Searcharxiv⌕ Search

arXiv subjects

Petr Ivankov

Publications and source records attributed to Petr Ivankov.

15 recordsLinked to original sources
arXiv2025

Algebraic topology of $C^*$-algebras

Any $C^*$-algebra can be regarded as a generalization of locally compact, Hausdorff topological space $\mathcal X$. From the commutative commutative Gelfand-Na\u{\i}mark theorem it follows that the spectrum of any commutative $C^*$-algebra is a locally compact, Hausdorff space which have the exact information of the $C^*$-algebra. Here we consider a Gelfand spaces of $C^*$-algebras which can be regarded as a generalization of the spectrum. In case of commutative $C^*$-algebras the Gelfand space coincides with the spectrum. Generally Gelfand spaces are not Hausdorff and provide more detailed information of noncommutative $C^*$-algebras. Usage of Gelfand spaces of $C^*$-algebra enables us to define some $C^*$-algebraic analogs of several notions of the classical algebraic topology, e.g. cohomology, fundamental group, multiplicative structure of $K$-theory and Adams operations

math.OA↗
arXiv2019

Noncommutative Geometry of Quantized Coverings

There are theories of coverings of $C^*$-algebras which can be included into a following list: coverings of commutative $C^*$-algebras, coverings of $C^*$-algebras of groupoids and foliations, coverings of noncommutative tori, the double covering of the quantum group $SO_q(3)$. This work is devoted to a single general theory which includes all theories of this list, i.e. we develop a system of axioms which can be applied for every element of the list. Otherwise since topological coverings are related to the set of geometric constructions one can obtain noncommutative generalizations of these constructions. Here the generalizations of the universal covering space, fundamental group, Hurewicz homomorphism, covering of the Riemannian manifold, flat connection are explained. The theory gives pure algebraic proof well known results of the topology and the differential geometry. Besides there are applications of the theory to (unbounded) operator spaces and this theme is also discussed here.

math.OA↗
arXiv2018

Coverings of Spectral Triples

It is well-known that any covering space of a Riemannian manifold has the natural structure of a Riemannian manifold. This article contains a noncommutative generalization of this fact. Since any Riemannian manifold with a Spin-structure defines a spectral triple, the spectral triple can be regarded as a noncommutative Spin-manifold. Similarly there is an algebraic construction which is a noncommutative generalization of topological covering. This article contains a construction of spectral triple on the "noncommutative covering space".

math.OA↗
arXiv2018

Quantization of noncompact coverings

The concept of quantization consists in replacing commutative quantities by noncommutative ones. In mathematical language an algebra of continuous functions on a locally compact topological space is replaced with a noncommutative $C^*$-algebra. Some classical topological notions have noncommutative generalizations. This article is concerned with a generalization of coverings.

math.OA↗
arXiv2017

Unoriented Spectral Triples

Any oriented Riemannian manifold with a Spin-structure defines a spectral triple, so the spectral triple can be regarded as a noncommutative Spin-manifold. Otherwise for any unoriented Riemannian manifold there is the two-fold covering by oriented Riemannian manifold. Moreover there are noncommutative generalizations of finite-fold coverings. This circumstances yield a notion of unoriented spectral triple which is covered by oriented one.

math.OA↗
arXiv2017

Coverings of foliation algebras

This article is devoted to the geometric construction which states a natural correspondence between topological coverings of a foliated manifolds and noncommutative coverings of the operator algebras. However this correspondence is not one to one because there are noncommutaive coverings of foliations which do not comply with discussed in this article construction.

math.OA↗
arXiv2016

Cyclic noncommutative covering projections

The Gelfand - Naĭmark theorem supplies the one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants can be defined by algebraical methods. This article contains a pure algebraical construction of (noncommutative) covering projections with finite cyclic groups of covering transformations.

math.OA↗
arXiv2015

Inverse Limits of Spectral Triples

Gelfand - Naĭmark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Similarly a spectral triple is a generalization of a Riemannian manifold. An (infinitely listed) covering of a Riemannian manifold has natural structure of Riemannian manifold. Here we will consider the noncommutative generalization of this result.

math.OA↗
arXiv2014

Infinite Noncommutative Covering Projections

Gelfand - Naĭmark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants may be defined by algebraic methods. For example Serre Swan theorem states that complex topological $K$-theory coincides with $K$-theory of $C^*$-algebras. This article devoted to the noncommutative generalization of infinite covering projections. Infinite covering projections of spectral triples are also discussed. It is shown that covering projection of foliation algebras can be constructed by topological coverings of foliations and isospectral deformations. Described an interrelationship between noncommutative covering projections and $K$-homology. The Dixmier trace of noncommutative covering projections is discussed.

math.OA↗
arXiv2014

The Unique Path Lifting for Noncommutative Covering Projections

This article contains a noncommutative generalization of the topological path lifting problem. Noncommutative geometry has no paths and even points. However there are paths of *-automorphisms. It is proven that paths of *-automorphisms comply with unique path lifting.

math.OA↗
arXiv2014

Noncommutative Generalization of Wilson Lines

A classical Wilson line is a cooresponedce between closed paths and elemets of a gauge group. However the noncommutative geometry does not have closed paths. But noncommutative geometry have good generalizations of both: the covering projection, and the group of covering transformations. These notions are used for a construction of noncommutative Wilson lines. Wilson lines can also be constructed as global pure gauge fields on the universal covering space. The noncommutative analog of this construction is also developed.

math.OA↗
arXiv2014

Noncommutative covering projections and $K$-homology

If $X$ is a topological space then there is a natural homomorphism $π_1(X)\rightarrow K_1(X)$ from a fundamental group to a $K_1$-homology group. Covering projections depend of fundamental group. So $K_1$-homology groups are interrelated with covering projections. This article is concerned with a noncommutative analogue of this interrelationship.

math.KT↗
arXiv2014

Finite covering projections of noncommutative torus

This article contains is concerned with noncommutative analogue of topological finitely listed covering projections. In my previous article I have already find a family of covering projections of the noncommutative torus. This article describes all covering projections of the noncommutative torus.

math.OA↗