Lemma 5

Let $\pi$ be an irreducible representation of A in a vector space E over K.

a) Let I be a two-sided ideal of A. If $\pi (I)\not=\{0\}$, then $\pi |I$ is irreducible;

b) Let $I_1$ and $I_2$ be two-sided ideals of A such that $\pi (I_1)\not= 0$ and $\pi (I_2)\not= 0$. Then $\pi (I_1I_2)\not= 0$.

The set of elements of E annihilated by $\pi (I)$ is stable under $\pi (A)$ and distinct from E, hence equal to 0. Therefore, if $\xi$ is a nonzero element of E, one has $\pi (I)\xi \not= 0$; since $\pi (I)\xi$ is stable under $\pi (A)$, one has $\pi (I)\xi = E$, which proves a). On the other hand, what precedes proves that $\pi (I_2)E = E$, $\pi (I_1)\pi (I_2)E = E$, hence $\pi (I_1I_2)\not= 0$, whence b).