Let $G$ be a compact group and $u: G \to B(H)$ be a strongly continuous unitary representation on the Hilbert space $H$. Then is $u: G \to B(H)$ strictly continuous?
That is, give $B(H)$ the topology induced by the $*$-isomorphism $M(B_0(H))\cong B(H)$. Explicitely, a net $(x_i)$ in $B(H)$ converges strictly to $x$ iff $\|x_i y -x y\|\to 0$ and $\|yx_i \to yx\| \to 0$ for all compact operators $y \in B_0(H)$. On bounded subsets this agrees with the $*$-strong topology.