Serre-Swan theorem for non-commutative C$^{*}$-algebras

A Serre-Swan theorem for bundles of bounded geometry

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. In this context it states that the category of Hilbert bundles of bounded geometry is equivalent to the category of operator *-modules over the operator *-algebra of continuously differentiable functions which vanish at infinity. Operator *-modules are generalizations of Hilbert C…

标准模型——交换情形：旋量、狄拉克算子与 de Rham 代数

The Standard Model - the Commutative Case: Spinors, Dirac Operator and de Rham Algebra

本文以模论方法综述经典场论的数学基础，涵盖旋量丛、狄拉克算子及 Connes 微分代数，并给出外代数丛与 Connes 微分代数同构的新证明。

Remark on Serre $C^*$-algebras

We study non-commutative algebraic geometry of Artin, Serre and Tate in terms of the operator algebras. Namely, we define the Serre $C^*$-algebra $\mathcal{A}_X$ of a projective variety $X$ as the norm-closure of a representation of the twisted homogeneous coordinate ring of $X$ by the linear operators on a Hilbert space $\mathcal{H}$. It is proved that $X$ is homeomorphic to the space of all irreducible representations of the crossed product of $\mathcal{A}_X$ by an automorphism of $\mathcal{A}_X$. The case of rational elliptic curves $X$ is considered in detail.

Noncommutative Local Systems

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 is concerned with generalization of local systems. The classical construction of local system implies an existence of a path groupoid. However the no…

On tea, donuts and non-commutative geometry

As many will agree, it feels good to complement a cup of tea by a donut or two. This sweet relationship is also a guiding principle of non-commutative geometry known as Serre Theorem. We explain the algebra behind this theorem and prove that elliptic curves are complementary to the so-called non-commutative tori.

Universal covering space of the noncommutative torus

Gelfand - Naĭmark theorem supplies contravariant functor from a category of commutative $C^*-$ algebras to a category of locally compact Hausdorff spaces. Therefore any commutative $C^*-$ algebra is an alternative representation of a topological space. Similarly a category of (noncommutative) $C^*-$ algebras can be regarded as a category of generalized (noncommutative) locally compact Hausdorff spaces. Generalizations of topological invariants may be defined by algebraic methods. For example Serre Swan theorem states that complex topological $K$ - theory coincides with $K$ - theory of $C^*$ - …