Exercise 8, § 1

Let A be an algebra, I a two-sided ideal of A, $J^I(A)$ the set of primitive ideals of A that do not contain I, and $\widehat{A}^I$ the set of irreducible representations $\pi \in \widehat{A}$ such that $\pi (I)\not= 0$.

a) If $\pi \in \widehat{A}^I$, we have $\pi |I\in \widehat{I}$ (lemma 5, a) of I, p. 12). If representations $\pi$ and $\pi '\in \widehat{A}$ are such that $\pi |I$ and $\pi '|I$ are equivalent, then $\pi$ and $\pi '$ are equivalent. (We may suppose that $\pi (x) =\pi '(x)$ for every $x\in I$. Then, for every $y\in A,\pi (y)$ and $\pi '(y)$ coincide on $\sum_{x\in I}$ Im$\pi (x)$, and this subspace is equal to the space of $\pi .$)

b) If $\pi \in \widehat{I},\pi$ extends to an element of $\widehat{A}^I$. (Let R be a maximal regular left ideal of I. Let $g$ be a right unit of I modulo R. Assuming that A admits a unit element, prove that the relation AR $+ A(1-g) = A$ would imply that $g^2\in R$. Hence AR $+ A(1-g)$ is contained in a maximal left ideal M of A. Prove that $M\cap I = R$ and M + I = A.)

c) Deduce from a) and b) that the mapping $I'\mapsto I'\cap I$ is a homeomorphism of $J^I(A)$ onto J(I), and that the mapping $\pi \mapsto \pi |I$ is a homeomorphism of $\widehat{A}^I$ onto $\widehat{I}$.