Automorphism of von Neumann algebras as point transformations

Abstract:

Given a concrete separable ${C^ \ast }$-algebra $\mathfrak {A}$ with unit and a faithful normal finite trace $\tau$ on $\mathfrak {A}''$, we introduce a notion of $\tau$-almost every state on $\mathfrak {A}$ which has the proper relationship with Segal’s noncommutative integration theory. We then prove that any $\ast$-automorphism of $\mathfrak {A}''$ is implemented by some point transformation in the state space of $\mathfrak {A}$, defined $\tau$-almost everywhere. This generalizes the classical result of von Neumann-Maharam.