φ-contractibility of some classes of Banach function algebras

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6543-6549
Author(s):  
Morteza Essmaili ◽  
Amir Sanatpour
Keyword(s):  
Metric Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Lipschitz Functions ◽  
Compact Metric Space ◽  
Function Algebras ◽  
Lipschitz Algebra ◽  
Banach Function Algebras

In this paper, we study ?-contractibility of natural Banach function algebras on a compact Hausdorff space. As a consequence, we characterize ?-contractibility of the Lipschitz algebra Lip(X,d?), for a compact metric space (X,d). We also characterize ?-contractibility of certain subalgebras of Lipschitz functions including rational Lipschitz algebras, analytic Lipschitz algebras and differentiable Lipschitz algebras.

scholarly journals Geometrical properties of the space of idempotent probability measures

2021 ◽  
Vol 22 (2) ◽  
pp. 399
Author(s):  
Kholsaid Fayzullayevich Kholturayev
Keyword(s):  
Metric Space ◽  
Category Theory ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Geometrical Properties ◽  
Idempotent Mathematics

Although traditional and idempotent mathematics are "parallel'', by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel''. At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism''). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism'' negation). Further for a compact Hausdorff space X we establish that the hereditary normality of I<sub>3</sub>(X)\ X implies the metrizability of X.

Maximality In Function Algebras

1970 ◽  
Vol 22 (5) ◽  
pp. 1002-1004 ◽  
Author(s):  
Robert G. Blumenthal
Keyword(s):  
Metric Space ◽  
Hausdorff Space ◽  
Function Algebra ◽  
General Function ◽  
Maximal Subalgebra ◽  
Maximal Subalgebras ◽  
Disc Algebra ◽  
Function Algebras ◽  
Closed Point ◽  
Uniformly Closed

In this paper we prove that the proper Dirichlet subalgebras of the disc algebra discovered by Browder and Wermer [1] are maximal subalgebras of the disc algebra (Theorem 2). We also give an extension to general function algebras of a theorem of Rudin [4] on the existence of maximal subalgebras of C(X). Theorem 1 implies that every function algebra defined on an uncountable metric space has a maximal subalgebra.A function algebra A on X is a uniformly closed, point-separating subalgebra of C(X), containing the constants, where X is a compact Hausdorff space. If A and B are function algebras on X, A ⊂ B, A ≠ B, we say A is a maximal subalgebra of B if whenever C is a function algebra on X with A ⊂ C ⊂ B, either C = A or C = B.

scholarly journals Some Dense Linear Subspaces of Extended Little Lipschitz Algebras

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Davood Alimohammadi ◽  
Sirous Moradi
Keyword(s):  
Compact Subset ◽  
Metric Space ◽  
Linear Subspace ◽  
Sufficient Condition ◽  
Compact Metric Space ◽  
Linear Subspaces ◽  
Lipschitz Algebra ◽  
Continuous Complex ◽  
Complex Valued

Let be a compact metric space. In 1987, Bade, Curtis, and Dales obtained a sufficient condition for density of a subspace of little Lipschitz algebra in this algebra and in particular showed that is dense in , whenever . Let be a compact subset of . We define new classes of Lipchitz algebras for and for , consisting of those continuous complex-valued functions on such that and , respectively. In this paper we obtain a sufficient condition for density of a linear subspace of extended little Lipschitz algebra in this algebra and in particular show that is dense in , whenever .

Extreme Points and Linear Isometries of the Banach Space of Lipschitz Functions

1968 ◽  
Vol 20 ◽  
pp. 1150-1164 ◽  
Author(s):  
Ashoke K. Roy
Keyword(s):  
Banach Space ◽  
Lipschitz Condition ◽  
Metric Space ◽  
Lipschitz Constant ◽  
Extreme Points ◽  
Lipschitz Functions ◽  
Compact Metric Space ◽  
Complex Valued ◽  
Linear Isometries

Let X be a compact metric space with metric d. A complex-valued function ƒ on X is said to satisfy a Lipschitz condition if, for all points x and y of X, there exists a constant K such thatThe smallest constant for which the above inequality holds is called the Lipschitz constant for ƒ and is denoted by ||ƒ||d, that is,

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

Hyperspaces of a CANR*

1961 ◽  
Vol 57 (4) ◽  
pp. 754-758
Author(s):  
D. Hammond Smith
Keyword(s):  
Metric Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Finite Topology ◽  
Closed Sets ◽  
Open Set

If X is a compact Hausdorff space we denote by S(X), and by C(X), the hyperspaces of X consisting of all non-empty closed sets, and all non-empty connected closed sets. The topology in each case is the finite topology of Michael ((6), Definition 1·7), in which a sub-base for the open sets is taken consisting of all sets of either of the forms {F|F ⊂ G} and {F|F ∩ G ≠ φ} (where G is any open set of X). Michael has shown that S(X) is also compact Hausdorff ((6), Theorem 4·9·6), and S(X) contains in an obvious way sets which are homeomorphic with C(X) and X itself. We recall that if Xis also a metric space, the topology induced on S(X) (and on C(X)) by Hausdorff's metric is the same as the finite topology ((6), Proposition 3·6).

On the Boundary and Tensor Product of Function Algebras

1966 ◽  
Vol 9 (1) ◽  
pp. 103-106
Author(s):  
A. S. Fox
Keyword(s):  
Tensor Product ◽  
Algebraic Structure ◽  
Compact Hausdorff Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Maximum Modulus ◽  
Function Algebras ◽  
The Family ◽  
Continuous Complex ◽  
Complex Valued

Let be an arbitrary family of continuous complex-valued functions defined on a compact Hausdorff space X. A closed subset B ⊆ X is called a boundary for if every attains its maximum modulus at some point of B. A boundary, B, is said to be minimal if there exists no boundary for properly contained in B. It can be shown that minimal boundaries exist regardless of the algebraic structure which may possess. Under certain conditions on the family , it can be shown that a unique minimal boundary for exists. In particular, this is the case if is a subalgebra or subspace of C(X) where X is compact and Hausdorff (see for example [2]). This unique minimal boundary for an algebra of functions is called the Silov boundary of .

scholarly journals Isomorphisms of Function Algebras and Algebras of Analytic Functions

1978 ◽  
Vol 21 (1) ◽  
pp. 61-71
Author(s):  
Bruce Lund
Keyword(s):  
Riemann Surface ◽  
Analytic Functions ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Finite Codimension ◽  
Open Riemann Surface ◽  
Analytic Boundary ◽  
Function Algebras

AbstractLet R be a finite open Riemann surface with analytic boundary Γ. Set and define is analytic on R}. Conditions are given on a function algebra A on a compact Hausdorff space X which imply that A is isomorphic to a subalgebra of A(R) of finite codimension.

scholarly journals BANACH ALGEBRAS OF VECTOR-VALUED FUNCTIONS

2013 ◽  
Vol 56 (2) ◽  
pp. 419-426 ◽  
Author(s):  
AZADEH NIKOU ◽  
ANTHONY G. O'FARRELL
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Function Algebra ◽  
Shilov Boundary ◽  
Function Algebras ◽  
Peak Points ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractWe introduce the concept of an E-valued function algebra, a type of Banach algebra that consists of continuous E-valued functions on some compact Hausdorff space, where E is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative E-valued function algebras. We give some specific examples.

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