Guillermo Cortiñas, Christian Valqui
We extend Cuntz-Quillen's excision theorem for algebras and pro-algebras in arbitrary Q-linear categories with tensor product.The excision theorems for the bivariant periodic cyclic cohomology of discrete,topological and bornological algebras and pro-algebras follow from this.
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This survey article on bivariant Kasparov theory and E-theory is mainly intended for readers with a background in homotopical algebra and category theory. We approach both bivariant K-theories via their universal properties and equip them with extra structure such as a tensor product and a triangulated category structure. We discuss the construction of the Baum-Connes assembly map via localisation of categories and explain how this is related to the purely topological construction by Davis and Lueck.
A higher categorical approach to graded bivariant $K$-theories for Real $C^*$-algebras
该研究针对局部紧第二可数群G作用下的分次实C*代数，给出经典KK理论与E理论的∞范畴细化，得到连接K理论谱的纤维序列，分析K0群自同态并解答民间问题，证明两类范畴非分次C*代数的Dwyer-Kan局部化。
Nonarchimedean bivariant K-theory
本文为非阿基米德bornological代数构建了双变K-理论，作为dagger同伦不变、矩阵稳定和切除函子的万有目标，并由此得到满足dagger同伦不变性、矩阵稳定性和切除性的稳定化过收敛解析K-理论。
Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^*$-algebras $\mathtt{SC^*_\infty}$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{SC^*_\infty}$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^*$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{SC^*_\infty}$ and give universal characterizations thereof. We construct a stable $\infty$-categorical model for bivariant connect…
Using the machinery of weak fibration categories due to Schlank and the first author, we construct a convenient model structure on the pro-category of separable $C^*$-algebras $\mathrm{Pro}(\mathtt{SC^*})$. The opposite of this model category models the $\infty$-category of pointed noncommutative spaces $\mathtt{N}\mathcal{S_*}$ defined by the third author. Our model structure on $\mathrm{Pro}(\mathtt{SC^*})$ extends the well-known category of fibrant objects structure on $\mathtt{SC^*}$. We show that the pro-category $\mathrm{Pro}(\mathtt{SC^*})$ also contains, as a full coreflective subcateg…
We describe a categorical framework for the classification of D-branes on noncommutative spaces using techniques from bivariant K-theory of C*-algebras. We present a new description of bivariant K-theory in terms of noncommutative correspondences which is nicely adapted to the study of T-duality in open string theory. We systematically use the diagram calculus for bivariant K-theory as detailed in our previous paper. We explicitly work out our theory for a number of examples of noncommutative manifolds.
It has been shown by Nistor that given any extension of associative algebras over C, the connecting morphism in periodic cyclic homology is compatible, under the Chern-Connes character, with the index morphism in lower algebraic K-theory. The proof relies on the abstract properties of cyclic theory, essentially excision, which does not provide explicit formulas a priori. Avoiding the use of excision, we explain in this article how to get explicit formulas in a wide range of situations. The method is connected to the renormalization procedure introduced in our previous work on the bivariant Che…
Equivariant periodic cyclic homology for ample groupoids
本文研究了ample群oid作用的双变等变周期性循环同调，证明了其在双变量中的同调不变性、稳定性与消去性，并证明了proper群oid作用的Green-Julg定理类比。
Categorical Milnor squares and K-theory of algebraic stacks
本文引入稳定∞-范畴的米尔诺平方概念，证明了代数K-理论将其映射为谱的笛卡尔平方的条件，并应用此结果证明了代数堆K-理论中的米尔诺消去和proper消去定理，推广了Weibel关于此类堆负K群消失的猜想。
We extend the definition of the bivariant $K$-theory $kk^{ban}$ from plain Banach algebras to Banach algebras equipped with an action of a locally compact Hausdorff group $G$. We also define a natural transformation from Lafforgue's theory $KK^{ban}_G$ into the new equivariant theory, overcoming some technical difficulties that are particular to the equivariant case. The categorical framework allows us to systematically define a descent homomorphism and to prove a Green-Julg theorem, a dual version of it and a generalised version that involves the action of a proper $G$-space. We also include …
We define and study equivariant periodic cyclic homology for locally compact groups. This can be viewed as a noncommutative generalization of equivariant de Rham cohomology. Although the construction resembles the Cuntz-Quillen approach to ordinary cyclic homology, a completely new feature in the equivariant setting is the fact that the basic ingredient in the theory is not a complex in the usual sense. As a consequence, in the equivariant context only the periodic cyclic theory can be defined in complete generality. Our definition recovers particular cases studied previously by various author…
We prove excision in entire and periodic cyclic cohomology and construct a Chern-Connes character for Fredholm modules over a C*-algebra without summability restrictions, taking values in a variant of Connes's entire cyclic cohomology. Before these results can be obtained, we have to sort out some fundamental questions about the class of algebras on which to define entire cyclic cohomology. The right domain of definition for entire cyclic cohomology is the category of complete bornological algebras. For these algebras, we define a bivariant cohomology theory, called analytic cyclic cohomology,…
Quillen introduced a new $K'_0$-theory of nonunital rings and showed that, under some assumptions (weaker than the existence of unity), this new theory agrees with the usual algebraic $K^{alg}_0$-theory. For a field $k$ of characteristic $0$, we introduce higher nonunital $K$-theory of $k$-algebras, denoted $KQ$, which extends Quillen's original definition of the $K'_0$ functor. We show that the $KQ$-theory is Morita invariant and satisfies excision connectively, in a suitable sense, on the category of idempotent $k$-algebras. Using these two properties we show that the $KQ$-theory agrees with…
We present a new categorical classification framework for D-brane charges on noncommutative manifolds using methods of bivariant K-theory. We describe several applications including an explicit formula for D-brane charge in cyclic homology, a refinement of open string T-duality, and a general criterion for cancellation of global worldsheet anomalies.
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