2
$\begingroup$

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)?

$\endgroup$

1 Answer 1

1
$\begingroup$

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 $C(S^2)\otimes M_2(\mathbb{C})$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .