Quotient representation

A quotient representation of a representation $\pi$ of a group (cf. Representation of a group), or algebra, $X$ is a representation $\rho$ of $X$ defined as follows. Let $E$ be the (topological) vector space of the representation $\pi$; then $\rho$ is a representation in a (topological) vector space $E/F$ that is the quotient space of $E$ by some invariant subspace $F$ of $\pi$ (cf. Invariant subspace of a representation), defined by the formula $\rho(x)(\xi+F)=\pi(x)\xi+F$ for all $x\in X$, $\xi\in E$. If $\pi$ is a continuous representation, then so is any quotient representation of it.