Let $\theta\in\text{Aut}(\frak{M},\tau)$, where $\frak{M}$ a von-Neumann algebra and $\tau$ a faith, finite, normal trace. Does $\theta$ preserve the norm structure? (Here assumed, that $\frak{M}$ acts on some fixed $\mathcal{H}$.)
If there is a way to express $\|a\|$ in terms of trace, that would be useful.
For example it holds at least that $\|a\|_{p}=\text{Sup}\{|\tau(ab)|:\|b\|_{q}=1\}$ for $p,q\in[1;\infty]$ with $1/p+1/q=1$ and $p<\infty$. Were this or something similar to in the case of $p=\infty$ hold, then one could find a positive answer.
(Note: here $\|a\|_{p}=\sqrt[p]{\tau(|a|^{p})}$ for $p<\infty$ and $\|a\|_{\infty}=\|a\|$.)
User60403 answered the question in a comment. Yes, because $*$-automorphisms of von Neumann algebras are always isometric.
More generally, if $\theta:A\to B$ is a $*$-homomorphism between C*-algebras $A$ and $B$, then $\|\theta(a)\|\leq \|a\|$ for all $a\in A$, and $\theta$ is isometric if it is injective.