Extension of an automorphism to its predual

Question

Let $M$ is a von Neumann algebra equipped with state $\varphi$ , $\alpha$ $\in$ $Aut(M)$ preserving state $\varphi$, then does $\alpha$ extends to $\alpha_{1}:L^{1}(M,\varphi)\rightarrow L^{1}(M, \varphi)$?

Answer 1

The space L$^1(M,φ)$ is independent of the choice of the weight φ. It is known as the predual of M, and is also denoted by $M_*$.

In Sakai's approach to von Neumann algebras, von Neumann algebras are defined as C*-algebras that admit a predual and morphisms of von Neumann algebras are defined as morphisms of C*-algebras that admit a predual. So the answer to your question is tautologically true. See Sakai's book “C*-algebras and W*-algebras”.