By J.-C. Bourin

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2 Extension to the von Neumann and C ∗ -algebras setting We do not wish to discuss the possible extensions of our results to the setting of operator algebras. Nevertheless we mention that in [8], versions of trace inequalities of Brown-Kosaki and Hansen-Pedersen are established in the framework of a C ∗ -algebra endowed with a densely defined, lower semicontinuous trace. We also note that the paper by Nelson [9] and that one by Fack and Kosaki [4] form a good presentation of the theory of noncommutative integration in semifinite von Neumann algebras.

2) diag(Bj ) = ⊕k Aj , 0 ≤ j ≤ n. Furthermore, we may require that B1 ∞ ≤ 1. 2. Let {Aj }nj=0 be hermitian operators on H with A0 ∞ ≤ 1. Then we can totally dilate them into a monotone family of hermitian operators {Bj }nj=0 on ⊕k H, k = 2n , in such a way that B0 ∞ ≤ 1. References [1] R. -C. Bourin, Singular values of compressions, restrictions and dilations, Linear Algebra Appl. 360 (2003) 259-272. -C. Bourin, Total dilations, Linear Algebra Appl. 368 (2003) 159-169. -D. -K. Li, Numerical ranges and dilations, Linear Multilinear Algebra 47 (2000) 35-48.

18]) to obtain Z −1 as [A − A1/2 C 2 A1/2 ]−1 A−1/2 CA1/2 [A1/2 C 2 A1/2 − A]−1 −1 1/2 −1/2 − A] A CA [A − A1/2 C 2 A1/2 ]−1 [A1/2 C 2 A1/2 that is Z −1 = and the proof is complete. 2. Let A, B be positive operators on H. The statement A ≥ I and B ≥ I is equivalent to each of the following: (1) For each p > 0, there exists a strictly positive operator Z on F ⊃ H such that A = ZH and B = (Z −p )H . (2) For each p > 0, there exists a strictly positive operator Z on ⊕4 H such that diag(Z) = ⊕4 A and diag(Z −p ) = ⊕4 B.