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.

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)}.

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 .

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).

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.

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.

scholarly journals MULTIPLICATIVELY SPECTRUM-PRESERVING MAPS OF FUNCTION ALGEBRAS. II

2005 ◽  
Vol 48 (1) ◽  
pp. 219-229 ◽  
Author(s):  
N. V. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Natural Extension ◽  
Ideal Space ◽  
Locally Compact ◽  
Function Algebras ◽  
Closed Point ◽  
Signum Function ◽  
Locally Compact Hausdorff Space

AbstractLet $\mathcal{A}$ be a closed, point-separating sub-algebra of $C_0(X)$, where $X$ is a locally compact Hausdorff space. Assume that $X$ is the maximal ideal space of $\mathcal{A}$. If $f\in\mathcal{A}$, the set $f(X)\cup\{0\}$ is denoted by $\sigma(f)$. After characterizing the points of the Choquet boundary as strong boundary points, we use this equivalence to provide a natural extension of the theorem in [10], which, in turn, was inspired by the main result in [6], by proving the ‘Main Theorem’: if $\varPhi:\mathcal{A}\rightarrow\mathcal{A}$ is a surjective map with the property that $\sigma(fg)=\sigma(\varPhi(f)\varPhi(g))$ for every pair of functions $f,g\in\mathcal{A}$, then there is an onto homeomorphism $\varLambda:X\rightarrow X$ and a signum function $\epsilon(x)$ on $X$ such that$$ \varPhi(f)(\varLambda(x))=\epsilon(x)f(x) $$for all $x\in X$ and $f\in\mathcal{A}$.AMS 2000 Mathematics subject classification: Primary 46J10; 46J20

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 The convolution algebraH1(R)

2010 ◽  
Vol 8 (2) ◽  
pp. 167-179 ◽  
Author(s):  
R. L. Johnson ◽  
C. R. Warner
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Tauberian Theorem ◽  
Singular Integrals ◽  
Approximate Identity ◽  
Maximal Ideal Space ◽  
Ideal Space

H1(R) is a Banach algebra which has better mapping properties under singular integrals thanL1(R) . We show that its approximate identity sequences are unbounded by constructing one unbounded approximate identity sequence {vn}. We introduce a Banach algebraQthat properly lies betweenH1andL1, and use it to show thatc(1 + lnn) ≤ ||vn||H1≤Cn1/2. We identify the maximal ideal space ofH1and give the appropriate version of Wiener's Tauberian theorem.

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

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