Exercise 25, § 2

Let $A$ be a complex Banach algebra and let $\pi$ be a nonzero irreducible representation of $A$ in a complex vector space $X$. Let $\xi_0$ be a nonzero element of $X$.

a) The annihilator $I$ of $\xi_0$ is a maximal regular left ideal of $A$; it is closed.

b) The mapping $x\mapsto x\xi_0$ defines by passing to the quotient an isomorphism $\varphi$ of the $A$-module $A/I$ onto the $A$-module $X$.

c) If the norm of the Banach space $A/I$ is transported by $\varphi$, then $X$ becomes a Banach space, and $\|\pi (x)\|\leqslant \|x\|$ for every $x\in A$.

d) If $A$ is primitive (that is to say that $\{0\}$ is a primitive ideal, cf. definition I, p. 11, cf. also A, VIII, p. 433, exerc. 6), then $A =\{0\}$ or $\mathbf{C}\cdot 1$ according as $A$ possesses or does not possess a unit element. (Use what precedes and Cor. 4 of I, p. 26).