The Kadison-Singer problem for the direct sum of matrix algebras. (English) Zbl 1270.46056
Summary: Let \(M _{n }\) denote the algebra of complex \(n \times n\) 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 \(D= \{\{x_n\}\in M: x_n\) is diagonal for all \(n\}\). We conjecture (contra R. V. Kadison and I. M. Singer [Am. J. Math. 81, 383–400 (1959; Zbl 0086.09704)]) 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\(^{\ast}\)) 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 \(^{\ast}\)-subalgebra of \(M\).
MSC:
| 46L30 | States of selfadjoint operator algebras |
Citations:
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