760 research outputs found
The coarse geometric Novikov conjecture and uniform convexity
The coarse geometric Novikov conjecture provides an algorithm to determine
when the higher index of an elliptic operator on a noncompact space is nonzero.
The purpose of this paper is to prove the coarse geometric Novikov conjecture
for spaces which admit a (coarse) uniform embedding into a uniformly convex
Banach space.Comment: 64 pages, to appear in Advances in Mathematic
Localizing subcategories in the Bootstrap category of separable C*-algebras
Using the classical universal coefficient theorem of Rosenberg-Schochet, we
prove a simple classification of all localizing subcategories of the Bootstrap
category of separable complex C*-algebras. Namely, they are in bijective
correspondence with subsets of the Zariski spectrum of the integers --
precisely as for the localizing subcategories of the derived category of
complexes of abelian groups. We provide corollaries of this fact and put it in
context with similar classifications available in the literature.Comment: 9 pages, simplified proof. Final version, to appear on J. of K-theor
Equivariant geometric K-homology for compact Lie group actions
Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant
geometric K-homology groups K^G_*(X), using an obvious equivariant version of
the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural
transformations to and from equivariant K-homology defined via KK-theory (the
"official" equivariant K-homology groups) and show that these are isomorphism.Comment: 25 pages. v2: some mistakes corrected, more detail added, Michael
Walter as author added. To appear in Abhandlungen aus dem Mathematischen
Seminar der Universit\"at Hambur
Real versus complex K-theory using Kasparov's bivariant KK-theory
In this paper, we use the KK-theory of Kasparov to prove exactness of
sequences relating the K-theory of a real C^*-algebra and of its
complexification (generalizing results of Boersema). We use this to relate the
real version of the Baum-Connes conjecture for a discrete group to its complex
counterpart. In particular, the complex Baum-Connes assembly map is an
isomorphism if and only if the real one is, thus reproving a result of Baum and
Karoubi. After inverting 2, the same is true for the injectivity or
surjectivity part alone.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-18.abs.htm
Algebraic Kasparov K-theory. I
This paper is to construct unstable, Morita stable and stable bivariant
algebraic Kasparov -theory spectra of -algebras. These are shown to be
homotopy invariant, excisive in each variable -theories. We prove that the
spectra represent universal unstable, Morita stable and stable bivariant
homology theories respectively.Comment: This is the final revised versio
Nonsplitting in Kirchberg's ideal-related KK-theory
A universal coefficient theorem in the setting of Kirchberg's ideal-related
KK-theory was obtained in the fundamental case of a C*-algebra with one
specified ideal by Bonkat and proved there to split, unnaturally, under certain
conditions. Employing certain K-theoretical information derivable from the
given operator algebras in a way introduced here, we shall demonstrate that
Bonkat's UCT does not split in general. Related methods lead to information on
the complexity of the K-theory which must be used to classify *-isomorphisms
for purely infinite C*-algebras with one non-trivial ideal.Comment: 14 pages, minor typos fixed, 5 figures adde
The noncommutative geometry of Yang-Mills fields
We generalize to topologically non-trivial gauge configurations the
description of the Einstein-Yang-Mills system in terms of a noncommutative
manifold, as was done previously by Chamseddine and Connes. Starting with an
algebra bundle and a connection thereon, we obtain a spectral triple, a
construction that can be related to the internal Kasparov product in unbounded
KK-theory. In the case that the algebra bundle is an endomorphism bundle, we
construct a PSU(N)-principal bundle for which it is an associated bundle. The
so-called internal fluctuations of the spectral triple are parametrized by
connections on this principal bundle and the spectral action gives the
Yang-Mills action for these gauge fields, minimally coupled to gravity.
Finally, we formulate a definition for a topological spectral action.Comment: 14 page
Equivariant representable K-theory
We interpret certain equivariant Kasparov groups as equivariant representable
K-theory groups. We compute these groups via a classifying space and as
K-theory groups of suitable sigma-C*-algebras. We also relate equivariant
vector bundles to these sigma-C*-algebras and provide sufficient conditions for
equivariant vector bundles to generate representable K-theory. Mostly we work
in the generality of locally compact groupoids with Haar system.Comment: Final version. Only minor corrections. 33 page
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