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

The Kadison–Singer problem for the direct sum of matrix algebras 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2011 by The Author(s)
Subject
Mathematics; Potential Theory; Operator Theory; Fourier Analysis; Econometrics; Calculus of Variations and Optimal Control; Optimization
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0109-1
Publisher site
See Article on Publisher Site

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.

Journal

PositivitySpringer Journals

Published: Feb 2, 2011

References

  • Classically normal pure states
    Akemann, C.A.; Weaver, N.

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