Let $\mathcal{A}$ be a C*-algebra and $\phi$ a state on $\mathcal{A}$. Then, it's not hard to see that $\mathcal{L} = \left\{ x : \phi(x^*x)=0 \right\}$ is a closed left ideal of $A$ and so $\mathcal{B}=\mathcal{L} \cap \mathcal{L^*}$ is a hereditary subalgebra of $\mathcal{A}$.
Can you help me to identify $\mathcal{B}$ in terms of $\phi$?
Just write down what $\mathcal L \cap \mathcal L^*$ is. You have $$ \mathcal L^* = \{x: \phi(xx^*)=0 \}, $$ as $x \in \mathcal L^*$ if and only if $x^* \in \mathcal L$ (and remembering that $(x^*)^*=x$). Then $$ \mathcal B = \{x: \phi(xx^*)=0 = \phi(x^*x) \}. $$