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
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.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Akemann, C., Anderson, J. & Tanbay, B. The Kadison–Singer problem for the direct sum of matrix algebras. Positivity 16, 53–66 (2012). https://doi.org/10.1007/s11117-010-0109-1
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DOI: https://doi.org/10.1007/s11117-010-0109-1