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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!

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    I don't know if$I$can say anything useful, but it's good to keep in mind that an ideal $I$ is maximal if and only if $\mathcal A/I$ is simple. Also, in a unital Banach algebra, the closure of a proper ideal is a proper ideal, which implies that maximal ideals are always closed. This means that the quotient is also a Banach algebra with norm $\|a+I\|=d(a,I)=\inf\limits_{x\in I}\|a-x\|$.2012-05-29

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Pick any $m\in {\mathbb N}$ (I am assuming ${\mathbb N}$ does not include zero). The constant function $d: A \to \{m\}$ satisfies all three conditions. Hence without further information or restrictions, a function satisfying all three conditions might not tell you anything about the original algebra.

By the way, your second condition is not what people usually mean by "submultiplicative".