C*-algebras complemented in their biduals

Question: Let $A$ be a C*-algebra that is complemented in its bidual $A^{**}$ by a *-homomorphism, that is, there exists a *-homomorphism $\pi\colon A^{**}\to A$ such that $\pi(a)=a$ for all $a\in A$ . Is $A$ a von Neumann algebra?

The converse is true: Let $A$ be a von Neumann algebra. Then $A$ has a (unique) isometric predual $A_*$ . Let $\kappa_{A_*}\colon A_*\to (A_*)^{**}$ be the natural inclusion of the Banach space $A_*$ in its bidual. We naturally identify the dual of $A_*$ with $A$ , and the dual of $(A_*)^{**}$ with $A^{**}$ . Then the transpose $\kappa_{A_*}^*\colon A^{**}=(A_*)^{***}\to (A_*)^*=A$ is a *-homomorphism that complements $A$ in $A^{**}$ .

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