Involution is not strongly continuous Let $H$ be a Hilbert space and consider on $B(H)$ the strong topology, i.e. the topology induced by the seminorms
$$x \mapsto \Vert x \xi \Vert, \quad  \xi \in H$$
I.e. a net $(x_\alpha)_\alpha$ converges to $x$, i.e. $x_\alpha \to x$ if and only if 
$$\forall \xi \in H: \quad \Vert (x_\alpha-x)\xi \Vert \to 0$$
I'm trying to show that $*: B(H) \to B(H): x \mapsto x^*$ is NOT strongly continuous.
I try to find a sequence $x_n$ in some $B(H)$ that converges strongly to $0$ but such that $x_n^*$ does not converge strongly to $0$, but I don't find one.

The typical example is the unilateral shift $S$. Since it is an isometry, all its powers are isometries. But the powers of its adjoint converge sot to zero. 
So one sequence that works is $\{S^{*n}\}$. You have, for any $x$, that $\|S^{*n}x\|\to0$, while $\|S^nx\|=\|x\|$ for all $n$. 
