Hilbert space on which $\mathcal A$ acts as a von Neumann algebra, then a selfadjoint $x\in\mathcal A$ is greater than or equal to zero if and only if $\langle x\xi,\xi\rangle\ge0$ for all $\xi\in\mathcal H$. We say that $x>0$ means that $x\ge0$ and $x$ is invertible (i.e. its spectrum is included in $(0,+\infty)$). We say $x\ge y$ if $x-y\ge0$, and similarly for “$>$.” In particular, it follows that $xx^*\leq\|x\|^2\cdot1_\mathcal A$ and $x^*x\leq\|x\|^2\cdot1_\mathcal A$. Clearly, for and $\varepsilon\in(0,+\infty)$, $xx^*<(\|x\|^2+\varepsilon)\cdot1_\mathcal A$, with strict inequality achieved only when $\varepsilon>0$.

As proved in [@Paulsen Lemma 3.1], $$\left(\begin{array}{cc}
1 & a\\
a^* & 1
\end{array}\right)\ge0\text{ in }M_2(\mathcal A)\iff \|a\|\leq1.$$ We claim that $$\left(\begin{array}{cc}
1 & a\\
a^* & 1
\end{array}\right)>0 \text{ in }M_2(\mathcal A)\iff \|a\|<1.$$ Indeed, if $\|a\|<1$, then $1-aa^*$ and $1-a^*a$ are invertible in $\mathcal A$ and $$\left(\begin{array}{cc}
1 & a\\
a^* & 1
\end{array}\right)^{-1}=\left(\begin{array}{cc}
(1-aa^*)^{-1} & -(1-aa^*)^{-1}a\\
-a^*(1-aa^*)^{-1} & (1-a^*a)^{-1}
\end{array}\right)\ \text{in }M_2(\mathcal A).$$ Conversely, if $\|a\|=1$, then for any $\varepsilon>0$ there exists $\xi_\varepsilon,\eta_\varepsilon\in\mathcal H$ of norm one such that $\langle a\eta_\varepsilon,\xi_\varepsilon\rangle>
1-\varepsilon.$ Then $$\left|\left\langle
\left(\begin{array}{cc}
1 & a\\
a^* & 1
\end{array}\right)\left[\begin{array}{c}
\xi_\varepsilon\\
\eta_\varepsilon
\end{array}\right],\left[\begin{array}{c}
\xi_\varepsilon\\
\eta_\varepsilon
\end{array}\right]
\right\rangle\right|=\|\xi_\varepsilon\|^2_2+\|\eta_\varepsilon\|^2_2
-2\langle a\eta_\varepsilon,\xi_\varepsilon\rangle<2\varepsilon,$$ so that zero belongs to the spectrum of $\left(\begin{array}{cc}
1 & a\\
