The KadisonSinger problem is the following statement: for any $\epsilon >0$, there exists $r\in \mathbb N$ such that for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a partition $(\mathcal P_s)_{1\le s\le r}$ of $\mathbb Z$, with $$ \max_{1\le s\le r}\Vert P_s(A\text{diag}A)P_s\Vert_{\mathcal B(\ell^2(\mathbb Z))}\le \epsilon \Vert A\text{diag}A\Vert_{\mathcal B(\ell^2(\mathbb Z))}\quad\tag {PC} $$ where $P_s=\sum_{j\in \mathcal P_s} p_j$ and $p_j$ is the orthogonal projection onto $e_j=(\delta_{j,k})_{k\in \mathbb Z}$; the point is that $r$ depends only on $\epsilon$. This problem is sometimes quoted as the KadisonSinger conjecture, a rather inaccurate denomination since these authors were inclined to think that the answer should be negative. We have given here the formulation of the Paving Conjecture, known to be equivalent to KS. Since (PC) seems to be now solved, I will read the paper and I withdraw my question.
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Virtually everything stated in the question is wrong. The KadisonSinger problem has been solved positively by Marcus, Spielman, and Srivastava, see: http://arxiv.org/pdf/1306.3969v3.pdf The question apparently also refers to an earlier paper of Akemann and me, in which assuming CH we falsified a different conjecture of Kadison and Singer. This problem also has to do with pure states on $B(H)$ so there is some connection with the wellknown KadisonSinger problem. But it's a different question. See: 

