Let
H
\mathcal {H}
be a separable Hilbert space and let
B
(
H
)
B(\mathcal {H})
be the
∗
*
-algebra of all bounded linear operators on
H
\mathcal {H}
. In the present paper, we prove that a positive/regular operator from
L
1
(
0
,
1
)
L_1(0,1)
into an arbitrary separable operator ideal in
B
(
H
)
B(\mathcal {H})
is necessarily Dunford–Pettis, extending and strengthening results 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
\mathcal {M}
and an arbitrary KB-ideal
C
E
C_E
in
B
(
H
)
B(\mathcal {H})
, the predual
M
∗
\mathcal {M}_*
of
M
\mathcal {M}
is not isomorphic to any subspace of
C
E
C_E
. This observation complements several earlier results.