Let $\mathcal{A}$ be an algebra. $d: \mathcal{A}\to \mathbb{N}$ is a function satisfying 1) $d(S+T)\le d(S)+d(T)$, 2)$d(ST)\le d(S)+d(T)$ and 3) $d(aS)=d(S)$ for all $a\in \mathbb{C}, S,T\in\mathcal{A}$.
I am wondering what else can we say about this function and what kind of information can this function give about the algebra?
For people who may be interested about the background of this function, I am thinking about the defect function in the almost invariant subspaces theory.
Thanks!