Homomorphisms Between Algebras of Continuous Functions

1989 ◽  
Vol 41 (1) ◽  
pp. 132-162 ◽  
Author(s):  
J. G. Llavona ◽  
J. A. Jaramillo
Keyword(s):  
Banach Space ◽  
Continuous Functions ◽  
Point Of View ◽  
Space Theory ◽  
Infinite Dimensional ◽  
Weakly Continuous ◽  
Dimensional Approximation ◽  
Weak Topologies ◽  
Bounded Sets ◽  
Uniformly Continuous

We are concerned in this paper with the study of homomorphisms between different algebras of continuous functions, especially the algebras of real functions which are either weakly continuous on bounded sets or weakly uniformly continuous on bounded sets on a Banach space (see definitions below).These spaces of weakly [uniformly] continuous functions appeared in relation with some questions in Infinite-dimensional Approximation Theory (see [4], [6], [11], [12], [13] and [16]); and since the structure of these function spaces is closely related with properties of different weak topologies (the bounded-weak and bounded-weak* topologies, respectively) and with the structure of Banach spaces on which they are defined, their study also presents interest from the point of view of Banach space theory, as can be seen in [2], [12] or [17].

scholarly journals On the M-hypercyclicity of cosine function on Banach spaces

2019 ◽  
Vol 38 (3) ◽  
pp. 133-140
Author(s):  
Abdelaziz Tajmouati ◽  
Abdeslam El Bakkali ◽  
Ahmed Toukmati
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Infinite Dimensional ◽  
Uniformly Continuous

In this paper we introduce and study the M-hypercyclicity of strongly continuous cosine function on separable complex Banach space, and we give the criteria for cosine function to be M-hypercyclic. We also prove that every separable infinite dimensional complex Banach space admits a uniformly continuous cosine function.

On the Structure of Finite Level and ω-Decomposable Borel Functions

2013 ◽  
Vol 78 (4) ◽  
pp. 1257-1287 ◽  
Author(s):  
Luca Motto Ros
Keyword(s):  
Banach Space ◽  
Elementary Proof ◽  
Borel Function ◽  
Continuous Functions ◽  
Full Description ◽  
Countable Union ◽  
Space Theory ◽  
Measurable Functions ◽  
Borel Functions

AbstractWe give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σα0-measurable functions (for every fixed 1 ≤ α < ω1). Moreover, we present some results concerning those Borel functions which are ω-decomposable into continuous functions (also called countably continuous functions in the literature): such results should be viewed as a contribution towards the goal of generalizing a remarkable theorem of Jayne and Rogers to all finite levels, and in fact they allow us to prove some restricted forms of such generalizations. We also analyze finite level Borel functions in terms of composition of simpler functions, and we finally present an application to Banach space theory.

SOME OBSERVATIONS ABOUT THE SPACE OF WEAKLY CONTINUOUS FUNCTIONS FROM A COMPACT SPACE INTO A BANACH SPACE

1992 ◽  
Vol 15 (4) ◽  
pp. 415-425 ◽  
Author(s):  
J. Arias De Reyna ◽  
J. Diestel ◽  
V. Lomonosov ◽  
L. Rodriguez-Piazza
Keyword(s):  
Banach Space ◽  
Compact Space ◽  
Continuous Functions ◽  
Weakly Continuous

Weak Continuity of a Composition Map Between Spaces of Compact Operators and Banach Valued Continuous Functions

1991 ◽  
Vol 34 (2) ◽  
pp. 145-146 ◽  
Author(s):  
Rajappa K. Asthagiri
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Compact Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Weak Continuity ◽  
Sequential Continuity ◽  
Bounded Sets

AbstractThis paper characterizes the Banach space E for the sequential continuity and the continuity on bounded sets of the composition map m: C(S, E)wk x K{E,F)wk —> C(S,F)wk. Here, K(E,F) denotes the Banach space of compact linear operators on the Banach space E to the Banach space F with the usual operator norm, and for any Banach space E, Ewk denote the Banach space E with the weak topology. Also we denote by C(S, E) the Banach space of E valued continuous functions on a nonvoid compact Hausdorff space S with sup norm.

scholarly journals Function spaces and the Mosco topology

1990 ◽  
Vol 42 (1) ◽  
pp. 7-19 ◽  
Author(s):  
Gerald Beer ◽  
Robert Tamaki
Keyword(s):  
Banach Spaces ◽  
Function Space ◽  
Weak Topology ◽  
Continuous Functions ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Compact Open Topology ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Weakly Continuous

Let X and Y be Banach spaces and let C(X, Y) be the functions from X to Y continuous with respect to the weak topology on X and the strong topology on Y. By the Mosco topology τM on C(X, Y) we mean the supremum of the Fell topologies determined by the weak and strong topologies on X × Y, where functions are identified with their graphs. The function space is Hausdorff if and only if both X and Y are reflexive. Moreover, τM coincides with the stronger compact-open topology on C(X, Y) provided X is reflexive and Y is finite dimensional. We also show convergence in either sense is properly weaker than continuous convergence, even for continuous linear functionals, whenever X is infinite dimensional. For real-valued weakly continuous functions, τM is the supremum of the Mosco epitopology and the Mosco hypotopology if and only if X is reflexive.

WEAKLY COMPACT HOMOMORPHISMS BETWEEN SMALL ALGEBRAS OF ANALYTIC FUNCTIONS

2001 ◽  
Vol 33 (6) ◽  
pp. 715-726 ◽  
Author(s):  
PABLO GALINDO ◽  
MIKAEL LINDSTRÖM
Keyword(s):  
Banach Space ◽  
Composition Operator ◽  
Analytic Functions ◽  
Approximation Property ◽  
Composition Operators ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Weakly Compact ◽  
The Relationship ◽  
Uniformly Continuous

The weak compactness of the composition operator CΦ(f) = f ○ Φ acting on the uniform algebra of analytic uniformly continuous functions on the unit ball of a Banach space with the approximation property is characterized in terms of Φ. The relationship between weak compactness and compactness of these composition operators and general homomorphisms is also discussed.

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