arXiv subjects

Ralf Meyer

Publications and source records attributed to Ralf Meyer.

105 records · Page 6Linked to original sources

arXiv2005

Euler characteristics and Gysin sequences for group actions on boundaries

Let G be a locally compact group, let X be a universal proper G-space, and let Z be a G-equivariant compactification of X that is H-equivariantly contractible for each compact subgroup H of G. Let W be the resulting boundary. Assuming the Baum-Connes conjecture for G with coefficients C and C(W), we construct an exact sequence that computes the map on K-theory induced by the embedding of the reduced group C*-algebra of G into the crossed product of G by C(W). This exact sequence involves the equivariant Euler characteristic of X, which we study using an abstract notion of Poincare duality in bivariant K-theory. As a consequence, if G is torsion-free and the Euler characteristic of the orbit space X/G is non-zero, then the unit element of the boundary crossed product is a torsion element whose order is equal to the absolute value of the Euler characteristic of X/G. Furthermore, we get a new proof of a theorem of Lueck and Rosenberg concerning the class of the de Rham operator in equivariant K-homology.

math.KT↗

arXiv2005

On a representation of the idele class group related to primes and zeros of L-functions

Let K be a global field. Using natural spaces of functions on the adele ring and the idele class group of K, we construct a virtual representation of the idele class group of K whose character is equal to the Weil distribution that occurs in Andre Weil's Explicit Formula. Hence this representation encodes information about the distribution of the prime ideals of K, and it is a spectral interpretation for the poles and zeros of the L-function. It is closely related to the spectral interpretation of Alain Connes.

math.NT↗

arXiv2005

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.

math.KT↗

arXiv2004

Embeddings of derived categories of bornological modules

Let A be an algebra with a countable basis and let B be, say, a Frechet algebra that contains A as a dense subalgebra. This embedding induces a functor from the derived category of B-modules to the derived category of A-modules. In many important examples, this functor is fully faithful. We study this property in some detail, giving several equivalent conditions, examples, and applications. To prepare for this, we explain carefully how to do homological algebra with modules over bornological algebras. We construct the derived category of bornological left A-modules and some standard derived functors, with special emphasis on the adjoint associativity between the tensor product and the internal Hom functor. We also discuss the category of essential modules over a non-unital algebra and its functoriality.

math.FA↗

arXiv2004

A descent principle for the Dirac dual Dirac method

Let G be a torsion free discrete group with a finite dimensional classifying space BG. We show that G has a dual Dirac morphism if and only if a certain coarse (co)-assembly map is an isomorphism. Hence the existence of a dual Dirac morphism for such G is a metric, that is, coarse, invariant of G. We obtain similar results for groups with torsion. The framework that we develop is also suitable for studying the Lipschitz and proper Lipschitz cohomology of Connes, Gromov and Moscovici.

math.OA↗

arXiv2004

Dualizing the coarse assembly map

We formulate and study a new coarse (co-)assembly map. It involves a modification of the Higson corona construction and produces a map dual in an appropriate sense to the standard coarse assembly map. The new assembly map is shown to be an isomorphism in many cases. For the underlying metric space of a group, the coarse co-assembly map is closely related to the existence of a dual Dirac morphism and thus to the Dirac dual Dirac method of attacking the Novikov conjecture.

math.OA↗

arXiv2004

Bornological versus topological analysis in metrizable spaces

Given a metrizable topological vector space, we can also use its von Neumann bornology or its bornology of precompact subsets to do analysis. We show that the bornological and topological approaches are equivalent for many problems. For instance, they yield the same concepts of convergence for sequences of points or linear operators, of continuity of functions, of completeness and completion. We also show that the bornological and topological versions of Grothendieck's approximation property are equivalent for Frechet spaces. These results are important for applications in noncommutative geometry. Finally, we investigate the class of ``smooth'' subalgebras appropriate for local cyclic homology and apply some of our results in this context.

math.FA↗

arXiv2003

Comparisons between periodic, analytic and local cyclic cohomology

We compute periodic, analytic and local cyclic cohomology for convolution algebras of compact Lie groups in order to exhibit differences between these theories. A surprising result is that the periodic and analytic cyclic cohomology of the smooth convolution algebras differ, although these algebras have finite homological dimension.

math.KT↗

arXiv2003

The cyclic homology and K-theory of certain adelic crossed products

The multiplicative group of a global field acts on its adele ring by multiplication. We consider the crossed product algebra of the resulting action on the space of Schwartz functions on the adele ring and compute its Hochschild, cyclic and periodic cyclic homology. We also compute the topological K-theory of the C*-algebra crossed product.

math.KT↗

arXiv2003

Smooth group representations on bornological vector spaces

We develop the basic theory of smooth representations of locally compact groups on bornological vector spaces. In this setup, we are able to formulate better general theorems than in the topological case. Still, smooth representations of totally disconnected groups on vector spaces and of Lie groups on Frechet spaces remain special cases of our theory. We identify smooth representations with essential modules over an appropriate convolution algebra. We examine smoothening functors on representations and modules and show that they agree if they are both defined. We establish the basic properties of induction and compact induction functors using adjoint functor techniques. We describe the center of the category of smooth representations.

math.FA↗

arXiv2003

A characterization of the Dirac Dual Dirac Method

Let G be a discrete, torsion free group with a finite dimensional classifying space BG. We show that the existence of a gamma-element for such G is a metric, that is, coarse, invariant of G. We also obtain results for groups with torsion. The method of proof involves showing that a group G possesses a gamma-element if and only if a certain coarse (co)-assembly map is an isomorphism.

math.KT↗

arXiv2000

Generalized fixed point algebras and square-integrable group actions

We analzye Rieffel's construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C*-algebra B. We construct a Hilbert module F over the reduced crossed product of G and B, using a pair (E, R), where E is an equivariant Hilbert module over B and R is a dense subspace of E with certain properties. The generalized fixed point algebra is the C*-algebra of compact operators on F. Any Hilbert module over the reduced crossed product arises by this construction for a pair (E, R) that is unique up to isomorphism. A necessary condition for the existence of R is that E be square-integrable. The consideration of square-integrable representations of Abelian groups on Hilbert space shows that this condition is not sufficient and that different choices for R may yield different generalized fixed point algebras. If B is proper in Kasparov's sense, there is a unique R with the required properties. Thus the generalized fixed point algebra only depends on E.

math.OA↗

arXiv2000

Equivariant Kasparov theory and generalized homomorphisms

Let G be a locally compact group. We describe elements of KK^G (A,B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal property of KK^G: It is the universal split exact stable homotopy functor. To describe a Kasparov triple (E, phi, F) by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form K(L^2G) otimes A' and more generally if the group action on A is proper in the sense of Rieffel and Exel.

math.KT↗

arXiv1999

Analytic cyclic cohomology

We prove excision in entire and periodic cyclic cohomology and construct a Chern-Connes character for Fredholm modules over a C*-algebra without summability restrictions, taking values in a variant of Connes's entire cyclic cohomology. Before these results can be obtained, we have to sort out some fundamental questions about the class of algebras on which to define entire cyclic cohomology. The right domain of definition for entire cyclic cohomology is the category of complete bornological algebras. For these algebras, we define a bivariant cohomology theory, called analytic cyclic cohomology, that contains Connes's entire cyclic cohomology as a special case. The definition of analytic cyclic cohomology is based on the Cuntz-Quillen approach to cyclic cohomology theories using tensor algebras and X-complexes. The appropriate completion of the tensor algebra that yields analytic cyclic cohomology can be understood using an appropriate notion of analytic nilpotence. In addition, we develop the elementary theory of analytic cyclic cohomology (smooth homotopy invariance, stability, Chern character in K-theory).

math.KT↗

arXiv1997

Finite dimensional quotients of commutative operator algebras

The matrix normed structure of the unitization of a (non-selfadjoint) operator algebra is determined by that of the original operator algebra. This yields a classification up to completely isometric isomorphism of two-dimensional unital operator algebras. This allows to define invariant distances on the spectrum of commutative operator algebras analogous to the Caratheodory distance for complex manifolds. Moreover, unitizations of two-dimensional operator algebras with zero multiplication provide a rich class of counterexamples. Especially, several badly behaved quotients of function algebras are exhibited. Recently, Arveson has developed a model theory for d-contractions. Quotients of the operator algebra of the d-shift are much more well-behaved than quotients of function algebras. Completely isometric representations of these quotients are obtained explicitly. This provides a generalization of Nevanlinna-Pick theory. An important property of quotients of the d-shift algebra is that their quotients of finite dimension r have completely isometric representations by rxr-matrices. Finally, the class of commutative operator algebras with this property is investigated.

funct-an↗