Nonformal deformations of the algebras of holomorphic functions on the polydisk and on the ball in C^n

Journal of Mathematical Analysis and Applications. 2025. Vol. 547. No. 2. Article 129371.

Pirkovskii A. Y.

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

Keywords: quantum polydisk, quantum ball, квантовый полидиск, квантовый шар, алгебра Фреше, Frechet algebra, nonformal deformation, неформальная деформация