Algebraically irreducible representation of a Banach algebra

Algebraic irreducibility excludes every proper nonzero invariant vector subspace, including those that are not closed. It is stronger than topological irreducibility, which excludes only closed invariant subspaces. For a nonzero irreducible action, every nonzero vector is cyclic in the algebraic sense: $A\xi =X$.

For countably many linearly independent vectors in an infinite-dimensional irreducible complex Banach module, there are elements $a_n$ satisfying the displayed annihilation conditions while the tail images $\{\pi (a_n\cdots a_1){\xi }_j:j\ge n\}$ remain linearly independent. Use the quotient norm on an irreducible Banach module to make each finite annihilator a closed Banach subspace. The Jacobson density theorem makes each condition of independence of finitely many tail images open and dense there; the Baire category theorem intersects these countably many conditions. This is stronger than prescribing only finitely many images once.

For a unital irreducible action and $\xi \ne 0$, 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.