Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTwisted cyclic theory, equivariant KK theory and KMS States
445
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Recently, examples of an index theory for KMS states of circle actions were discovered, cite{CPR2,CRT}. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C*-algebra A to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in cite{CPR2,CRT} as special cases. Next we use the Araki-Woods III_lambda representations of the Fermion algebra to show that there are examples which are not Cuntz-Krieger systems.
rate research
Read More
By considering the general properties of approximate units in differentiable algebras, we are able to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, and the lifting of Kasparov products to the unbounded category. In particular, by strengthening Kasparovs technical theorem, we show that given any two composable KK-classes, we can find unbounded representatives whose product can be constructed to yield an unbounded representative of the Kasparov product.
We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish grad
This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with the Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on the Cuntz algebra. We introduce a modified $K_1$-group of the Cuntz algebra so as to pair with this twisted cocycle. As a corollary we obtain a noncommutative geometry interpretation for Arakis notion of relative entropy in this example. We also note the connection of this example to the theory of noncommutative manifolds.
We give a new construction of the equivariant $K$-theory of group actions (cf. Barwick et al.), producing an infinite loop $G$-space for each Waldhausen category with $G$-action, for a finite group $G$. On the category $R(X)$ of retractive spaces over a $G$-space $X$, this produces an equivariant lift of Waldhausens functor $A(X)$, and we show that the $H$-fixed points are the bivariant $A$-theory of the fibration $X_{hH}to BH$. We then use the framework of spectral Mackey functors to produce a second equivariant refinement $A_G(X)$ whose fixed points have tom Dieck type splittings. We expect this second definition to be suitable for an equivariant generalization of the parametrized $h$-cobordism theorem.
We extend the new perturbation formula of equilibrium states by Hastings to KMS states of general $W^*$-dynamical systems.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
