorf space in which Ω is embedded in a natural fashion, so that to every g ∈ H∞ (Ω) there corresponds the Gelfand transform ĝ ∈ C(∑ (Ω)); ĝ|Ω = g.

Deficiencies of Certain Real Uniform Algebras

1975 ◽  
Vol 27 (1) ◽  
pp. 121-132
Author(s):  
B. V. Limaye ◽  
R. R. Simha
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Linear Span ◽  
Commutative Banach Algebra ◽  
Uniform Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
The Real ◽  
Uniform Algebras ◽  
Real Linear

Let U be a complex uniform algebra, Z and dZ its maximal ideal space and its Šilov boundary, respectively. The Dirichlet (respectively Arens-Singer) deficiency of U is the codimension in CR(∂Z) of the closure of Re U (respectively of the real linear span of log|U-1|). Algebras with finite Dirichlet deficiency have many interesting properties, especially when the Arens-Singer deficiency is zero. (See, e.g. [5].) By a real uniform algebra we mean a real commutative Banach algebra A with identity 1, and norm ‖ ‖ such that ‖f2‖ = ‖f‖2 for each fin A

scholarly journals PROPERTIES OF CERTAIN SUBALGEBRAS OF DALES-DAVIE ALGEBRAS

2007 ◽  
Vol 49 (2) ◽  
pp. 225-233 ◽  
Author(s):  
M. ABTAHI ◽  
T. G. HONARY
Keyword(s):  
Maximal Ideal ◽  
Rational Functions ◽  
Maximal Ideal Space ◽  
Analytic Extension ◽  
Ideal Space ◽  
Differentiable Functions ◽  
Function Algebras ◽  
Infinitely Differentiable Functions ◽  
Interesting Class ◽  
Compact Plane

AbstractWe study an interesting class of Banach function algebras of infinitely differentiable functions on perfect, compact plane sets. These algebras were introduced by H. G. Dales and A. M. Davie in 1973, called Dales-Davie algebras and denoted by D(X, M), where X is a perfect, compact plane set and M = {Mn}∞n = 0 is a sequence of positive numbers such that M0 = 1 and (m + n)!/Mm+n ≤ (m!/Mm)(n!/Mn) for m, n ∈ N. Let d = lim sup(n!/Mn)1/n and Xd = {z ∈ C : dist(z, X) ≤ d}. We show that, under certain conditions on X, every f ∈ D(X, M) has an analytic extension to Xd. Let DP [DR]) be the subalgebra of all f ∈ D(X, M) that can be approximated by the restriction to X of polynomials [rational functions with poles off X]. We show that the maximal ideal space of DP is $X^_d$, the polynomial convex hull of Xd, and the maximal ideal space of DR is Xd. Using some formulae from combinatorial analysis, we find the maximal ideal space of certain subalgebras of Dales-Davie algebras.

scholarly journals Interpolation problem for l1 and a uniform algebra

2002 ◽  
Vol 72 (1) ◽  
pp. 1-12
Author(s):  
Takahiko Nakazi
Keyword(s):  
Maximal Ideal ◽  
Interpolation Problem ◽  
Uniform Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Interpolating Sequence ◽  
Uniform Algebras ◽  
Interpolating Sequences

AbstractLet A be a uniform algebra and M(A) the maximal ideal space of A. A sequence {an} in M(A) is called l1-interpolating if for every sequence (αn) in l1 there exists a function f in A such that f (an) = αn for all n. In this paper, an l1-interpolating sequence is studied for an arbitrary uniform algebra. For some special uniform algebras, an l1-interpolating sequence is equivalent to a familiar l-interpolating sequence. However, in general these two interpolating sequences may be different from each other.

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