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?

  • $\begingroup$ Of course. Take $a_0=0$. $\endgroup$ – Martin Argerami Jul 27 '18 at 21:49
  • $\begingroup$ Actually,I want to get the conclusion that $\{\phi(a_0)x_{i}\}$ is norm convergent to a nonzero element. $\endgroup$ – math112358 Jul 28 '18 at 6:26
  • $\begingroup$ I have reedited the question. $\endgroup$ – math112358 Jul 28 '18 at 18:13
  • $\begingroup$ I suppose taking some point of an approximate unit might get you somewhere, with some non-degeneracy added to your homomorphiam? $\endgroup$ – Munk Jul 28 '18 at 22:54

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 $v=0$.


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