234 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
Differential sensitivity of brainstem vs cortical astrocytes to changes in pH reveals functional regional specialization of astroglia
Astrocytes might function as brain interoceptors capable of detecting different (chemo)sensory modalities and transmitting sensory information to the relevant neural networks controlling vital functions. For example, astrocytes which reside near the ventral surface of the brainstem (central respiratory chemosensitive area) respond to physiological decreases in pH with vigorous elevations in intracellular Ca(2+) and release of ATP. ATP transmits astroglial excitation to the brainstem respiratory network and contributes to adaptive changes in lung ventilation. Here we show that in terms of pH-sensitivity ventral brainstem astrocytes are clearly distinct from astrocytes residing in the cerebral cortex. We monitored vesicular fusion in cultured rat brainstem astrocytes using total internal reflection fluorescence microscopy and found that approximately 35% of them respond to acidification with an increased rate of exocytosis of ATP-containing vesicular compartments. These fusion events require intracellular Ca(2+) signaling and are independent of autocrine ATP actions. In contrast, the rate of vesicular fusion in cultured cortical astrocytes is not affected by changes in pH. Compared to cortical astrocytes, ventral brainstem astrocytes display higher levels of expression of genes encoding proteins associated with ATP vesicular transport and fusion, including vesicle-associated membrane protein-3 and vesicular nucleotide transporter. These results suggest that astrocytes residing in different parts of the rat brain are functionally specialized. In contrast to cortical astrocytes, astrocytes of the brainstem chemosensitive area(s) possess signaling properties which are functionally relevant – they are able to sense changes in pH and respond to acidification with enhanced vesicular release of ATP
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
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
Localisation and colocalisation of KK-theory at sets of primes
Given a set of prime numbers S, we localise equivariant bivariant Kasparov
theory at S and compare this localisation with Kasparov theory by an exact
sequence. More precisely, we define the localisation at S to be KK^G(A,B)
tensored with the ring of S-integers Z[S^-1]. We study the properties of the
resulting variants of Kasparov theory.Comment: 16 page
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
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
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
Diagonalizing operators over continuous fields of C*-algebras
It is well known that in the commutative case, i.e. for being a
commutative C*-algebra, compact selfadjoint operators acting on the Hilbert
C*-module (= continuous families of such operators , ) can
be diagonalized if we pass to a bigger W*-algebra which can be obtained from by completing it with respect to the weak
topology. Unlike the "eigenvectors", which have coordinates from , the
"eigenvalues" are continuous, i.e. lie in the C*-algebra . We discuss here
the non-commutative analog of this well-known fact. Here the "eigenvalues" are
defined not uniquely but in some cases they can also be taken from the initial
C*-algebra instead of the bigger W*-algebra. We prove here that such is the
case for some continuous fields of real rank zero C*-algebras over a
one-dimensional manifold and give an example of a C*-algebra for which the
"eigenvalues" cannot be chosen from , i.e. are discontinuous. The main point
of the proof is connected with a problem on almost commuting operators. We
prove that for some C*-algebras if is a selfadjoint, is a
unitary and if the norm of their commutant is small enough then one can
connect with the unity by a path so that the norm of
would be also small along this path.Comment: 21 pages, LaTeX 2.09, no figure
A Short Survey of Noncommutative Geometry
We give a survey of selected topics in noncommutative geometry, with some
emphasis on those directly related to physics, including our recent work with
Dirk Kreimer on renormalization and the Riemann-Hilbert problem. We discuss at
length two issues. The first is the relevance of the paradigm of geometric
space, based on spectral considerations, which is central in the theory. As a
simple illustration of the spectral formulation of geometry in the ordinary
commutative case, we give a polynomial equation for geometries on the four
dimensional sphere with fixed volume. The equation involves an idempotent e,
playing the role of the instanton, and the Dirac operator D. It expresses the
gamma five matrix as the pairing between the operator theoretic chern
characters of e and D. It is of degree five in the idempotent and four in the
Dirac operator which only appears through its commutant with the idempotent. It
determines both the sphere and all its metrics with fixed volume form.
We also show using the noncommutative analogue of the Polyakov action, how to
obtain the noncommutative metric (in spectral form) on the noncommutative tori
from the formal naive metric. We conclude on some questions related to string
theory.Comment: Invited lecture for JMP 2000, 45
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