A Lie triple derivation of an associative algebra M is a linear map
L
:
M
→
M
L:M \to M
such that
\[
L
[
[
X
,
Y
]
,
Z
]
=
[
[
L
(
X
)
,
Y
]
,
Z
]
+
[
[
X
,
L
(
Y
)
]
,
Z
]
+
[
[
X
,
Y
]
,
L
(
Z
)
]
L[[X,Y],Z] = [ {[L(X),Y],Z} ] + [ {[X,L(Y)],Z} ] + [ {[X,Y],L(Z)} ]
\]
for all
X
,
Y
,
Z
∈
M
X,Y,Z \in M
. (Here
[
X
,
Y
]
=
X
Y
−
Y
X
[X,Y] = XY - YX
and [M, M] is the linear subspace of M generated by such terms.) We show that if M is a von Neumann algebra with no central abelian summands then there exists an operator
A
∈
M
A \in M
such that
L
(
X
)
=
[
A
,
X
]
+
λ
(
X
)
L(X) = [A,X] + \lambda (X)
where
λ
:
M
→
Z
M
\lambda :M \to {Z_M}
is a linear map which annihilates brackets of operators in M.