Periodic Cyclic Cohomology Research Articles

Local index theory for operators associated with Lie groupoid actions

We develop a local index theory for a class of operators associated with non-proper and non-isometric actions of Lie groupoids on smooth submersions. Such actions imply the existence of a short exact sequence of algebras, relating these operators to their non-commutative symbol. We then compute the connecting map induced by this extension on periodic cyclic cohomology. When cyclic cohomology is localized at appropriate isotropic submanifolds of the groupoid in question, we find that the connecting map is expressed in terms of an explicit Wodzicki-type residue formula, which involves the jets of non-commutative symbols at the fixed-point set of the action.

The asymptotic Connes–Moscovici characteristic map and the index cocycles

We show that the (even and odd) index cocycles for theta-summable Fredholm modules are in the image of the Connes-Moscovici characteristic map. To show this, we first define a new range of asymptotic cohomologies, and then we extend the Connes-Moscovici characteristic map to our setting. The ordinary periodic cyclic cohomology and the entire cyclic cohomology appear as two instances of this setup. We then construct an asymptotic characteristic class, defined independently from the underlying Fredholm module. Paired with the $K$-theory, the image of this class under the characteristic map yields a non-zero scalar multiple of the index in the even case, and the spectral flow in the odd case.

Tracing cyclic homology pairings under twisting of graded algebras

We give a description of cyclic cohomology and its pairing with K-groups for 2-cocycle deformation of algebras graded over discrete groups. The proof relies on a realization of monodromy for the Gauss–Manin connection on periodic cyclic cohomology in terms of the cup product action of group cohomology.

Transverse noncommutative geometry of foliations

We define an L2-signature for proper actions on spaces of leaves of transversely oriented foliations with bounded geometry. This is achieved by using the Connes fibration to reduce the problem to the case of Riemannian bifoliations where we show that any transversely elliptic first order operator in an appropriate Beals–Greiner calculus, satisfying the usual axioms, gives rise to a semi-finite spectral triple over the crossed product algebra of the foliation by the action, and hence a periodic cyclic cohomology class through the Connes–Chern character. The Connes–Moscovici hypoelliptic signature operator yields an example of such a triple and gives the differential definition of our “L2-signature”. For Galois coverings of bounded geometry foliations, we also define an Atiyah–Connes semi-finite spectral triple which generalizes to Riemannian bifoliations the Atiyah approach to the L2-index theorem. The compatibility of the two spectral triples with respect to Morita equivalence is proven, and by using an Atiyah-type theorem proven in [7], we deduce some integrality results for Riemannian foliations with torsion-free monodromy groupoids.

The Gauss–Manin connection for the cyclic homology of smooth deformations, and noncommutative tori

Given a smooth deformation of topological algebras, we define Getzler’s Gauss–Manin connection on the periodic cyclic homology of the corresponding smooth field of algebras. Basic properties are investigated including the interaction with the Chern–Connes pairing with K -theory. We use the Gauss–Manin connection to prove a rigidity result for periodic cyclic cohomology of Banach algebras with finite weak bidimension. Then we illustrate the Gauss–Manin connection for the deformation of noncommutative tori. We use the Gauss–Manin connection to identify the periodic cyclic homology of a noncommutative torus with that of the commutative torus via a parallel translation isomorphism.We explicitly calculate the parallel translation maps and use them to describe the behavior of the Chern–Connes pairing under this deformation.

Generalized Connes–Chern characters inKK-theory with an application to weak invariants of topological insulators

We use constructive bounded Kasparov [Formula: see text]-theory to investigate the numerical invariants stemming from the internal Kasparov products [Formula: see text], [Formula: see text], where the last morphism is provided by a tracial state. For the class of properly defined finitely-summable Kasparov [Formula: see text]-cycles, the invariants are given by the pairing of [Formula: see text]-theory of [Formula: see text] with an element of the periodic cyclic cohomology of [Formula: see text], which we call the generalized Connes–Chern character. When [Formula: see text] is a twisted crossed product of [Formula: see text] by [Formula: see text], [Formula: see text], we derive a local formula for the character corresponding to the fundamental class of a properly defined Dirac cycle. Furthermore, when [Formula: see text], with [Formula: see text] the algebra of continuous functions over a disorder configuration space, we show that the numerical invariants are connected to the weak topological invariants of the complex classes of topological insulators, defined in the physics literature. The end products are generalized index theorems for these weak invariants, which enable us to predict the range of the invariants and to identify regimes of strong disorder in which the invariants remain stable. The latter will be reported in a subsequent publication.

Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

First, referring to our previous work, 'Hopf cyclic cohomology in braided monoidal categories', we reduce the restriction of the ambient category C being symmetric. We let C to be non-symmetric but assume only the restriction, ψ 2 = id, on the braid map correspond to the Hopf algebra H, which is the main player in the theory. We define a family of examples of such desired braided Hopf algebras, H, living in the category of anyonic vector spaces. Next, on one hand, we will prove that these anyonic Hopf algebras are the enveloping (Hopf) algebras of particular quantum Lie algebras, which we will construct. On the other hand, we will show that, analogous to the non-super and the super case, the well known relationship between the periodic Hopf cyclic cohomology of an enveloping (super) algebra and the (super) Lie algebra homology also holds for these particular quantum Lie algebras.

Cyclic homology for Hom-associative algebras

In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Hom-associative setting.

Index of elliptic operators for diffeomorphisms of manifolds

We develop an elliptic theory for operators associated with a diffeomorphism of a closed smooth manifold. The aim of the present paper is to obtain an index formula for such operators in terms of topological invariants of the manifold and the symbol of the operator. The symbol in this situation is an element of a certain crossed product. We express the index as the pairing of the class in K-theory defined by the symbol and the Todd class in periodic cyclic cohomology of the crossed product.

Smooth Crossed Products of Rieffel’s Deformations

Assume A is a Frechet algebra equipped with a smooth isometric action of a vector group V, and consider Rieffel's deformation A_J of A. We construct an explicit isomorphism between the smooth crossed products V\ltimes\A_J and V\ltimes\A. When combined with the Elliott-Natsume-Nest isomorphism, this immediately implies that the periodic cyclic cohomology is invariant under deformation. Specializing to the case of smooth subalgebras of C*-algebras, we also get a simple proof of equivalence of Rieffel's and Kasprzak's approaches to deformation.

Cyclic cohomology after the excision theorem of Cuntz and Quillen

AbstractThe excision theorem of Cuntz and Quillen established the existence of a six term exact sequence in the bivariant periodic cyclic cohomology HP*(–,–) associated with an arbitrary algebra extension 0 →S→P→Q→ 0. This remarkable result enabled far reaching developments in the purely algebraic periodic cyclic cohomology. It also provided a new formalism that led to the creation of new versions of this theory for topological and bornological algebras. In this article we outline some of the developments that resulted from this breakthrough.

K-theory, cyclic cohomology and pairings for quantum Heisenberg manifolds

The C*-algebras called quantum Heisenberg manifolds (QHMs) were introduced by Rieffel in 1989 as strict deformation quantizations of Heisenberg manifolds. It was later shown that they are also examples of generalized crossed products. In this article, we compute the pairings of K-theory and cyclic cohomology on the QHM. Combining these calculations with other results proved elsewhere, we also determine the periodic cyclic homology and cohomology of these algebras, and obtain explicit bases of the periodic cyclic cohomology of the QHM. We further isolate bases of periodic cyclic homology, expressed as Chern characters of the K-theory.

Cyclic Cocycles on Twisted Convolution Algebras

We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper \'etale groupoids, Tu and Xu provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to a construction of Mathai and Stevenson. When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras. The results in this article were originally published in the author's Ph.D. thesis.

Excision in Bivariant Periodic Cyclic Cohomology: A Categorical Approach

We extend Cuntz-Quillen's excision theorem for algebras and pro-algebras in arbitrary Q-linear categories with tensor product.The excision theorems for the bivariant periodic cyclic cohomology of discrete,topological and bornological algebras and pro-algebras follow from this.

Diffeotopy functors of ind-algebras and local cyclic cohomology

We introduce a new bivariant cyclic theory for topological algebras, called local cyclic cohomology. It is obtained from bivariant periodic cyclic cohomology by an appropriate modification, which turns it into a deformation invariant bifunctor on the stable diffeotopy category of topological ind-algebras. We set up homological tools which allow the explicit calculation of local cyclic cohomology. The theory turns out to be well behaved for Banach- and C^* -algebras and possesses many similarities with Kasparov's bivariant operator K-theory. In particular, there exists a multiplicative bivariant Chern-Connes character from bivariant K-theory to bivariant local cyclic cohomology.

Excision in entire cyclic cohomology

We prove that entire and periodic cyclic cohomology satisfy excision for extensions of bornological algebras with a bounded linear section. That is, for such an extension we obtain a six term exact sequence in cohomology.

Periodic cyclic homology of Iwahori–Hecke algebras

In this Note we explain how to determine the periodic cyclic homology of the Iwahori–Hecke algebras H q , for q∈ C ∗ not a proper root of unity (by a proper root of unity, we shall mean a root of unity other than 1). Our method is based on a general result on periodic cyclic homology, which states that any “weakly spectrum preserving” morphism of finite type algebras induces an isomorphism in periodic cyclic cohomology.

Excision in cyclic homology theories

In this paper we establish excision for Z/2Z-graded cyclic homology theories, including periodic, entire, asymptotic and local bivariant cyclic cohomology. Our work is a modification of the approach of Cuntz-Quillen [CQ] to excision in bivariant periodic cyclic homology and makes also use of the work of Wodzicki [Wo]. We study the dimension shift of periodic cyclic cohomology under the boundary map associated to an extension of algebras and obtain the possible values of this dimension shift. Excision in entire cyclic cohomology is used to calculate these groups in a number of hitherto unknown cases. Finally a bivariant and multiplicative Chern-Connes character on Kasparov’s bivariant K-theory [Ka] with values in bivariant local cyclic cohomology is constructed. To put our approach into perspective we recall the previous work on excision in cyclic homology. Ever since the invention of cyclic homology by Connes and independently by Tsygan the problem of excision played a central role in the theory. It is concerned with the question of to what extent an extension

Periodic cyclic cohomology Chern character for pseudomanifolds with one singular stratum

We compute the periodic cyclic cohomology Chern character of an admissible pseudomanifold X † X^{\dagger } with one singular stratum. As a corollary, we obtain the index theorem and spectral flow for signature operators.

Periodic cyclic cohomology of group rings

We generalize previous results ([2], [3], [4], etc.) relative to the cyclic homology and cohomology of the group algebra of G. In many cases, we express them in terms of the (co)homology of the discrete groups(u) = Z (u)/C (u), where 〈 u〉 runs through the set of conjugacy classes of G and where Z(u) (resp. C(u)) denotes the centralizer of u (resp. the cyclic group generated by u).