Algebra of matrix valued functions on the sphere I was watching a video of a lecture by Alain Connes, and at around 8:00 he very briefly mentions a way to think about the algebra of 2x2 matrix functions over the 2-sphere, i.e. the maps from $S^2 \rightarrow M_2(\mathbb{C})$.
First he forms the monomials $m_1 Ym_2Y\dots m_n Y$ where $Y$ is some element with $Y^2=1$, and the $m_i$ have the interpretation as constant matrices over the sphere. Then he claims that algebra generated by these monomials is isomorphic to the 2x2 matrix valued functions over the sphere.
I am wondering how this claim actually works. Is $Y$ itself interpreted as some matrix valued function? How do you see the isomorphism?
If this is too difficult to explain in an answer does anyone know of a reference where this is shown (it is not shown in the paper he is discussing as far as I can tell)?
 A: I did end up finding and deciphering a paper that discussed this. The statement I wanted to understand is that the algebra $\mathcal{A}$ generated by $Y$ and $M_2(\mathbb{C})$ is isomorphic to $C(S^2)\otimes M_2(\mathbb{C})$.
We can decompose $Y$ as $Y=\sum \varepsilon_{ij} Y^{ij}$, where 
$\varepsilon_{ij}$ is the usual basis for $M_2(\mathbb{C})\subset \mathcal{A}$, and $Y^{ij}\in \mathcal{A}$ are elements that commute with all $M_2(\mathbb{C})$.
Some extra conditions on $Y$ are that $Y^\star=Y$ is hermitian and $Y$ satisfies a trace condition $$\langle Y\rangle\equiv\sum_k Y^{kk}=0.$$
Then we can write $Y$ in matrix form
$$
Y = \left(\begin{array}{cc}
  a     & b^\star\\
  b &  -a \end{array}\right),
$$
where $a^\star=a$ is hermitian.
Multiplying this matrix by itself and using the condition $Y^2=1$, we find that $a,b,b^\star$ all commute with each other and
$$
a^2 + b^\star b = 1.
$$
Since the algebra generated by $a,b,b^\star$ is commutative it is isomorphic to some $C(X)$, and by the above condition $X$ must be some compact subset of $S^2$.
In fact we can represent the generators of the algebra by spherical harmonics, so it is indeed isomorphic to $C(S^2)$
$$
a\cong \cos \theta,\quad b\cong e^{i\phi}\sin \theta.
$$
This shows that the algebra generated by $Y^{ij}$ is isomorphic to functions on a sphere. We can recover the elements $Y^{ij}$ in the algebra $\mathcal{A}$, e.g.
$$
a= \varepsilon_{11}Y\varepsilon_{11}-\varepsilon_{22}Y\varepsilon_{22},
$$
so the whole algebra does indeed end up being isomorphic to $C(S^2)\otimes M_2(\mathbb{C})$.
