Arens-Michael envelopes, homological epimorphisms, and relatively quasifree algebras

One of the leading ideas of noncommutative geometry is to consider noncommutative algebras as algebras of functions of noncommutative spaces. This principle is based on the observation that many important classical geometric objects and results, translated in the language of commutative algebra, remain valid in the noncommutative setup. On the list of areas of geometry with noncommutative analogues, there is an object whose noncommutative version is still not sufficiently well developed, namely, complex analytic geometry. It seems that one of the problems is that it is not clear which are the noncommutative analogues of the algebras of holomorphic functions on complex varieties. The Arens-Michael envelope of an associative algebra is a reasonable candidate to serve for this purpose. It is the completion with respect to all submultiplicative seminorms. For instance, the Arens-Michael envelope of the polynomial algebra is the algebra of entire functions. The paper under review studies Arens-Michael envelopes and their homological properties. It introduces analytic analogues of some classical ring-theoretic constructs: Ore extensions, Laurent extensions, and tensor algebras. For some finitely generated algebras, the author explicitly describes their Arens-Michael envelopes as algebras of noncommutative power series, and shows that the embeddings of such algebras in their Arens-Michael envelopes are homological epimorphisms. For that purpose he introduces and studies the concepts of relative homological epimorphism and relatively quasi-free algebra. The above results hold for multiparameter quantum affine spaces and quantum tori, quantum Weyl algebras, algebras of quantum 2times 2 matrices, and universal enveloping algebras of some Lie algebras of small dimensions.