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,

2-Local Isometries on Spaces of Lipschitz Functions

2011 ◽  
Vol 54 (4) ◽  
pp. 680-692 ◽  
Author(s):  
A. Jiménez-Vargas ◽  
Moisés Villegas-Vallecillos
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Isometry Group ◽  
Lipschitz Constant ◽  
Lipschitz Functions ◽  
Linear Isometry ◽  
Local Isometry ◽  
Surjective Isometry ◽  
Image Position

AbstractLet (X, d) be a metric space, and let Lip(X) denote the Banach space of all scalar-valued bounded Lipschitz functions ƒ on X endowed with one of the natural normswhere L(ƒ) is the Lipschitz constant of ƒ. It is said that the isometry group of Lip(X) is canonical if every surjective linear isometry of Lip(X) is induced by a surjective isometry of X. In this paper we prove that if X is bounded separable and the isometry group of Lip(X) is canonical, then every 2-local isometry of Lip(X) is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of Lip(X) when X is bounded.

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 .

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 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 .

scholarly journals Linear Isometries between Real Banach Algebras of Continuous Complex-Valued Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Davood Alimohammadi ◽  
Hadis Pazandeh
Keyword(s):  
Composition Operators ◽  
Banach Algebras ◽  
Extreme Points ◽  
Function Algebra ◽  
Hausdorff Spaces ◽  
Uniform Function ◽  
Weighted Composition ◽  
Continuous Complex ◽  
Complex Valued ◽  
Linear Isometries

Let and be compact Hausdorff spaces, and let and be topological involutions on and , respectively. In 1991, Kulkarni and Arundhathi characterized linear isometries from a real uniform function algebra on (, ) onto a real uniform function algebra on (, ) applying their Choquet boundaries and showed that these mappings are weighted composition operators. In this paper, we characterize all onto linear isometries and certain into linear isometries between and applying the extreme points in the unit balls of and .

scholarly journals The Daugavet property for spaces of Lipschitz functions

2007 ◽  
Vol 101 (2) ◽  
pp. 261 ◽  
Author(s):  
Yevgen Ivakhno ◽  
Vladimir Kadets ◽  
Dirk Werner
Keyword(s):  
Metric Space ◽  
Lipschitz Functions ◽  
Compact Metric Space ◽  
Daugavet Property

For a compact metric space $K$ the space $\mathrm{Lip}(K)$ has the Daugavet property if and only if the norm of every $f \in \mathrm{Lip}(K)$ is attained locally. If $K$ is a subset of an $L_p$-space, $1<p<\infty$, this is equivalent to the convexity of $K$.

scholarly journals A note on a Residual Subset of Lipschitz Functions on Metric Spaces

2014 ◽  
Vol 58 (3) ◽  
pp. 631-636
Author(s):  
Fabio Cavalletti
Keyword(s):  
Banach Space ◽  
Probability Measure ◽  
Metric Spaces ◽  
Lipschitz Constant ◽  
Lipschitz Functions ◽  
Residual Subset ◽  
Lipschitz Maps ◽  
Almost Everywhere ◽  
Bounded Functions ◽  
Separable Metric Space

AbstractLet (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of real-valued Lipschitz functions with non-zero pointwise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous to a result proved for real-valued Lipschitz maps defined on ℝ2 by Alberti et al.

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

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 $\textrm {C}(K)$ .

Convergence of successive approximations to a stochastic fixed point

1980 ◽  
Vol 17 (1) ◽  
pp. 297-299
Author(s):  
Arun P. Sanghvi
Keyword(s):  
Fixed Point ◽  
Recent Result ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Successive Approximations ◽  
Compact Metric Space

This paper describes some sufficient conditions that ensure the convergence of successive random applications of a family of mappings {Γα : α ∈ A} on a compact metric space (X, d) to a stochastic fixed point. The results are similar in spirit to a recent result of Yahav (1975).

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