Integral Equations 0378-620X/88/010028-2151.50+0.20/0 and Operator Theory (c) 1988 Birkh~user Verlag, Basel Vol. ii (1988) MATR/X NORM INEQUALITIES AND THE RELATIVE DIXMIER PROPERTY Kenneth Berman, Herbert Halpern, Victor Kaftal (*) and Gary Weiss (**) 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 {Pro} for which IIZpmxpmll < (1/k)llx]l. 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 21/4k -1/2. .~1 I N T R O D U C T I O N A long standing open problem, ftrst discussed by Kadison and Singer in 1959, is whether every pure state on the atomic masa D of diagonal operators on a separable Hilbert space H has a unique extension to a pure state of B(H) (extension property for the embedding of D into B(H)). An equivalent problem is whether every element x of B(H) has the Dixmier property relative to D, i.e., whether the norm closed convex hull K(x) = co-{uxu* I u~D,u unitary} has non-empty intersection with D , in which case K(x) 0 D = {E(x)} , where E(x) denotes the diagonal of x ( relative Dixmier property for the embedding of D into B(H) ). The distance (~(x) from E(x) to K(x) or equivalently, the distance from 0 to K(x - E(x)), can be measured using decompositions of the identity into a sum of finitely many mutually orthogonal diagonal projections ( diagonal decompositions or d.d. for short ), i.e. we have: or(x) = inf { IIZpm(X- E(x))pmll I {Pm} is a d.d. }. Partially supported by grants (*) of the Taft Foundation, (**) NSF.