论文标题
紧凑型量子空间的乘法K理论
The multiplicative K-theory of compact quantum spaces
论文作者
论文摘要
在本文中，我们按K理论类型完善紧凑型量子空间的分类。我们通过引入一个乘法k理论函数，以用于在增强环类别中采用值的Unital c*代数，以及对标准K理论函数的自然转换，从而将值属于Abelian群体类别。乘法K理论类型完善了K理论类型，该类型在CW-Waldhausen类别的框架中定义。在我们的完善中，我们需要形态的兼容性与我们称为K-Topology的其他结构。这是Grothendieck拓扑的非共同版本，涵盖了由紧凑的量子主束和由合适的k连续图关联的基地给出的家族。在经典的紧凑型Hausdorff空间的情况下，K-Topology和K-continition分别降低了通常的拓扑和连续性，并且我们在K理论中的非交易性对应物减少了拓扑K理论中的标准产品。作为应用程序，我们表明，复杂的投影空间的标准CW复合结构的非同态量化属于相同的乘法K理论类型。此外，对于这些复杂的量子射击空间，我们证明了根据截短的多项式对K理论环的Atiyah-TODD计算的非共同概括。
In this paper, we refine the classification of compact quantum spaces by K-theory type. We do it by introducing a multiplicative K-theory functor for unital C*-algebras taking values in the category of augmented rings, together with a natural transformation to the standard K-theory functor, taking values in the category of abelian groups. The multiplicative K-theory type refines the K-theory type, defined in the framework of cw-Waldhausen categories. In our refinement, we require compatibility of morphisms with an additional structure which we call k-topology. This is a noncommutative version of the Grothendieck topology with covering families given by compact quantum principal bundles and bases related by suitable k-continuous maps. In the classical case of compact Hausdorff spaces, k-topology and k-continuity reduce to the usual topology and continuity, respectively, and our noncommutative counterpart of the product in K-theory reduces to the standard product in topological K-theory. As an application, we show that non-isomorphic quantizations of the standard CW-complex structure of a complex projective space belong to the same multiplicative K-theory type. In addition, for these complex quantum projective spaces, we prove a noncommutative generalization of the Atiyah--Todd calculation of the K-theory ring in terms of truncated polynomials.
