237 research outputs found
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
Operator *-correspondences in analysis and geometry
An operator *-algebra is a non-selfadjoint operator algebra with completely
isometric involution. We show that any operator *-algebra admits a faithful
representation on a Hilbert space in such a way that the involution coincides
with the operator adjoint up to conjugation by a symmetry. We introduce
operator *-correspondences as a general class of inner product modules over
operator *-algebras and prove a similar representation theorem for them. From
this we derive the existence of linking operator *-algebras for operator
*-correspondences. We illustrate the relevance of this class of inner product
modules by providing numerous examples arising from noncommutative geometry.Comment: 31 pages. This work originated from the MFO workshop "Operator spaces
and noncommutative geometry in interaction
Adenoviral vectors for highly selective gene expression in central serotonergic neurons reveal quantal characteristics of serotonin release in the rat brain
<p>Abstract</p> <p>Background</p> <p>5-hydroxytryptamine (5 HT, serotonin) is one of the key neuromodulators in mammalian brain, but many fundamental properties of serotonergic neurones and 5 HT release remain unknown. The objective of this study was to generate an adenoviral vector system for selective targeting of serotonergic neurones and apply it to study quantal characteristics of 5 HT release in the rat brain.</p> <p>Results</p> <p>We have generated adenoviral vectors which incorporate a 3.6 kb fragment of the rat tryptophan hydroxylase-2 (TPH-2) gene which selectively (97% co-localisation with TPH-2) target raphe serotonergic neurones. In order to enhance the level of expression a two-step transcriptional amplification strategy was employed. This allowed direct visualization of serotonergic neurones by EGFP fluorescence. Using these vectors we have performed initial characterization of EGFP-expressing serotonergic neurones in rat organotypic brain slice cultures. Fluorescent serotonergic neurones were identified and studied using patch clamp and confocal Ca<sup>2+ </sup>imaging and had features consistent with those previously reported using post-hoc identification approaches. Fine processes of serotonergic neurones could also be visualized in un-fixed tissue and morphometric analysis suggested two putative types of axonal varicosities. We used micro-amperometry to analyse the quantal characteristics of 5 HT release and found that central 5 HT exocytosis occurs predominantly in quanta of ~28000 molecules from varicosities and ~34000 molecules from cell bodies. In addition, in somata, we observed a minority of large release events discharging on average ~800000 molecules.</p> <p>Conclusion</p> <p>For the first time quantal release of 5 HT from somato-dendritic compartments and axonal varicosities in mammalian brain has been demonstrated directly and characterised. Release from somato-dendritic and axonal compartments might have different physiological functions. Novel vectors generated in this study open a host of new experimental opportunities and will greatly facilitate further studies of the central serotonergic system.</p
The Baum-Connes Conjecture via Localisation of Categories
We redefine the Baum-Connes assembly map using simplicial approximation in
the equivariant Kasparov category. This new interpretation is ideal for
studying functorial properties and gives analogues of the assembly maps for all
equivariant homology theories, not just for the K-theory of the crossed
product. We extend many of the known techniques for proving the Baum-Connes
conjecture to this more general setting
Uniformization and an Index Theorem for Elliptic Operators Associated with Diffeomorphisms of a Manifold
We consider the index problem for a wide class of nonlocal elliptic operators
on a smooth closed manifold, namely differential operators with shifts induced
by the action of an isometric diffeomorphism. The key to the solution is the
method of uniformization: We assign to the nonlocal problem a
pseudodifferential operator with the same index, acting in sections of an
infinite-dimensional vector bundle on a compact manifold. We then determine the
index in terms of topological invariants of the symbol, using the Atiyah-Singer
index theorem.Comment: 16 pages, no figure
Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles
We apply some methods of homology and K-theory to special classes of branes
wrapping homologically nontrivial cycles. We treat the classification of
four-geometries in terms of compact stabilizers (by analogy with Thurston's
classification of three-geometries) and derive the K-amenability of Lie groups
associated with locally symmetric spaces listed in this case. More complicated
examples of T-duality and topology change from fluxes are also considered. We
analyse D-branes and fluxes in type II string theory on with torsion flux and demonstrate in details
the conjectured T-duality to with no flux. In the
simple case of , T-dualizing the circles reduces to
duality between with
flux and with no flux.Comment: 27 pages, tex file, no figure
Fluxes, Brane Charges and Chern Morphisms of Hyperbolic Geometry
The purpose of this paper is to provide the reader with a collection of
results which can be found in the mathematical literature and to apply them to
hyperbolic spaces that may have a role in physical theories. Specifically we
apply K-theory methods for the calculation of brane charges and RR-fields on
hyperbolic spaces (and orbifolds thereof). It is known that by tensoring
K-groups with the rationals, K-theory can be mapped to rational cohomology by
means of the Chern character isomorphisms. The Chern character allows one to
relate the analytic Dirac index with a topological index, which can be
expressed in terms of cohomological characteristic classes. We obtain explicit
formulas for Chern character, spectral invariants, and the index of a twisted
Dirac operator associated with real hyperbolic spaces. Some notes for a
bivariant version of topological K-theory (KK-theory) with its connection to
the index of the twisted Dirac operator and twisted cohomology of hyperbolic
spaces are given. Finally we concentrate on lower K-groups useful for
description of torsion charges.Comment: 26 pages, no figures, LATEX. To appear in the Classical and Quantum
Gravit
Some Remarks on Group Bundles and C*-dynamical systems
We introduce the notion of fibred action of a group bundle on a C(X)-algebra.
By using such a notion, a characterization in terms of induced C*-bundles is
given for C*-dynamical systems such that the relative commutant of the
fixed-point algebra is minimal (i.e., it is generated by the centre of the
given C*-algebra and the centre of the fixed-point algebra). A class of
examples in the setting of the Cuntz algebra is given, and connections with
superselection structures with nontrivial centre are discussed.Comment: 22 pages; to appear on Comm. Math. Phy
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