The above lemma is from Paulsen's book.
I have two questions:
1.The author set $A=\pi(a)$, we can view $\pi$ as a injective $*$-homomorphism, then $\|a\|=\|A\|\leq 1$, \begin{bmatrix} I& A\\ A^*& I\end{bmatrix} is positive , we only have \begin{bmatrix} I& \pi(a)\\ \pi(a)^*& I\end{bmatrix} is positive, how to prove that \begin{bmatrix} 1& a\\ a^*& 1\end{bmatrix} is positive?
2.If $\|A\|\geq 1$, how to conclude that there exist unit vectors $x$ and $y$ such that $\langle Ay,x \rangle <-1$
$M_2(A)$ is a unital subalgebra of $M_2(\mathbb B(\mathcal H))$, thus for any $x\in M_2(A)$, the spectrum is independent of the algebra, that is, $\sigma_{M_2(A)}(x)=\sigma_{M_2(\mathbb B(\mathcal H))}(x)$. Since $\begin{bmatrix} 1& a\\ a^*& 1\end{bmatrix}$ is self-adjoint, and $$\sigma_{M_2(A)}\left(\begin{bmatrix} 1& a\\ a^*& 1\end{bmatrix}\right)=\sigma_{M_2(\mathbb B(\mathcal H))}\left(\begin{bmatrix} I& \pi(a)\\ \pi(a)^*& I\end{bmatrix}\right)\subset[0,\infty),$$ it follows that $\begin{bmatrix} 1& a\\ a^*& 1\end{bmatrix}$ is positive.
For your second question, Since $\|A\|>1$, there exists $y\in\mathcal H$ with $\|y\|=1$ such that $\|Ay\|>1$. Letting $x_0=\frac{1}{\|Ay\|}Ay$, we have $$\langle Ay,x_0\rangle=\|Ay\|>1.$$ Now let $x=-x_0$. Then $$\langle Ay,x\rangle=-\langle Ay,x_0\rangle<-1.$$