arXiv subjects

Anar Dosi

Publications and source records attributed to Anar Dosi.

8 recordsLinked to original sources

arXiv2025

Noncommutative localizations of a contractive quantum plane

In the present paper we investigate the localizations in the sense of J. L. Taylor of the Arens-Michael-Fréchet algebras associated with noncommutative analytic spaces of a contractive q-plane representing its formal geometry. It turns out that all noncommutative Fréchet algebras obtained by the Fréchet algebra structure sheaves over open subsets from the topology bases are indeed localizations. That topological homology property of the structure sheaves results in the key properties of Taylor and Putinar spectra of the left Banach q-modules over the algebras of global sections.

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arXiv2025

Homological framework of noncommutative complex analytic geometry and functional calculus

In the paper we propose topological homology framework of noncommutative complex analytic geometries of Fréchet algebras, and investigate the related functional calculus and spectral mapping properties. It turns out that an ideal analytic geometry of a Fréchet algebra A can be described in terms of a Čech category over A. The functional calculus problem within a particular Čech A-category, and a left Fréchet A-module X is solved in term of the homological spectrum of X with respect to that category. As an application, we use the formal q-geometry of a contractive operator q-plane, and solve the related noncommutative holomorphic functional calculus problem. The related spectrum is reduced to Putinar spectrum of a Fréchet q-module. In the case of a Banach q-module we come up with the closure of its Taylor spectrum.

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arXiv2024

Noncommutative complex analytic geometry of a contractive quantum plane

In the paper we investigate the Banach space representations of Manin's quantum q-plane for |q| is not 1. The Arens-Michael envelope of the quantum plane is extended up to a Frechet algebra presheaf over its spectrum. The obtained ringed space represents the geometry of the quantum plane as a union of two irreducible components being copies of the complex plane equipped with the q-topology and the disk topology, respectively. It turns out that the Frechet algebra presheaf is commutative modulo its Jacobson radical, which is decomposed into a topological direct sum. The related noncommutative functional calculus problem and the spectral mapping property are solved in terms of the noncommutative Harte spectrum.

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arXiv2024

Taylor spectrum of a Banach module over the quantum plane

In the paper we investigate the joint spectra of Banach space representations of the quantum q-plane called Banach q-modules. Based on the transversality relation from the topological homology of the trivial modules versus given a left Banach q-module, we introduce the joint (essential) spectra of a Banach q-module. In particular, we have the well defined Taylor joint spectrum of a Banach q-module. The noncommutative projection q-property is proved for the Taylor spectrum, which stands out the conventional projection property in the commutative case. It is provided the key examples of the Banach q-modules, which do not possesses nether forward nor backward projection properties.

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arXiv2023

Projective positivity of the function systems

The present paper is devoted to the projective positivity in the category of function systems, which plays a key role in the quantization problems of the operator systems. The main result of the paper asserts that every unital star-normed space can be equipped with the projective positivity. The geometry of the related state spaces is described in the case of Lp-spaces, Schatten matrix spaces, and Lp-spaces of a finite von Neumann algebra.

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arXiv2022

Deformation quantization of projective schemes and differential operators

The paper is devoted to noncommutative projective schemes within Kapranov's framework of noncommutative algebraic geometry. We classify all noncommutative projective schemes obtained from the differential chains in the universal enveloping algebra of the free nilpotent Lie algebra of index q generated by x_{0},...,x_{n}. The construction proposed allows us to provide a new method of deformation quantization of the commutative projective schemes within the considered framework.

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arXiv2021

Algebraic spectral theory and Serre multiplicity formula

The present paper is devoted to an algebraic treatment of the joint spectral theory within the framework of Noetherian modules over an algebra finite extension of an algebraically closed field. We prove the spectral mapping theorem and analyze the index of tuples in purely algebraic case. The index function over tuples from the coordinate ring of a variety is naturally extended up to a numerical Tor-polynomial. Based on Serre's multiplicity formula, we deduce that Tor-polynomial is just the Samuel polynomial of the local algebra.

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arXiv2019

Separable morphisms of operator Hilbert systems, Pietsch factorizations and entanglement breaking maps

In this paper we investigate operator Hilbert systems and their separable morphisms. We prove that the operator Hilbert space of Pisier is an operator system, which possesses the self-duality property. It is established a link between unital positive maps and Pietch factorizations, which allows us to describe all separable morphisms from an abelian C*-algebra to an operator Hilbert system. Finally, we prove a key property of entanglement breaking maps that involves operator Hilbert systems.

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