  Consider the compact quantum group $U_q(2)$, where $q$ is a non-zero complex
deformation parameter such that $|q|\neq 1$. Let $C(U_q(2))$ denote the
underlying $C^*$-algebra of the compact quantum group $U_q(2)$. We prove that
if $q$ is a non-real complex number and $q^\prime$ is real, then the underlying
$C^*$-algebras $C(U_q(2))$ and $C(U_{q^\prime}(2))$ are non-isomorphic. This is
in sharp contrast with the case of braided $SU_q(2)$, introduced earlier by
Woronowicz et al., where $q$ is a non-zero complex deformation parameter. In
another direction, on a geometric aspect of $U_q(2)$, we introduce torus action
on the $C^*$-algebra $C(U_q(2))$ and obtain a $C^*$-dynamical system
$(C(U_q(2)),\mathbb{T}^3,\alpha)$. We construct a $\mathbb{T}^3$-equivariant
spectral triple for $U_q(2)$ that is even and $3^+$-summable. It is shown that
the Dirac operator is K-homologically nontrivial.
