SOT-isomorphic C*-algebras

Question

Suppose that $A, B \subset B(\mathcal{H})$ are $C^*$-algebras.

Assume that $\{p_n\} \subset B(\mathcal{H})$ is a monotone sequence of projections such that:

1. $p_n \rightarrow 1$ in strong operator topology
2. $\forall n$ $p_n A p_n \simeq p_n B p_n$
3. Each $p_n A p_n$ is finite dimensional

Is it true that $A \simeq B$ ?

Answer 1

No.

For any such sequence $\{p_n\}$ with each $p_n$ of finite rank, take $A=B(\mathcal H)$ and $$ B=\overline{ \{b\in B(\mathcal H):\ \exists n,\ b=p_nbp_n\}}.$$

Then $p_nAp_n=p_nBp_n$ for all $n$, but $B$ is separable while $A$ is non-separable.