This is part of an exercise problem (5.I) in Wegge-Olsen's book "K-theory and $C^*$-algebras".
There he defines the Bott projection for $\mathbb{R}^2$ by $B:\mathbb{R}^2\rightarrow\mathbb{M}_2$, $B(z)=B(x,y)=\frac{1}{1+|z|^2}\begin{pmatrix} |z|^2 & z \\ \bar{z} & 1 \end{pmatrix}$, $z=x+iy$. Then he writes that the Bott projection is constant in infinity in the sense that $B(z)\rightarrow\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ for $|z|\rightarrow\infty$; and that $B(0)=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$. Thus the Bott projection may be considered a function in the unitization of $S^{2} \mathbb{M}_{2}$, where $S^2$ denotes double suspension.
It looks like the statements about $B(\infty)$ and $B(0)$ are wrong (if someone can confirm this). I also don't quite understand the last statement. Hope that someone can explain it to me.
Next, he defines a map $u_0:\mathbb{R}^2\rightarrow\mathbb{M}_2$ by $u_0(x,y)=\frac{1}{\sqrt{1+|x|^2+|y|^2}}\begin{pmatrix} x+iy & -1 \\ 1 & x-iy \end{pmatrix}$, and asks to prove that $u_0$ is unitary, and that $B=u_0\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}u_0^*$.
Can someone please help me with the part about it being unitary?
Since the value of $B(\infty)$ is not a multiple of the identity matrix, the function $B$ does not seem to belong to the unitization of $S^2\mathbb M_2$ as claimed, but rather to $\mathbb M_2$ of the unitization of $S^2\mathbb C$.