scholarly journals Extreme Points of the Unit Ball in the Dual Space of Some Real Subspaces of Banach Spaces of Lipschitz Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Davood Alimohammadi ◽  
Hadis Pazandeh
Keyword(s):  
Hausdorff Space ◽  
Extreme Points ◽  
Complex Banach Space ◽  
Lipschitz Functions ◽  
Compact Metric Space ◽  
Closed Subalgebra ◽  
Real Linear ◽  
Complex Valued ◽  
Uniformly Closed

Let be a compact Hausdorff space, be a continuous involution on and denote the uniformly closed real subalgebra of consisting of all for which . Let be a compact metric space and let denote the complex Banach space of complex-valued Lipschitz functions of order on under the norm , where . For , the closed subalgebra of consisting of all for which as , denotes by . Let be a Lipschitz involution on and define for and for . In this paper, we give a characterization of extreme points of , where is a real linear subspace of or which contains 1, in particular, or .

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,

A Note on Strongly Regular Function Algebras

1969 ◽  
Vol 21 ◽  
pp. 912-914 ◽  
Author(s):  
Donald R. Wilken
Keyword(s):  
Hausdorff Space ◽  
Regular Function ◽  
Function Algebra ◽  
Continuous Functions ◽  
Function Algebras ◽  
Closed Subalgebra ◽  
Strongly Regular ◽  
Complex Valued ◽  
Uniformly Closed ◽  
The Ideal

Let A be a uniformly closed subalgebra of C(X), the algebra of all complex-valued continuous functions on a compact Hausdorff space X. If A separates the points of X and contains the constant functions, A is called a function algebra. The algebra A is said to be strongly regular on X if it has the following property.Property. For each f in A, each point x in X, and every , there is a neighbourhood U of x and a function g in A with g(y) = f(x) for all y in U and for all y in X.That is, each function in A is uniformly approximate on X by functions in A which are constant near any point of X. Stated in terms of ideals, strong regularity means that, for each x, the ideal of functions vanishing in a neighbourhood of x is uniformly dense in the maximal ideal at x.

HOMOMORPHISMS OF BANACH ALGEBRAS WITH RANGE IN C1[0, 1]

1994 ◽  
Vol 05 (02) ◽  
pp. 201-212 ◽  
Author(s):  
HERBERT KAMOWITZ ◽  
STEPHEN SCHEINBERG
Keyword(s):  
Analytic Functions ◽  
Compact Hausdorff Space ◽  
Unit Disc ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Locally Compact ◽  
Disc Algebra ◽  
Uniform Algebras ◽  
Closed Subalgebra ◽  
Uniformly Closed

Many commutative semisimple Banach algebras B including B = C (X), X compact, and B = L1 (G), G locally compact, have the property that every homomorphism from B into C1[0, 1] is compact. In this paper we consider this property for uniform algebras. Several examples of homomorphisms from somewhat complicated algebras of analytic functions to C1[0, 1] are shown to be compact. This, together with the fact that every homomorphism from the disc algebra and from the algebra H∞ (∆), ∆ = unit disc, to C1[0, 1] is compact, led to the conjecture that perhaps every homomorphism from a uniform algebra into C1[0, 1] is compact. The main result to which we devote the second half of this paper, is to construct a compact Hausdorff space X, a uniformly closed subalgebra [Formula: see text] of C (X), and an arc ϕ: [0, 1] → X such that the transformation T defined by Tf = f ◦ ϕ is a (bounded) homomorphism of [Formula: see text] into C1[0, 1] which is not compact.

scholarly journals Generalized Numerical Index of Function Algebras

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Han Ju Lee
Keyword(s):  
Banach Space ◽  
Function Algebra ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Function Algebras ◽  
Bounded Complex ◽  
Closed Subalgebra ◽  
Bounded Continuous Functions ◽  
Complex Valued ◽  
Numerical Index

Let X be a complex Banach space and Cb(Ω:X) be the Banach space of all bounded continuous functions from a Hausdorff space Ω to X, equipped with sup norm. A closed subspace A of Cb(Ω:X) is said to be an X-valued function algebra if it satisfies the following three conditions: (i) A≔{x⁎∘f:f∈A,  x⁎∈X⁎} is a closed subalgebra of Cb(Ω), the Banach space of all bounded complex-valued continuous functions; (ii) ϕ⊗x∈A for all ϕ∈A and x∈X; and (iii) ϕf∈A for every ϕ∈A and for every f∈A. It is shown that k-homogeneous polynomial and analytic numerical index of certain X-valued function algebras are the same as those of X.

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.

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.

A Characterization of the Algebra of Functions Vanishing at Infinity

1969 ◽  
Vol 21 ◽  
pp. 751-754 ◽  
Author(s):  
Robert E. Mullins
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Compact ◽  
Algebra Of Functions ◽  
Continuous Complex ◽  
Complex Valued ◽  
Locally Compact Hausdorff Space

1. In this paper, X will always denote a locally compact Hausdorff space, C0(X) the algebra of all complex-valued continuous functions vanishing at infinity on X and B(X) the algebra of all bounded continuous complex-valued functions defined on X. If X is compact, C0(X) is identical to B (X) and all the results of this paper are obvious. Therefore, we will assume at the outset that X is not compact. If A represents an algebra of functions, AR will denote the algebra of all real-valued functions in A.

Isometries of self-adjoint complex function spaces

1989 ◽  
Vol 105 (1) ◽  
pp. 133-138 ◽  
Author(s):  
A. J. Ellis
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Complex Function ◽  
Supremum Norm ◽  
Affine Functions ◽  
Continuous Complex ◽  
Image Position ◽  
Complex Valued ◽  
Uniformly Closed

By a complex function space A we will mean a uniformly closed linear space of continuous complex-valued functions on a compact Hausdorff space X, such that A contains constants and separates the points of X. We denote by S the state-spaceendowed with the w*-topology. If A is self-adjoint then it is well known (cf. [1]) that A is naturally isometrically isomorphic to , and re A is naturally isometrically isomorphic to A(S), where (respectively A(S)) denotes the Banach space of all complex-valued (respectively real-valued) continuous affine functions on S with the supremum norm.

scholarly journals On equicontinuous factors of flows on locally path-connected compact spaces

2020 ◽  
pp. 1-18
Author(s):  
NIKOLAI EDEKO
Keyword(s):  
Lie Group ◽  
Metric Space ◽  
Betti Number ◽  
Alternative Proof ◽  
Simple Consequence ◽  
Compact Metric Space ◽  
Invariant Functions ◽  
Compact Spaces ◽  
Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

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