$*-$ isomorphism between two compact spaces $K(H)$ and $K(H')$

Question

The following is a theorem of Murphy's C*-algebras and operator theory:

I think we can write the proof more easily than Murphy's. After show that $E'$ is an orthonormal basis for $H'$, define unitary linear map $u:H\to H'$ such that $u(e)= \bar e$. then for $e\in E$$$Adu(p_e) = up_eu^*=q_e=\phi(p_e)$$ From it $Adu$ and $\phi$ are equal at rank one projection $\xi\otimes \xi $, for all $\xi \in(H)_{\|.\|=1}$. Also we know that $F(H)$(the space of finite rank operators ) is generated by the rank one projections, so $\phi$ and $Adu$ are equal on $K(H)$.

Is my claim correct? Thanks.

Answer 1

We know that $Adu(x\otimes x)=\phi(x\otimes x)$ for $x\in E$, but we do not know that for $x$ being from the linear span of $E$, since $\{y\otimes y, y\in span(E)\}\not\subset \overline{span}(x\otimes x, x\in E)$. For instance, we do not know that for $x=e_1+e_2$, so we cannot get all rank one projections at this point.