Continuous functions on the quantum ball and $C^*$-algebra bundles

I will introduce a family of $C^*$-algebras depending on a parameter $q$, where $0<q<1$. These algebras were introduced by L. L. Vaksman as quantum analogues of the algebra of continuous functions on the $n$-dimensional ball $\mathbb B^n$. I will concentrate on the one-dimensional case in which one can better see some similarities between the quantized algebras and the usual algebra $C(\mathbb D)$ of continuous functions on the disk. It is natural to expect the algebras of continuous functions on the quantum ball to be a continuous deformation of $C(\mathbb B^n)$. I will explain a way to formalize this via continuous $C^*$-algebra bundles and construct an upper semicontinuous $C^*$-algebra bundle with Vaksman algebras as fibers over $0<q<1$ and $C(\mathbb D)$ over $1$. I will also talk about my attempts to prove its continuity.