Searcharxiv⌕ Search

arXiv subjects

A. Yu. Pirkovskii

Publications and source records attributed to A. Yu. Pirkovskii.

18 recordsLinked to original sources
arXiv2024

Holomorphic functions on the quantum polydisk and on the quantum ball

We introduce and study noncommutative (or ``quantized'') versions of the algebras of holomorphic functions on the polydisk and on the ball in $\mathbb C^n$. Specifically, for each $q\in\mathbb C\setminus\{ 0\}$ we construct Fréchet algebras $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ such that for $q=1$ they are isomorphic to the algebras of holomorphic functions on the open polydisk $\mathbb D^n$ and on the open ball $\mathbb B^n$, respectively. In the case where $0<q<1$, we establish a relation between our holomorphic quantum ball algebra $\mathcal O_q(\mathbb B^n)$ and L. L. Vaksman's algebra $C_q(\bar{\mathbb B}^n)$ of continuous functions on the closed quantum ball. Finally, we show that $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ are not isomorphic provided that $|q|=1$ and $n\ge 2$. This result can be interpreted as a $q$-analog of Poincaré's theorem, which asserts that $\mathbb D^n$ and $\mathbb B^n$ are not biholomorphically equivalent unless $n=1$. This paper replaces the first part of Version 1: arXiv:1508.05768v1 [math.FA].

math.FA↗
arXiv2021

Dense quasi-free subalgebras of the Toeplitz algebra

We introduce a family of dense subalgebras of the Toeplitz algebra and give conditions under which our algebras are quasi-free. As a corollary, we show that the smooth Toeplitz algebra introduced by Cuntz is quasi-free.

math.OA↗
arXiv2021

Open embeddings and pseudoflat epimorphisms

We characterize open embeddings of Stein spaces and of $C^\infty$-manifolds in terms of certain flatness-type conditions on the respective homomorphisms of function algebras.

math.FA↗
arXiv2013

Quantum polydisk, quantum ball, and a q-analog of Poincaré's theorem

The classical Poincaré theorem (1907) asserts that the polydisk $\mathbb D^n$ and the ball $\mathbb B^n$ in $\mathbb C^n$ are not biholomorphically equivalent for $n\ge 2$. Equivalently, this means that the Fréchet algebras $\mathcal O(\mathbb D^n)$ and $\mathcal O(\mathbb B^n)$ of holomorphic functions are not topologically isomorphic. Our goal is to prove a noncommutative version of the above result. Given $q\in\mathbb C\setminus\{ 0\}$, we define two noncommutative power series algebras $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$, which can be viewed as $q$-analogs of $\mathcal O(\mathbb D^n)$ and $\mathcal O(\mathbb B^n)$, respectively. Both $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ are the completions of the algebraic quantum affine space $\mathcal O_q^{\mathrm{reg}}(\mathbb C^n)$ w.r.t. certain families of seminorms. In the case where $0<q<1$, the algebra $\mathcal O_q(\mathbb B^n)$ admits an equivalent definition related to L. L. Vaksman's algebra of continuous functions on the closed quantum ball. We show that both $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ can be interpreted as Fréchet algebra deformations (in a suitable sense) of $\mathcal O(\mathbb D^n)$ and $\mathcal O(\mathbb B^n)$, respectively. Our main result is that $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ are not isomorphic if $n\ge 2$ and $|q|=1$, but are isomorphic if $|q|\ne 1$.

math.FA↗
arXiv2013

Holomorphically finitely generated algebras

We introduce and study holomorphically finitely generated (HFG) Fréchet algebras, which are analytic counterparts of affine (i.e., finitely generated) $\mathbb C$-algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some natural constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions on the quantum affine space and on the quantum torus), the algebras of holomorphic functions on the free polydisk, on the quantum polydisk, and on the quantum polyannulus.

math.FA↗
arXiv2012

Noncommutative analogues of Stein spaces of finite embedding dimension

We introduce and study holomorphically finitely generated (HFG) Fréchet algebras, which are analytic counterparts of affine (i.e., finitely generated) $\mathbb C$-algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some standard constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions on the quantum affine space and on the quantum torus), the algebras of holomorphic functions on the free polydisk, on the quantum polydisk, and on the quantum ball. We further concentrate on the algebras of holomorphic functions on the quantum polydisk and on the quantum ball and show that they are isomorphic, in contrast to the classical case. Finally, we interpret our algebras as Fréchet algebra deformations of the classical algebras of holomorphic functions on the polydisk and on the ball in $\mathbb C^n$.

math.FA↗
arXiv2012

Homological dimensions of modules of holomorphic functions on submanifolds of Stein manifolds

Let X be a Stein manifold, and let Y be a closed complex submanifold of X. Denote by O(X) the algebra of holomorphic functions on X. We show that the weak (i.e., flat) homological dimension of O(Y) as a Fr'echet O(X)-module equals the codimension of Y in X. In the case where X and Y are of Liouville type, the same formula is proved for the projective homological dimension of O(Y) over O(X). On the other hand, we show that if X is of Liouville type and Y is hyperconvex, then the projective homological dimension of O(Y) over O(X) equals the dimension of X.

math.FA↗
arXiv2010

The Arens-Michael envelope of a smash product

Given a Hopf algebra H and an H-module algebra A, we explicitly describe the Arens-Michael envelope of the smash product A#H in terms of the Arens-Michael envelope of H and a certain completion of A. We also give an example (Manin's quantum plane) showing that the result fails for non-Hopf bialgebras.

math.FA↗
arXiv2009

Homological dimensions of smooth and complex analytic quantum tori

We survey some results on homological dimensions of the algebraic, complex analytic, and smooth quantum tori. Our main theorem states, in particular, that the smooth and the complex analytic quantum n-tori have global dimension n. This contrasts with the result of McConnell and Pettit (1988) who proved that, in the generic case, the algebraic quantum n-torus has global dimension 1. In this connection we also formulate some general theorems on homological dimensions of nuclear Fréchet algebras.

math.FA↗
arXiv2008

Homological dimensions of K"othe algebras

Given a metrizable K"othe algebra $λ(P)$, we compute the global dimension, the weak global dimension, the bidimension, and the weak bidimension of $λ(P)$ in terms of the K"othe set $P$.

math.FA↗
arXiv2006

Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities

Let A be a locally m-convex Fréchet algebra. We give a necessary and sufficient condition for a cyclic Fréchet A-module X=A_+/I to be strictly flat, generalizing thereby a criterion of Helemskii and Sheinberg. To this end, we introduce a notion of "locally bounded approximate identity" (a locally b.a.i. for short), and we show that X is strictly flat if and only if the ideal I has a right locally b.a.i. Next we apply this result to amenable algebras and show that a locally m-convex Fréchet algebra A is amenable if and only if A is isomorphic to a reduced inverse limit of amenable Banach algebras. We also extend a number of characterizations of amenability obtained by Johnson and by Helemskii and Sheinberg to the setting of locally m-convex Fréchet algebras. As a corollary, we show that Connes and Haagerup's theorem on amenable C^*-algebras and Sheinberg's theorem on amenable uniform algebras hold in the Fréchet algebra case. We also show that a quasinormable locally m-convex Fréchet algebra has a locally b.a.i. if and only if it has a b.a.i. On the other hand, we give an example of a commutative, locally m-convex Fréchet-Montel algebra which has a locally b.a.i., but does not have a b.a.i. Some of the results of this paper were announced in ArXiv preprint math.FA/0511132.

math.FA↗
arXiv2006

Strictly flat cyclic Fréchet modules and approximate identities

Let A be a locally m-convex Fréchet algebra. We give a necessary and sufficient condition for a cyclic Fréchet A-module X=A_+/I to be strictly flat, generalizing thereby a criterion of Helemskii and Sheinberg. To this end, we introduce a notion of locally bounded approximate identity (a.i.), and we show that X is strictly flat if and only if the ideal I has a right locally bounded a.i. An example is given of a commutative locally m-convex Fréchet algebra that has a locally bounded a.i., but does not have a bounded a.i. On the other hand, we show that a quasinormable locally m-convex Fréchet algebra has a locally bounded a.i. if and only if it has a bounded a.i. Some applications to amenable Fréchet algebras are also given.

math.FA↗
arXiv2006

Approximate characterizations of projectivity and injectivity for Banach modules

We characterize projective and injective Banach modules in approximate terms, generalizing thereby a characterization of contractible Banach algebras given by F. Ghahramani and R. J. Loy. As a corollary, we show that each uniformly approximately amenable Banach algebra is amenable. Some applications to homological dimensions of Banach modules and algebras are also given.

math.FA↗
arXiv2004

Stably flat completions of universal enveloping algebras

We study localizations (in the sense of J. L. Taylor) of the universal enveloping algebra, U(g), of a complex Lie algebra g. Specifically, let f : U(g) --> H be a homomorphism to some well-behaved topological Hopf algebra H. We formulate some conditions on the dual algebra, H', that are sufficient for H to be stably flat over U(g) (i.e., for f to be a localization). As an application, we prove that the Arens-Michael envelope of U(g) is stably flat over U(g) provided g admits a positive grading. We also show that Goodman's weighted completions of U(g) are stably flat over U(g) for each nilpotent Lie algebra g, and that Rashevskii's hyperenveloping algebra is stably flat over U(g) for arbitrary g. Finally, Litvinov's algebra A(G) of analytic functionals on the corresponding connected, simply connected complex Lie group G is shown to be stably flat over U(g) precisely when g is solvable.

math.FA↗
arXiv2004

Arens-Michael enveloping algebras and analytic smash products

Let g be a finite-dimensional complex Lie algebra, and let U(g) be its universal enveloping algebra. We prove that if \hat{U}(g), the Arens-Michael envelope of U(g), is stably flat over U(g) (i.e., if the canonical homomorphism U(g)-->\hat{U}(g) is a localization in the sense of Taylor), then g is solvable. To this end, given a cocommutative Hopf algebra H and an H-module algebra A, we explicitly describe the Arens-Michael envelope of the smash product A#H as an ``analytic smash product'' of their completions w.r.t. certain families of seminorms.

math.FA↗
arXiv2002

Biprojectivity and biflatness for convolution algebras of nuclear operators

For a locally compact group G, the convolution product on the space N(L^p(G)) of nuclear operators was defined by Neufang. We study homological properties of the convolution algebra N(L^p(G)) and relate them with some properties of the group G, such as compactness, finiteness, discreteness, and amenability.

math.FA↗