Let $V\subset B(H,K)$ be closed subspace. Let $A$ be $C^{\ast}$-algebra generated by $VV^*$.
Is it true that $A$ must have approximate identity of the form $e_{\lambda}f_{\lambda}^*$ for $e_{\lambda} \in V$ and $f_{\lambda}^* \in V^*$
It seems it should not be true but I can’t really see the counterexample. Any ideas?
Its not true. Let $V\subseteq C([0,1])$ be the one-dimensional subspace spanned by the function $\sqrt x$, so $VV^*$ is the span of the function $x$ in $C([0,1])$. The $C^*$ algebra generated by this element is isomorphic to $C(\,(0,1]\,)$ (eg by Stone-Weierstraß), but $VV^*$ itself does not contain an approximate unit of the algebra ($VV^*$ is one-dimensional).
This scenario is slightly different from your formulation, but just represent $C([0,1])$ faithfully into $B(H)$ for some Hilbert space $H$ so you may view $V\subseteq B(H,H)$.