limit of sequence of bounded operators

Question

Suppose $\phi_n:A\rightarrow B(H_n)$ is a sequence of nonzero representations, where $A$ is a nonunital $C^*$-algebra,$H_n$ is a Hilbert space, and $P_n$ is a sequence of projections on $H_n$. Does there exist $a_0$ such that $\{P_n\phi_n(a_0)\}$ is norm convergent?

Answer 1

This makes no sense as written. The elements $P_n\phi_n(a_0)$ are in different Hilbert spaces, if the $H_n$ are different.

If the Hilbert space is fixed, the answer is no unless the sequence is zero. Take $A=c_0$, $\phi_n(a)=a(1)\,I\in B(\ell^2(\mathbb N))$. Take $\{P_n\}$ to a sequence of pairwise orthogonal projections.

Then, if $a(1)=0$, we have $P_n\phi_n(a_0)=0$ for all $n$. But if $a(1)\ne0$, then $P_n\phi(a_0)=a(1)\,P_n$ is a sequence with no convergent subsequences.