A noncommutative gretsky-ostroy theorem and its applications

Let 1-t be a separable Hilbert space and let B(1-t) be the *-algebra of all bounded linear operators on 1-t. In the present paper, we prove that a positive/regular operator from L1(0, 1) into an arbitrary separable operator ideal in B(1-t) is necessarily Dunford-Pettis, extending and strengthening re-sults due to Gretsky and Ostroy [Glasgow Math. J. 28 (1986), pp. 113-114], and Holub [Proc. Amer. Math. Soc. 104 (1988), pp. 89-95]. Consequently, for an arbitrary atomless von Neumann algebra M and an arbitrary KB-ideal CE in B(1-t), the predual M* of M is not isomorphic to any subspace of CE. This observation complements several earlier results.