# Failure of Schur's lemma for topological group representations

Is there an example of $$G$$, $$\rho$$ as below?

• $$G$$ is a locally compact group.

• $$\rho$$ is a continuous representation of $$G$$ on a Hilbert space $$V$$. This means that we have a homomorphism from $$G$$ to the group of bounded linear operators on $$V$$ with bounded inverse, such that $$G \times V \rightarrow V$$ is continuous.

• Schur's lemma fails for $$\rho$$.