= Quotient norm on an irreducible Banach module
{title2=$\|\pi(a)\xi\|_0=\inf_{b\in L_\xi}\|a+b\|$}
For a <unital> irreducible action and $\xi\ne0$, the annihilator $L_\xi=\{a:\pi(a)\xi=0\}$ is a <maximal left ideal> and is <closed>. Thus $A/L_\xi$ supplies the displayed complete <norm> on the module, with contractive action. Every <module endomorphism> is bounded for this quotient <norm>: its value at the cyclic <vector> is $a\xi$, so it is induced by bounded right multiplication by $a$ on the quotient. The <Gelfand-Mazur theorem> then makes the module's commutant consist of scalar operators. This constructs one compatible <norm>; it does not by itself establish <continuity> in a different preassigned <norm>.
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