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.

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 $A_q$?

Fix a real number $0<q<1$. We consider the standard Podles 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.

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 $\lVert a\rVert$ and $\lVert b\rVert$ in $A_q$?

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.

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 $A_q$?

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.

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 $A_q$?