Twisted K-theory, K-homology and bivariant Chern-Connes type character of some infinite dimensional spaces

We study the twisted K-theory and K-homology of some infinite dimensional spaces, like SU(\infty), in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable \sigma-C^*-algebras that generalizes both twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern--Connes type character taking values in bivariant local cyclic homology. We analyse the structure of the dual Chern--Connes character from (analytic) K-homology to local cyclic cohomology under some reasonable hypotheses. We also investigate the twisted periodic cyclic homology via locally convex algebras and the local cyclic homology via C^*-algebras (in the compact case).Comment: v2. 32 pages, acknowledgements added and some minor changes made (including the title); v3. 34 pages, some changes following the referee's suggestions; v4. references updated, exposition improved, to appear in Kyoto J. Mat

Lecture Notes on Noncommutative Algebraic Geometry and Noncommutative Tori

The first part of these notes gives an introduction to noncommutative projective geometry after Artin--Zhang. The second part provides an overview of the work of Polishchuk that reconciles noncommutative two-tori having real multiplication with the Artin--Zhang setting.Comment: Final version - exposition improved; a proof of the derived equivalence added (Prop. 3.8). To appear in the proceedings volume of the "International Workshop on Noncommutative Geometry", IPM, Tehran 200

C∗C^*-algebraic drawings of dendroidal sets

In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. In this article we introduce the concept of a $C^*$-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves on $C^*$-algebras. We show that the construction is functorial and, in fact, it is the left adjoint of a Quillen adjunction between combinatorial model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable $\infty$-categories, which is the primary motivation behind this article. As a consequence we obtain a single mechanism to construct bivariant homology theories in both paradigms. We propose a (conjectural) roadmap to harmonize algebraic and analytic (or topological) bivariant K-theory. Finally, a method to analyse graph algebras in terms of trees is sketched.Comment: 28 pages; v2 expanded version with some improvements; v3 revised and added references; v4 some changes according to the suggestions of the referees (to appear in Algebr. Geom. Topol.

Symmetric monoidal noncommutative spectra, strongly self-absorbing C∗C^*-algebras, and bivariant homology

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 connective E-theory and compute the connective E-theory groups of $\mathcal{O}_\infty$-stable $C^*$-algebras. We also introduce and study the nonconnective version of Quillen's nonunital K'-theory in the framework of stable $\infty$-categories. This is done in order to promote our earlier result relating topological $\mathbb{T}$-duality to noncommutative motives to the $\infty$-categorical setup. Finally, we carry out some computations in the case of stable and $\mathcal{O}_\infty$-stable $C^*$-algebras.Comment: 26 pages; v2 revised in accordance with arXiv:1412.8370, corrections in Sections 3 and 4; v3 minor changes, to appear in J. Noncommut. Geo

On the Generating Hypothesis in Noncommutative Stable Homotopy

Freyd's Generating Hypothesis is an important problem in topology with deep structural consequences for finite stable homotopy. Due to its complexity some recent work has examined analogous questions in various other triangulated categories. In this short note we analyze the question in noncommutative stable homotopy, which is a canonical generalization of finite stable homotopy. Along the way we also discuss Spanier--Whitehead duality in this extended setup.Comment: 6 pages; v2 added a Section on Matrix Generating Hypothesis; v3 included Spanier--Whitehead duality discussion and removed some unnecessary material, to appear in Math. Scand; v4 updated references and metadat