Fix a real number $0<q<1$. We consider the standard Podle\'s sphere $A_q$ as the universal unit $C^*$-algebra generated by $a$ and $b$ with relations
\begin{equation*}
\begin{split}
&a=a^*,~ ab=q^2ba, ~ab^*=q^{-2}b^*a,\\
&bb^*=q^{-2}a(1-a), ~b^*b=a(1-q^2a).
\end{split}
\end{equation*}
See for example Section 3.1 [Geometry of Quantum Spheres][1].

It seems to me that $A_q$ is only defined by the universal property. Although certain properties of $A_q$ can by obtained, I know very little about its elements. I even have the following basic question:

>>Can we compute the norms $||a||$ and $||b||$ in $A_q$?


  [1]: https://arxiv.org/abs/math/0501240