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The Josefson-Nissenzweig theorem and filters on
Witold Marciszewski1, Damian Sobota2
1Institute of Mathematics and Computer Science, University of Warsaw, Warsaw, Poland.
Archive for Mathematical Logic
|September 16, 2024
Summary
This study introduces a new class of Tychonoff spaces and investigates their properties using concepts from Banach space theory. The research establishes conditions under which certain function spaces contain specific measure sequences, impacting the structure of continuous function spaces.
Area of Science:
- Topology
- Banach Space Theory
- Measure Theory
Background:
- The Josefson-Nissenzweig theorem is a cornerstone in Banach space theory, concerning the existence of certain sequences of measures.
- Tychonoff spaces are fundamental in general topology, and simpler non-discrete examples are of particular interest.
- Understanding the structure of spaces of continuous functions is crucial in functional analysis.
Purpose of the Study:
- To introduce and study a class of simple non-discrete Tychonoff spaces.
- To investigate the relationship between these spaces and the existence of specific measure sequences in their duals.
- To explore implications for complemented function spaces and the properties of continuous function spaces over compact Hausdorff spaces.
Main Methods:
- Topological construction of specific Tychonoff spaces based on free filters.
- Utilizing sequences of normalized finitely supported signed measures.
- Applying concepts of dual ideals and Katětov preorders.
- Investigating properties of bounded continuous real-valued functions and function spaces like C(K).
Main Results:
- Characterization of the existence of a specific measure sequence in the studied spaces based on the dual ideal's relation to the asymptotic density ideal.
- Demonstration that if a Tychonoff space contains a homeomorphic copy of these spaces, its bounded continuous function space contains a complemented copy of a pointwise-convergent sequence space.
- Proof that if a compact Hausdorff space contains a homeomorphic copy of these spaces, its space of continuous functions C(K) is not a Grothendieck space.
Conclusions:
- The study provides a new perspective on Tychonoff spaces and their connection to Banach space theory.
- The results offer generalizations of known theorems regarding Grothendieck spaces and function spaces.
- This work contributes to the understanding of topological and measure-theoretic properties influencing functional analytic structures.
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