Let $A$ be a $C^*$ algebra, $\phi$ a positive linear functional on $A$. Put a pre-inner product on $A$ by $\langle x,y \rangle _{\phi} = \phi(y^*x)$. Let $$N_\phi := \{x \in A \, : \, \phi(x^*x) = 0 \}$$ then $N_\phi$ is a closed left ideal.
How is this so?
I think more fundamentally, I am really un familiar with the notion of positive elments of $C^*$ algebras. It would be nice if someone recommends me a reference for the notions behind this proof.
Let $x\in N_\phi$, $a\in A$. Then, using that $a^*a\leq \|a\|^2 I$ and that the map $b\longmapsto x^*bx$ preserves positivity, $$ \phi((ax)^*ax)=\phi(x^*a^*ax)\leq \|a\|^2\phi(x^*x)=0. $$
The basics of positive elements in C$^*$-algebras appear in every book on the topic (like Murphy, Davidson, Kadison-Ringrose, Blackadar, etc.), and others too (like Conway's Functional Analysis).