Representation of a Banach algebra
= Representation of a Banach algebra
{title2=$\pi:A\to\operatorname{End}_{\mathbb C}(X)$}
A representation of a complex <Banach algebra> is a complex-linear multiplicative action on a <vector space>. For a <unital> action require $\pi(1)=I$; a nonunital algebra can be handled by extending the action to its <unitization>. A nonzero action is algebraically irreducible when it has no nonzero proper invariant <vector> subspace, equivalently when it defines a <simple module>.