Matrix norm inequalities and the relative Dixmier property

If x is a selfadjoint matrix with zero diagonal and non-negative entries, then there exists a decomposition of the identity into k diagonal orthogonal projections {p_m} for which $$\parallel \sum p_m xp_m \parallel \leqslant (1/k)\parallel x\parallel $$ From this follows that all bounded matrices with non-negative entries satisfy the relative Dixmier property or, equivalently, the Kadison Singer extension property. This inequality fails for large Hadamard matrices. However a similar inequality holds for all matrices with respect to the Hilbert-Schmidt norm with constant k^?1/2 and for Hadamard matrices with respect to the Schatten 4-norm with constant 2^1/4k^?1/2.