Product of analytic elements for a group of automorphisms in a von Neumann algebra.

Question

How can we prove that the product of two analytic elements for a given group of automorphisms is also an analytic element for that group?

Answer 1

If $(\alpha_t)$ is an automorphism group of $A$ and $a,b\in A$ are analytic, then $z\mapsto \alpha_z(a)\alpha_z(b)$ is an analytic function (as a product of analytic functions). Since it coincides on $\mathbb R$ with $t\mapsto \alpha_t(a)\alpha_t(b)=\alpha_t(ab)$, the element $ab$ is analytic and $\alpha_z(ab)=\alpha_z(a)\alpha_z(b)$ for all $z\in\mathbb C$.