Question regarding $C^*$ algebra generated by a product

Question

Let $V=M_{1,\infty}$,the row Hilbert space. Suppose $W$ denotes the $C^*-$ algebra generated by $V^*V=\{x^*y : x,y \in V \}$

Is it true that $W= K(l_2)$, space of compact operators on $l_2$?

I can see that $V^*V$ is nothing but $M_\infty$ but I cannot see the exact claim. Please help.

Answer 1

Since the row Hilbert operator space is $\{e_{1j}:\ j\in\mathbb N\}\subset B(\ell^2(\mathbb N))$, your C$^*$-algebra $W$ is generated by the matrix units $e_{kj}=e_{1k}^*e_{1j}$. So $W=K(\ell^2(\mathbb N))$, the compact operators.