Convergence of a net in a Hilbert space

Question

Suppose $\phi:A\rightarrow B(H)$ is a nonzero $*$ homomorphism, where $A$ is a nonunital $C^*$ algebra, $H$ is a Hilbert space, $\{x_{i}\}$ is a net of unit vectors in $H$, does there exist $a_0\in A$ such that $\{\phi(a_0)x_{i}\}$ is norm convergent to a nonzero element?

Answer 1

No. For instance, suppose $(x_i)$ is a net which accumulates at every point in the unit sphere of $H$ (for instance, if $H$ is separable, you could just take a sequence formed by a countable dense subset). If $a\in B(H)$ is such that $(ax_i)$ converges to some vector $v$, then by continuity we must have $ax=v$ for all $x$ in the unit sphere. This is obviously impossible by linearity of $a$ unless $v=0$.