My doubts are probably very trivial, but I am not 100% sure about that, so any help would be much appreciated. Let's suppose we have a C*-algebra $\mathcal{A}$ and $\tau$ is a *-automorphism. Let's moreover define the subset $\mathcal{A} \supset \mathcal{C} = (a \in \mathcal{A}: \|a\| =1 )$. My questions are:
- Is $\mathcal{C}$ a sub C*-algebra?
- In order to be a *-automorphism, does $\tau$ need to have $\mathcal{A}$ as domain or is it possible to have a subset of $\mathcal{A}$? As you may guess, I was thinking if it is possible to have $\mathcal{C}$ as domain of $\tau$.
Subalgebras are in particular subspaces, so they need to be closed under scalar multiplication. Your set $\mathcal C$ is a sphere, which is far from being a subspace.
An automorphism has by definition the whole algebra as its domain. Now, you can always restrict a function to a subset of its domain, so it is perfectly reasonable to consider the restriction of $\tau$ to $\mathcal C$. And since every $*$-monomorphism of C$^*$-algebras is isometric, you have that $\tau(\mathcal C)=\mathcal C$.