I have a question regarding a remark in my lecture. We defined the Nullspace $\mathcal{N}_{\phi}$ of positive form $\phi$ on a $C^*$-algebra $\mathcal{A}$ as $\mathcal{N}_{\phi}=\{A\in\mathcal{A}\,|\,\phi(A^*A)=0\}$. This space is closed and a left ideal. It was mentioned that we can therefore consider the quotient space $\mathcal{A}/\mathcal{N}_{\phi}$. Why do we need these properties? (I thought about the equivalence relation, but I didn't see it)
Thanks for your help.
The fact that $\mathcal N_\phi$ is a left ideal is what allows you to show that it is a subspace. You don't need the language, though, as what one does is to use Cauchy-Schwarz to show that $$ \mathcal N_\phi=\{A:\ \phi(BA)=0\ \text{ for all } B\}. $$ This lets you show that $\mathcal N_\phi$ is a closed subspace, and hence the quotient space and the inner product are well-defined.