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 .