Filters in C*-Algebras

TL;DR

This paper explores the structure of states on C*-algebras, their connection to filters, and implications for the Kadison-Singer conjecture, including the existence of maximal filters and towers under various set-theoretic assumptions.

Contribution

It establishes the equivalence of the Kadison-Singer conjecture to a weak paving conjecture and analyzes the existence of maximal filters and towers in the Calkin algebra.

Findings

Proves the equivalence of the Kadison-Singer conjecture to a weak paving conjecture.

Shows the existence of maximal centred filters for certain ultrafilters.

Demonstrates the consistency of non-existence of maximal centred filters and the preservation of towers under embeddings.

Abstract

In this paper we analyze states on C*-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to investigate the Kadison-Singer conjecture, proving its equivalence to an apparently quite weak paving conjecture and the existence of unique maximal centred extensions of projections coming from ultrafilters on the natural numbers. We then prove that Reid's positive answer to this for q-points in fact also holds for rapid p-points, and that maximal centred filters are obtained in this case. We then show that consistently such maximal centred filters do not exist at all meaning that, for every pure state on the Calkin algebra, there exist projections p and q on which the state is 1, even though it is bounded strictly below 1 for projections below both p and q. Lastly we investigate towers, using cardinal invariant equalities to construct towers on the natural numbers that do and do not remain towers when canonically embedded into the Calkin algebra. Finally we show that consistently all towers on the natural numbers remain towers under this embedding.