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Register a new userNonunital spectral triples and metric completeness in unbounded KK-theory
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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.
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We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $theta$-deformations. In particular we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).
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.
We prove two results about nonunital index theory left open by [CGRS2]. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow.
Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero. Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem. Denote by ${{KK}}_e(C,A)^{++}$ the set of those $kappa$ for which $kappa(K_0(C)_+setminus{0})subset K_0(A)_+setminus{0}$ and $kappa([1_C])=[1_A]$. Suppose that $kappain {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $phi: Cto A$ such that $[phi]=kappa.$ Suppose that $C$ is a unital AH-algebra and $lambda: mathrm{T}(A)to mathrm{T}_{mathtt{f}}(C)$ is a continuous affine map for which $tau(kappa([p]))=lambda(tau)(p)$ for all projections $p$ in all matrix algebras of $C$ and any $tauin mathrm{T}(A),$ where $mathrm{T}(A)$ is the simplex of tracial states of $A$ and $mathrm{T}_{mathtt{f}}(C)$ is the convex set of faithful tracial states of $C.$ We prove that there is a unital monomorphism $phi: Cto A$ such that $phi$ induces both $kappa$ and $lambda.$ Suppose that $h: Cto A$ is a unital monomorphism and $gamma in mathrm{Hom}(Kone(C), aff(A)).$ We show that there exists a unital monomorphism $phi: Cto A$ such that $[phi]=[h]$ in ${KK}(C,A),$ $taucirc phi=taucirc h$ for all tracial states $tau$ and the associated rotation map can be given by $gamma.$ Applications to classification of simple C*-algebras are also given.
In the founding paper on unbounded $KK$-theory it was established by Baaj and Julg that the bounded transform, which associates a class in $KK$-theory to any unbounded Kasparov module, is a surjective homomorphism (under a separability assumption). In this paper, we provide an equivalence relation on unbounded Kasparov modules and we thereby describe the kernel of the bounded transform. This allows us to introduce a notion of topological unbounded $KK$-theory, which becomes isomorphic to $KK$-theory via the bounded transform. The equivalence relation is formulated entirely at the level of unbounded Kasparov modules and consists of homotopies together with an extra degeneracy condition. Our degenerate unbounded Kasparov modules are called spectrally decomposable since they admit a decomposition into a part with positive spectrum and a part with negative spectrum.
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