Complementability of von Neumann algebras

Question

Is every von Neumann algebra complemented in its bidual? It is certainly true for commutative von Neumann algebras as their spectrum is hyperstonian. Is it 1-complemented?

Answer 1

Every dual Banach space $X^{\ast}$ is complemented in its bidual $X^{\ast\ast\ast}$ with a projection of norm $1$: Indeed, the adjoint $(\iota_X)^{\ast}:X^{\ast\ast\ast} \to X^{\ast}$ of the canonical inclusion $\iota_{X}: X \to X^{\ast\ast}$ is left inverse to the canonical inclusion $\iota_{X^{\ast}}: X^{\ast} \to X^{\ast\ast\ast}$, thus $p = \iota_{X^{\ast}} (\iota_X)^{\ast}$ is a norm one projection onto $\iota_{X^{\ast}}(X^{\ast})$. Since by Sakai's theorem a von Neumann algebra is precisely a $C^{\ast}$-algebra whose underlying Banach space is a dual space, a positive answer to both your questions follows immediately.