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

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.