Lemma 41.5.2.label Let $A$ be a $C^{*}$-algebra and $x, y \in A$, then
\[\begin{bmatrix}0&x \\ x^{*}&y\end{bmatrix} \ge 0\]
if and only if $x = 0$ and $y \ge 0$.
Proof. ($\Rightarrow$): Using the Gelfand-Naimark Theorem, assume without loss of generality that there exists a complex Hilbert space $H$ such that $A \subset B(H)$ is a $C^{*}$-subalgebra. In which case, for any $\xi, \eta \in H$,
\begin{align*}\angles{\begin{bmatrix}0&x \\ x^{*}&y\end{bmatrix}\begin{bmatrix}\xi \\ \eta\end{bmatrix}, \begin{bmatrix}\xi \\ \eta\end{bmatrix}}_{H^2}&= \angles{\begin{bmatrix}x\eta \\ x^{*}\xi + y\eta\end{bmatrix}, \begin{bmatrix}\xi \\ \eta\end{bmatrix}}_{H^2}\\&= \dpn{x^*\xi, \eta}{H}+ \dpn{x\eta, \xi}{H}+ \dpn{y\eta, \eta}{H}\\&= 2\text{Re}(\dpn{x^*\eta, \xi}{H}) + \dpn{y\eta, \eta}{H}\end{align*}
If $x \ne 0$, then there exists $\xi_{0}, \eta_{0} \in H$ such that $\dpn{x^*\eta_0, \xi_0}{H}\ne 0$. Therefore
\[\inf_{\xi \in H}\text{Re}(\dpn{x^*\eta_0, \xi}{H}) = -\infty\]
and the given matrix is not positive.$\square$
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