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 Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

Spectral synthesis on zero-dimensional locally compact Abelian groups

2019 ◽  
pp. 450-456
Author(s):  
Sergey S. Platonov
Keyword(s):  
Invariant Subspace ◽  
Linear Subspace ◽  
Spectral Synthesis ◽  
Linear Span ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Closed Linear Subspace ◽  
Compact Abelian Groups ◽  
Continuous Complex ◽  
Complex Valued

Let G be a zero-dimensional locally compact Abelian group whose elements are compact, C(G) the space of continuous complex-valued functions on the group G. A closed linear subspace H⊆ C(G) is called invariant subspace, if it is invariant with respect to translations τ_y ∶ f(x) ↦ f(x + y), y ∈ G. We prove that any invariant subspace H admits spectral synthesis, which means that H coincides with the closure of the linear span of all characters of the group G contained in H.

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.

Chaos to Multiple Mappings

2017 ◽  
Vol 27 (08) ◽  
pp. 1750119 ◽  
Author(s):  
Lidong Wang ◽  
Yingcui Zhao ◽  
Yuelin Gao ◽  
Heng Liu
Keyword(s):  
Metric Space ◽  
Hausdorff Metric ◽  
Strong Sense ◽  
Sufficient Condition ◽  
Compact Metric Space ◽  
Distributional Chaos ◽  
Nonwandering Set

Let [Formula: see text] be a compact metric space and [Formula: see text] be an [Formula: see text]-tuple of continuous selfmaps on [Formula: see text]. This paper investigates Hausdorff metric Li–Yorke chaos, distributional chaos and distributional chaos in a sequence from a set-valued view. On the basis of this research, we draw the main conclusions as follows: (i) If [Formula: see text] has a distributionally chaotic pair, especially [Formula: see text] is distributionally chaotic, the strongly nonwandering set [Formula: see text] contains at least two points. (ii) We give a sufficient condition for [Formula: see text] to be distributionally chaotic in a sequence and chaotic in the strong sense of Li–Yorke. Finally, an example to verify (ii) is given.

scholarly journals Random products of maps synchronizing on average

2020 ◽  
Vol 22 (08) ◽  
pp. 1950086
Author(s):  
Edgar Matias ◽  
Ítalo Melo
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Random Product ◽  
Random Products ◽  
Necessary And Sufficient

We present a necessary and sufficient condition for a random product of maps on a compact metric space to be (strongly) synchronizing on average.

LIMITS OF HOMOMORPHISMS WITH FINITE-DIMENSIONAL RANGE

2005 ◽  
Vol 16 (07) ◽  
pp. 807-821 ◽  
Author(s):  
SHANWEN HU ◽  
HUAXIN LIN ◽  
YIFENG XUE
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
C Algebra ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Unital Simple ◽  
Dimensional Range

Let X be a compact metric space and A be a unital simple C*-algebra with TR (A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.

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 Sums of two-dimensional spectral triples

2007 ◽  
Vol 100 (1) ◽  
pp. 35 ◽  
Author(s):  
Erik Christensen ◽  
Cristina Ivan
Keyword(s):  
Metric Space ◽  
Spectral Triple ◽  
Two Dimensional ◽  
Explicit Computation ◽  
Lebesgue Integral ◽  
Compact Metric Space ◽  
Spectral Triples ◽  
Complex Functions ◽  
Continuous Complex ◽  
Dixmier Trace

We study countable sums of two-dimensional modules for the continuous complex functions on a compact metric space and show that it is possible to construct a spectral triple which gives the original metric back. This spectral triple will be finitely summable for any positive parameter. We also construct a sum of two-dimensional modules which reflects some aspects of the topological dimensions of the compact metric space, but this will only give the metric back approximately. At the end we make an explicit computation of the last module for the unit interval in $\mathsf R$. The metric is recovered exactly, the Dixmier trace induces a multiple of the Lebesgue integral and the growth of the number of eigenvalues $N(\Lambda)$ bounded by $\Lambda$ behaves, such that $N(\Lambda)/\Lambda$ is bounded, but without limit for $\Lambda\to \infty$.

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 $\textrm {C}(K)$ .

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