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Positivity

, Volume 16, Issue 1, pp 53–66 | Cite as

The Kadison–Singer problem for the direct sum of matrix algebras

  • Charles Akemann
  • Joel Anderson
  • Betul Tanbay
Open Access
Article

Abstract

Let M n denote the algebra of complex n × n matrices and write M for the direct sum of the M n . So a typical element of M has the form
$$ x = x_1\oplus x_2 \cdots \oplus x_n \oplus \cdots, $$
where \({x_n \in M_n}\) and \({\|x\| = \sup_n\|x_n\|}\). We set \({D= \{\{x_n\}\in M: x_n\,{\rm is\,diagonal\,for\,all}\,N\}}\). We conjecture (contra Kadison and Singer in Am J Math 81:383–400, 1959) that every pure state of D extends uniquely to a pure state of M. This is known for the normal pure states of D, and we show that this is true for a (weak*) open, dense subset of all the singular pure states of D. We also show that (assuming the Continuum hypothesis) M has pure states that are not multiplicative on any maximal abelian *-subalgebra of M.

Keywords

Kadison–Singer problem Matrix algebras Operator algebras 

Mathematics Subject Classification (2000)

Primary 46L 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of MathematicsPenn State UniversityUniversity ParkUSA
  3. 3.Department of MathematicsBogazici UniversityBebek, IstanbulTurkey

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