Question: Let
be a C*-algebra that is complemented in its bidual
by a *-homomorphism, that is, there exists a *-homomorphism
such that
for all
. Is
a von Neumann algebra?
The converse is true: Let be a von Neumann algebra. Then
has a (unique) isometric predual
. Let
be the natural inclusion of the Banach space
in its bidual. We naturally identify the dual of
with
, and the dual of
with
. Then the transpose
is a *-homomorphism that complements
in
.