Describing the algebra of functions on $S^2$

Question

Chapter 2 of the book "Elements of Noncommutative Geometry" claims that the $C^*$-algebra of functions on $S^2$ can be described as an algebra with 3 generators a,b,c all with norm 1, where $a,b$ are positive and $c^*c = 4ab.$ However it never explicitly tells you which functions on the sphere these are. What are they?

Answer 1

They are the ($\ell=1$)-spherical harmonics; some help is provided in the reference just after the statement (which is in German):

$$ a=-\sqrt{\frac{8 \pi}{3}}Y_{1,1} = \mathrm{e}^{\mathrm{i}\phi}\sin \theta, \,\,b=\sqrt{\frac{8 \pi}{3}}Y_{1,-1} = \mathrm{e}^{\mathrm{-i}\phi}\sin \theta,\,\, c:=c_+-c_-=\cos \theta=\sqrt{\frac{4\pi}{3}}Y_{1,0}. $$ where $c_-+c_+=1$. The relation is actually $ab=4c_-c_+$ for this choice of normalization (which can be redefined).