Proposition 3

Let $I\in J(A)$. In order that $\{I\}$ be closed in J(A), it is necessary and sufficient that I be a maximal primitive ideal.

In fact, the closure of $\{I\}$ consists of the primitive ideals of A containing I.

The relation

“ $\pi_1,\pi_2$ are representations of A which are isomorphic ”

is an equivalence relation with respect to $\pi_1$ and $\pi_2$. For every representation $\pi$ of A, we shall denote by cl($\pi$ ) the equivalence class of $\pi$, which is therefore a representation of A isomorphic to $\pi$, such that two representations $\pi_1$ and $\pi_2$ are isomorphic if and only if cl($\pi_1$) $=$ cl($\pi_2$). One says that cl($\pi$ ) is the class of $\pi$.

Let $\mathfrak{c}$ be the cardinal of A. Let $\pi$ be a nonzero irreducible representation of A in a K-vector space E. Let $\xi$ be a nonzero element of E. Since $\pi (A)\xi = E$ (lemma 4), the dimension of E is $\leqslant \mathfrak{c}$ (A, II, p. 97, corollaire). The relation

“ $\lambda$ is a class of irreducible representations of A

in a K-vector space of dimension $\leqslant \mathfrak{c}$ ”

is collectivizing in $\lambda$ (E, II, p. 3). In fact, every vector space of dimension $\leqslant \mathfrak{c}$ is isomorphic to a space $K^B$ where B is a subset of A (A, II, p. 25, déf. 10), and the assertion then follows from E, II, p. 47.

We denote by $\widehat{A}$ the set of classes of irreducible, non-trivial, representations of A. From what precedes, for every non-trivial irreducible representation $\pi$ of A, there exists a unique representation $\widehat{\pi}\in \widehat{A}$ which is isomorphic to $\pi$.

The mapping of $\widehat{A}$ into J(A) which associates to $\pi$ its kernel is surjective. If A is commutative, it follows from the fact that the primitive ideals are the regular maximal ideals that this mapping is a bijection.

We endow $\widehat{A}$ with the inverse image topology of that of J(A) by the mapping $\widehat{A}\rightarrow J(A)$.