Algebraic vs. continuous representations of topological groups

Question

Let $G$ be a locally compact (Hausdorff) topological group and $\rho:G\to \operatorname{GL}(V)$ a complex continuous representation of $G$ in some locally convex topological vector space $V$. I'm looking for examples of $\rho$ such that

1. $\rho$ is topologically irreducible but not algebraically irreducible. That is, the unique complemented $G$-invariant subspaces of $V$ are $V$ and $\{0\}$, but there is a proper non-trivial $G$-invariant subspace of $V$ (that is obviously not complemented).
2. $\rho$ is topologically idecomposable but not algebraically idecomposable. That is, $V$ cannot be written as a topological direct sum of non-trivial $G$-invariant subspaces, but it can be written as an algebraic direct sum of non-trivial $G$-invariant subspaces.

If possible, I'd love to know such examples in the case where $V$ is a Banach o Hilbert space. I didn't find such examples in the literature. I would also love to know criteria for determining when an algebraic notion (irreducibility, idecomposability) implies its topological counterpart.