# 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

, Volume 16 (1) – Feb 2, 2011
14 pages

Publisher
Springer Journals
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.

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations