A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space

In 1975 A. Connes proved the fundamental result that injective factors on a separable Hilbert space are hyperfinite. In this paper a new proof of this result is presented in which the most technical parts of Connes proof are avoided. Particularly the proof does not rely on automorphism group theory. The starting point in this approach is Wassermann's simple proof of injective ? semidiscrete together with Choi and Effros' characterization of semidiscrete von Neumann algebras as those von Neumann algebras N for which the identity map on N has an approximate completely positive factorization through n × n-matrices.     

La conjecture de Baum–Connes à coefficients pour le groupe Sp(n,1)The Baum–Connes conjecture with coefficients for the group Sp(n,1)

We show that the Lie groups Sp(n,1) satisfy the Baum–Connes conjecture with arbitrary coefficients. The main tool is the construction, due to Cowling, of a family of uniformly bounded representations. To cite this article: P. Julg, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 533–538.     

Character Connes amenability of dual Banach algebras

We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A^** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.     

On Connes’ trace formula for the Hankel transformation of order ?1/2

A local Hankel transformation of order ?1/2 is defined for every finite place of the field of rational numbers. Its inversion formula and the Plancherel type theorem are obtained. A Connes type trace formula is given for each local Hankel transformation of order ?1/2. An S-local Connes type trace formula is derived for the S-local Hankel transformation of order ?1/2. These formulas are generalizations of Connes?? corresponding trace formulas in 1999.     

The correlation numerical range of a matrix and Connes’ embedding problem

We define a new numerical range of an $n\times n$ complex matrix in terms of correlation matrices and develop some of its properties. We also define a related numerical range that arises from Connes’ famous embedding problem.     

Higher index theory for certain expanders and Gromov monster groups,I

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum–Connes assembly map is injective, but not surjective, for the associated metric space X.Expanders with this girth property are a necessary ingredient in the construction of the so-called ‘Gromov monster’ groups that (coarsely) contain expanders in their Cayley graphs. We use this connection to show that the Baum–Connes assembly map with certain coefficients is injective but not surjective for these groups. Using the results of the second paper in this series, we also show that the maximal Baum–Connes assembly map with these coefficients is an isomorphism.     

On a generalised Connes-Hochschild-Kostant-Rosenberg theorem

The central result of this paper is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalising some fundamental results of Connes and Hochschild-Kostant-Rosenberg. The Connes-Chern character is also identified here with the twisted Chern character.     

K-theory of equivariant quantization

Using an equivariant version of Connes? Thom isomorphism, we prove that equivariant K-theory is invariant under strict deformation quantization for a compact Lie group action.     

Laplacians and Spectrum for Singular Foliations

The author surveys Connes＇ results on the longitudinal Laplace operator along a （regular） foliation and its spectrum, and discusses their generalization to any singular foliation on a compact manifold. Namely, it is proved that the Laplacian of a singular foliation is an essentially self-adjoint operator （unbounded） and has the same spectrum in every （faithful） representation, in particular, in L2 of the manifold and L2 of a leaf. The author also discusses briefly the relation of the Baum-Connes assembly map with the calculation of the spectrum.     

Étude de l'algèbre de Lie double des arbres enracinés décorés

The double Lie algebra LD of rooted trees decorated by a set D is introduced, generalising the construction of Connes and Kreimer. It is shown that it is a simple Lie algebra. Its derivations and its automorphisms are described, as well as some central extensions. Finally, the category of lowest weight modules is introduced and studied.     

Rankin-Cohen brackets and formal quantization

In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results [A. Connes, H. Moscovici, Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J. 4 (1) (2004) 111-130, 311]. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation [D. Zagier, Modular forms and differential operators, in: K.G. Ramanathan Memorial Issue, Proc. Indian Acad. Sci. Math. Sci. 104 (1) (1994) 57-75] of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author [X. Tang, Deformation quantization of pseudo (symplectic) Poisson groupoids, Geom. Funct. Anal. 16 (3) (2006) 731-766], we reconstruct a universal deformation formula of the Hopf algebra H₁ associated to codimension one foliations. In the end, we prove that the first Rankin-Cohen bracket RC₁ defines a noncommutative Poisson structure for an arbitrary H₁ action.     

Applications of the Connes Spectrum to C1-dynamical systems,III

For a C¹-dynamical system (A, G, α) we show that the crossed product C¹-algebra is induced from a simple C¹-algebra equipped with an action of the Connes Spectrum, provided that A is G-simple and all isotropy subgroups of G under the action on the primitive ideal space of A are discrete. We then study the Borchers Spectrum of α and characterize its annihilator in G as the group of locally derivable automorphisms, under the assumption that the Arveson Spectrum of α is compact modulo the Borchers Spectrum. Finally a properly outer automorphism α is characterized by a series of equivalent conditions, one of which says that α is not close to the inner automorphisms on any ideal, another that α is not universally weakly inner on any ideal, and a third that the Borchers Spectrum of α on any invariant hereditary C¹-subalgebra is non-zero. This characterization leads to the conclusion that α is aperiodic (i.e., every non-zero power is properly outer) precisely when the Connes Spectrum of α is the full circle group.     

Dixmier traces on noncompact isospectral deformations

We extend the isospectral deformations of Connes, Landi and Dubois-Violette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group Rl. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of functions on the manifold. We show that this relation persists for actions of Rl, under mild restrictions on the geometry of the manifold which guarantee the Dixmier traceability of those operators.     

The cyclic and simplicial cohomology of the Cuntz semigroup algebra

The main objective of this paper is to determine the simplicial and cyclic cohomology groups of the Cuntz semigroup algebra ${?}^1({\mathbf{S}}_m)$. We also determine the simplicial and cyclic cohomology of the tensor algebra of a Banach space, a class which includes the algebra on the free semigroup on m-generators ${?}^1({\mathbf{FS}}_m)$. In order to do so, we first establish some general results which can be used when studying simplicial and cyclic cohomology of Banach algebras in general. We then turn our attention to ${?}^1({\mathbf{S}}_m)$, showing that the cyclic cohomology groups of degree n vanish when n is odd and are one-dimensional when n is even ($n?2$). Using the Connes–Tzygan exact sequence, these results are used to show that the simplicial cohomology groups of degree n vanish for $n?1$. A similar strategy is used for the tensor algebra of a Banach space.     

Measurable operators and the asymptotics of heat kernels and zeta functions

In this note we answer some questions inspired by the introduction in Connes (1988, 1994) [6], [7] of the notion of measurable operators using Dixmier traces. These questions concern the relationship of measurability to the asymptotics of ζ-functions and heat kernels. The answers have remained elusive for some 15 years.     

The local index formula in semifinite Von Neumann algebras I: Spectral flow

We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self-adjoint Breuer-Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is ‘almost’ a (b,B)-cocycle in the cyclic cohomology of A.     

Bivariant Chern character and longitudinal index

In this paper we consider a family of Dirac-type operators on fibration P→B equivariant with respect to an action of an étale groupoid. Such a family defines an element in the bivariant K theory. We compute the action of the bivariant Chern character of this element on the image of Connes' map Φ in the cyclic cohomology. A particular case of this result is Connes' index theorem for étale groupoids [A. Connes, Noncommutative Geometry, Academic Press, 1994] in the case of fibrations.     

Twisted Equivariant KK-Theory and the Baum–Connes Conjecture for Group Extensions

In this paper we study the stability of the Baum–Connes conjecture with coefficients undertaking group extensions. For this, it is necessary to extend Kasparov's equivariant KK-theory to an equivariant theory for twisted actions of groups on C ^*-algebras. As a consequence of our stability results, we are able to reduce the problem of whether closed subgroups of connected groups satisfy the Baum–Connes conjecture, with coefficients to the special case of center-free semi-simple Lie groups.     

Higher index theory for certain expanders and Gromov monster groups,II

In this paper, the second of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs has girth tending to infinity, then the maximal coarse Baum–Connes assembly map is an isomorphism for the associated metric space X. As discussed in the first paper in this series, this has applications to the Baum–Connes conjecture for ‘Gromov monster’ groups.We also introduce a new property, ‘geometric property (T)’. For the metric space associated to a sequence of graphs, this property is an obstruction to the maximal coarse assembly map being an isomorphism. This enables us to distinguish between expanders with girth tending to infinity, and, for example, those constructed from property (T) groups.     

Scaling group flow and Lefschetz trace formula for laminated spaces with p-adic transversal

In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function (resp. the zeta function ζY(s) of a smooth projective curve Y over a finite field Fq, q=pf)) one could possibly associate a foliated Riemannian laminated space (SQ,F,g,?t) (resp. (SY,F,g,?t)) endowed with an action of a flow ?t whose primitive compact orbits should correspond to the primes of Q (resp. Y). Precise conjectures were stated in our report [E. Leichtnam, An invitation to Deninger's work on arithmetic zeta functions, in: Geometry, Spectral Theory, Groups, and Dynamics, in: Contemp. Math. vol. 387, Amer. Math. Soc., Providence, RI, 2005, pp. 201-236] on Deninger's work. The existence of such a foliated space and flow ?t is still unknown except when Y is an elliptic curve (see Deninger [C. Deninger, On the nature of explicit formulas in analytic number theory, a simple example, in: Number Theoretic Methods, Iizuka, 2001, in: Dev. Math., vol. 8, Kluwer Acad. Publ., Dordrecht, 2002, pp. 97-118]). Being motivated by this latter case, we introduce a class of foliated laminated spaces () where L is locally , D being an open disk of C. Assuming that the leafwise harmonic forms on L are locally constant transversally, we prove a Lefschetz trace formula for the flow ?t acting on the leafwise Hodge cohomology (0?j?2) of (S,F) that is very similar to the explicit formula for the zeta function of a (general) smooth curve over Fq. We also prove that the eigenvalues of the infinitesimal generator of the action of ?t on have real part equal to .Moreover, we suggest in a precise way that the flow ?t should be induced by a renormalization group flow “à la K. Wilson”. We show that when Y is an elliptic curve over Fq this is indeed the case. It would be very interesting to establish a precise connection between our results and those of Connes (p. 553 in [A. Connes, Noncommutative Geometry Year 2000, in: Special Volume GAFA 2000 Part II, pp. 481-559], p. 90 in [A. Connes, Symétries Galoisiennes et Renormalisation, in: Séminaire Bourbaphy, Octobre 2002, pp. 75-91]) and Connes-Marcolli [A. Connes, M. Marcolli, Q-lattices: quantum statistical mechanics and Galois theory, in: Frontiers in Number Theory, Physics and Geometry, vol. I, Springer-Verlag, 2006, pp. 269-350; A. Connes, M. Marcolli, From physics to number theory via noncommutative geometry. Part II: renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory, in: Frontiers in Number Theory, Physics and Geometry, vol. II, Springer-Verlag, 2006, pp. 617-713] on the Galois interpretation of the renormalization group.