I have questions regarding part of E. Christopher Lance. Ergodic Theorems for Convex Sets and Operator Algebras, Page 202.
More precisely, in the paragraph starting with "We assume familiarity with..." the 7th line (the adjoint mapping $\alpha^*$) and the 10th line (automorphism $\alpha_*$).
I would like to know how $\alpha^*$ and $\alpha_*$ are defined.
If $\alpha$ is an automorphism of $\mathfrak{A}$, then $$\alpha^*\colon S(\mathfrak{A})\to S(\mathfrak{A}),\qquad \alpha^*(\tau)=\tau\circ\alpha.$$
If $\mathfrak{A}$ is a von Neumann algebra, the predual $\mathfrak{A}_*$ is the space of ultraweakly continuous linear functionals on $\mathfrak{A}$. $\alpha_*$ is given similarly: $$\alpha_*(f)=f\circ\alpha$$ It is stated that "$\alpha$ is the adjoint of $\alpha_*$". This is because a von Neumann algebra $\mathfrak{A}$ is canonically identified with the dual $(\mathfrak{A}_*)^*$, in which case $\alpha$ is really just the adjoint of $\alpha_*$ in the sense of Banach spaces.