Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}\_{q}}(2) $ is defined to be the universal unital $ C^{\*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the following five relations:
\begin{align}
\alpha^{\*} \alpha + \beta^{\*} \beta & = 1, \\
\alpha \alpha^{\*} + q^{2} \beta \beta^{\*} & = 1, \\
\beta^{\*} \beta & = \beta \beta^{\*}, \\
\alpha \beta & = q \beta \alpha, \\
\alpha \beta^{\*} & = q \beta^{\*} \alpha.
\end{align}
In his paper [*Twisted $ \text{SU}(2) $ Group. An Example of a Non-Commutative Differential Calculus*](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB8QFjAA&url=https%3A%2F%2Fwww.ems-ph.org%2Fjournals%2Fshow_pdf.php%3Fissn%3D0034-5318%26vol%3D23%26iss%3D1%26rank%3D5&ei=NZWTVcOoBM71oASw_ILoCg&usg=AFQjCNE7msZDaxPfyjZWHn7ajMPtH6ILag&sig2=KWZzMsTHNwfzdhb5qtZ_qQ&bvm=bv.96952980,d.cGU), Woronowicz proves that there is a $ \* $-isomorphism between $ {\text{SU}\_{q}}(2) $ and $ {\text{SU}\_{0}}(2) $. Part of his proof proceeds as follows. He begins with the claim that
\begin{align}
 \sigma(\alpha^{\*} \alpha)
& \subseteq \left\{ 0,1 - q^{2},1 - q^{4},1 - q^{6},\ldots,1 \right\} ~ \text{and} \\
 \sigma(\beta^{\*} \beta)
& \subseteq \left\{ 0,\ldots,q^{6},q^{4},q^{2},1 \right\}.
\end{align}
Next, he chooses arbitrary continuous functions $ f,g: [0,1] \to \Bbb{R} $ such that:


* $ f(0) = 0 $ and $ f(t) = 1 $ for all $ t \in \left[ 1 - q^{2},1 \right] $.
* $ g(1) = 1 $ and $ g(t) = 0 $ for all $ t \in \left[ 0,q^{2} \right] $.


Letting $ a = \alpha ~ f(\alpha^{\*} \alpha) $ and $ b = \beta ~ g(\beta^{\*} \beta) $, he then says that $ (a,b) $ is a generating pair for $ {\text{SU}\_{0}}(2) $, i.e.,
\begin{align}
a^{\*} a + b^{\*} b & = 1, \\
a a^{\*} & = 1, \\
b^{\*} b & = b b^{\*}, \\
a b & = 0, \\
a b^{\*} & = 0.
\end{align}


The problem is, none of this seems to work. For example, let us try to verify that $ a a^{\*} = 1 $. Observe that
\begin{align}
 a a^{\*}
& = [\alpha ~ f(\alpha^{\*} \alpha)] [\alpha ~ f(\alpha^{\*} \alpha)]^{\*} \\
& = [\alpha ~ f(\alpha^{\*} \alpha)] [f(\alpha^{\*} \alpha)^{\*} ~ \alpha^{\*}] \\
& = [\alpha ~ f(\alpha^{\*} \alpha)]
 \left[ \overline{f}(\alpha^{\*} \alpha) ~ \alpha^{\*} \right] \qquad
 (\text{By the continuous functional calculus.}) \\
& = [\alpha ~ f(\alpha^{\*} \alpha)] [f(\alpha^{\*} \alpha) ~ \alpha^{\*}] \qquad
 \left( \text{As $ \overline{f} = f $.} \right) \\
& = [f(\alpha \alpha^{\*}) ~ \alpha] [\alpha^{\*} ~ f(\alpha \alpha^{\*})] \qquad
 (\text{As $ \alpha ~ p(\alpha^{\*} \alpha) = p(\alpha \alpha^{\*}) ~ \alpha $ for every
 polynomial $ p $.}) \\
& = [f(\alpha \alpha^{\*})] (\alpha \alpha^{\*}) [f(\alpha \alpha^{\*})] \\
& = h(\alpha \alpha^{\*}),
\end{align}
where $ h: [0,1] \to \Bbb{R} $ is defined by $ h(t) \stackrel{\text{df}}{=} t [f(t)]^{2} $ for all $ t \in [0,1] $. However,
$$
\sigma(\alpha \alpha^{\*}) \cup \{ 0 \} = \sigma(\alpha^{\*} \alpha) \cup \{ 0 \},
$$
so $ h|\_{\sigma(\alpha \alpha^{\*})} $ is the identity function on $ \sigma(\alpha \alpha^{\*}) $, which yields $ h(\alpha \alpha^{\*}) = \alpha \alpha^{\*} $ instead of $ h(\alpha \alpha^{\*}) = 1 $.



> 
> **Question.** For those who are familiar with this topic, am I hallucinating, or is something not right here?
> 
> 
> 


**Note:** Woronowicz’s argument is given on Page 179, in Section A2, of his paper.



A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $
As Jan suggests, there's a typo in the definition of $f$. It should read
$$
f(t) = \begin{cases}\frac{1}{\sqrt{t}} & (t \ge 1 - q^2)\\ 0 & (t = 0).\end{cases}
$$
Since $\alpha \alpha^\* \ge 1-q^2$ (this follows from usual realization as in p. 123), this will imply that $a^\*$ is an isometry. And $\alpha^\* \alpha + \beta^\* \beta = 1$ implies that the eigenspace of $\beta^\* \beta$ for the eigenvalue $1$ is precisely the kernel of $\alpha$, which implies $a^\* a + b^\* b = 1$.


Morally, at $q=0$ you have the Toeplitz algebra as the 'quantum limit' of the deformation quantization of the $2$-dimensional symplectic leaves
$$
\left\{\begin{pmatrix}\bar{z} \lambda & \sqrt{1-|z|^2}\lambda\\-\sqrt{1-|z|^2}\bar{\lambda}&z\bar{\lambda}\end{pmatrix} \mid |z| < 1 \right\} \quad (|\lambda|=1)
$$ 
inside $SU(2)$ together with boundary $U(1)$, and $C(SU\_q(2))$ is an elaborate patching of these. The $q$-disc algebra generated by an operator $Z\_q$ satisfying $\|Z\_q\|=1$ and $1-Z\_q^\* Z\_q = q^2 (1-Z\_qZ\_q^\*)$ interpolates the Toeplitz algebra and the usual function algebra on the closed disc, but these algebras are in fact isomorphic to each other for $|q|<1$.


