Let
M and
N be von Neumann algebras such that
N⊂
M'. Let
Z=
N∩
M and ρ be any normal positive linear functional of (
M∪
N)''. There exists a unique mapping
FρNM from
M into
N satisfying
(1/2)ρ(F
ρNM(
Q1)
Q2+
Q2FρNM(
Q1))=ρ(
Q1Q2)
for all
Q1∈
M,
Q2∈
N and
s(
FρNM(
Q1)){≤}
sN(ρ), where
s denotes the support and
sN denotes the support in
N. The mapping
FρNM is
Z-linear, positive and transposed-
n-positive, of norm 1 and continuous on the unit ball weakly and strongly.
As an application, a generalization of a clustering theorem for an asymptotically abelian case is given.
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