28 research outputs found
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
-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
-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an
object in the category of presheaves on -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 -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 -algebras, and bivariant homology
Continuing our project on noncommutative (stable) homotopy we construct
symmetric monoidal -categorical models for separable -algebras
and noncommutative spectra using the
framework of Higher Algebra due to Lurie. We study smashing (co)localizations
of and with respect to strongly
self-absorbing -algebras. We analyse the homotopy categories of the
localizations of and give universal characterizations
thereof. We construct a stable -categorical model for bivariant
connective E-theory and compute the connective E-theory groups of
-stable -algebras. We also introduce and study the
nonconnective version of Quillen's nonunital K'-theory in the framework of
stable -categories. This is done in order to promote our earlier result
relating topological -duality to noncommutative motives to the
-categorical setup. Finally, we carry out some computations in the case
of stable and -stable -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