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Filters in C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Tristan Matthew Bice
Tristan Matthew Bice*
Affiliation:
Mathematical Logic Group, Kobe University, Natural Sciences Building No. 3, Hyogo Prefecture, Japan, e-mail: tristan.bice@gmail.com
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Abstract

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In this paper we analyze states on ${{\text{C}}^{*}}$-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 exists a pair of projections on which the state is 1, even though the state is bounded strictly below 1 for projections below this pair. Next, 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.

Keywords

46L0303E35C*-algebrasstates, Kadinson-Singer conjectureultrafilterstowers
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 3 , 01 June 2013 , pp. 485 - 509
Copyright
Copyright © Canadian Mathematical Society 2013

References

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