Nonformal deformations of the algebras of holomorphic functions on the polydisk and on the ball in Cn

We construct Fr & eacute;chet a(Cx)-algebras adef(Dn) and adef(Bn) which may be interpreted as nonformal (or, more exactly, holomorphic) deformations of the algebras a(Dn) and a(Bn) of holomorphic functions on the polydisk Dn subset of Cn and on the ball Bn subset of Cn, respectively. The fibers of our algebras over q is an element of Cx are isomorphic to the previously introduced "quantum polydisk" and "quantum ball" algebras, aq(Dn) and aq(Bn). We show that the algebras adef(Dn) and adef(Bn) yield continuous Fr & eacute;chet algebra bundles over Cx which are strict deformation quantizations (in Rieffel's sense) of Dnand Bn. We also give a noncommutative power series interpretation of adef(Dn) and apply it to showing that adef(Dn) is not topologically projective (and a fortiori is not topologically free) over a(Cx). Finally, we consider respective formal deformations of a(Dn) and a(Bn), and we show that they can be obtained from the holomorphic deformations by extension of scalars. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.