Nonformal deformations of the algebras of holomorphic functions on the polydisk and on the ball in ${\mathbb{C}}^n$

We construct Fréchet $\mathcal{O}({\mathbb{C}}^{\times })$-algebras ${\mathcal{O}}_{\mathrm{d}\mathrm{e}\mathrm{f}}({\mathbb{D}}^n)$ and ${\mathcal{O}}_{\mathrm{d}\mathrm{e}\mathrm{f}}({\mathbb{B}}^n)$ which may be interpreted as nonformal (or, more exactly, holomorphic) deformations of the algebras $\mathcal{O}({\mathbb{D}}^n)$ and $\mathcal{O}({\mathbb{B}}^n)$ of holomorphic functions on the polydisk ${\mathbb{D}}^n\subset {\mathbb{C}}^n$ and on the ball ${\mathbb{B}}^n\subset {\mathbb{C}}^n$, respectively. The fibers of our algebras over $q\in {\mathbb{C}}^{\times }$ are isomorphic to the previously introduced quantum polydisk'' and quantum ball’’ algebras, ${\mathcal{O}}_q({\mathbb{D}}^n)$ and ${\mathcal{O}}_q({\mathbb{B}}^n)$. We show that the algebras ${\mathcal{O}}_{\mathrm{d}\mathrm{e}\mathrm{f}}({\mathbb{D}}^n)$ and ${\mathcal{O}}_{\mathrm{d}\mathrm{e}\mathrm{f}}({\mathbb{B}}^n)$ yield continuous Fréchet algebra bundles over ${\mathbb{C}}^{\times }$ which are strict deformation quantizations (in Rieffel’s sense) of ${\mathbb{D}}^n$ and ${\mathbb{B}}^n$. We also give a noncommutative power series interpretation of ${\mathcal{O}}_{\mathrm{d}\mathrm{e}\mathrm{f}}({\mathbb{D}}^n)$ and apply it to showing that ${\mathcal{O}}_{\mathrm{d}\mathrm{e}\mathrm{f}}({\mathbb{D}}^n)$ is not topologically projective (and a fortiori is not topologically free) over $\mathcal{O}({\mathbb{C}}^{\times })$. Finally, we consider respective formal deformations of $\mathcal{O}({\mathbb{D}}^n)$ and $\mathcal{O}({\mathbb{B}}^n)$, and we show that they can be obtained from the holomorphic deformations by extension of scalars.