A self Morita equivalence over an algebra B, given by a B-bimodule E, is thought of as a line bundle over B. The corresponding Pimsner algebra OE is then the total space algebra of a noncommutative principal circle bundle over B. A natural Gysin-like sequence relates the KK-theories of OE and of B. Interesting examples come from OE a quantum lens space over B a quantum weighted projective line (with arbitrary weights). The KK-theory of these spaces is explicitly computed and natural generators are exhibited.

/pdf/pimsner-algebras-and-gysin-sequences-from-principal-circle-5243utlyx0.pdf

Pimsner algebras and Gysin sequences from principal circle actions

© Cambridge University Press, 2016 We show how the fine structure in shift–tail equivalence, appearing in the non-commutative geometry of Cuntz–Krieger algebras developed by the first two listed authors, has an analogue in a wide range of other Cuntz–Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz–Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third listed author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz and Cuntz–Krieger algebras and for Cuntz–Pimsner algebras associated to vector bundles twisted by an equicontinuous (Formula presented.)-automorphism.

/pdf/shift-tail-equivalence-and-an-unbounded-representative-of-3q002nn3d8.pdf

Shift-tail equivalence and an unbounded representative of the Cuntz-Pimsner extension

We give a construction allowing to lift spectral triples to crossed products by Hilbert bimodules. The spectral triple one obtains is a concrete unbounded representative of the Kasparov product of the spectral triple and the Pimsner-Toeplitz extension associated to the crossed product by the Hilbert bimodule. To prove that the lifted spectral triple is the above-mentioned Kasparov product, we rely on operator-$*$-algebras and connexions.

Spectral Triples and Generalized Crossed Products

Given a spectral triple $(A,H,D)$ and a $C^*$-dynamical system $(\mathbf{A}, G, \alpha)$ where $A$ is dense in $\mathbf{A}$ and $G$ is a locally compact group, we extend the triple to a triplet $(\mathcal{B},\mathcal{H},\mathcal{D})$ on the crossed product $G \ltimes_{\alpha, \text{red}} \mathbf{A}$ which can be promoted to a modular-type twisted spectral triple within a general procedure exemplified by two cases: the $C^*$-algebra of the affine group and the conformal group acting on a complete Riemannian spin manifold.

https://www.ems-ph.org/journals/show_pdf.php?issn=1661-6952&vol=10&iss=1&rank=3

Crossed product extensions of spectral triples

A reconstruction theorem for Connes–Landi deformations of commutative spectral triples

In this article, we give a general construction of spectral triples from certain Lie group actions on unital C ∗ -algebras. If the group G is compact and the action is ergodic, we actually obtain a real and finitely summable spectral triple which satisfies the first order condition of Connes’ axioms. This provides a link between the “algebraic” existence of ergodic action and the “analytic” finite summability property of the unbounded selfadjoint operator. More generally, for compact G we carefully establish that our (symmetric) unbounded operator is essentially selfadjoint. Our results are illustrated by a host of examples – including noncommutative tori and quantum Heisenberg manifolds.

/pdf/ergodic-actions-and-spectral-triples-1g9y7el1f0.pdf

Ergodic Actions and Spectral Triples

We define smooth generalized crossed products and prove six-term exact sequences of Pimsner-Voiculescu type. This sequence may, in particular, be applied to smooth subalgebras of the Quantum Heisenberg Manifolds in order to compute the generators of their cyclic cohomology. Our proof is based on a combination of arguments from the setting of (Cuntz-)Pimsner algebras and the Toeplitz proof of Bott-periodicity.

https://www.ems-ph.org/journals/show_pdf.php?issn=1661-6952&vol=7&iss=2&rank=7

Six-term exact sequences for smooth generalized crossed products

Using an appropriate notion of locally convex Kasparov modules, we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras; examples are obtained from spectral triples. Our considerations are based on the action of algebraic K-theory on these functors, and involve compatibility properties of the induction process with this action, and with Kasparov-type products. This is based on an appropriate interpretation of the Connes-Skandalis connection formalism. As an application, we prove Bott periodicity and a Thom isomorphism for algebras of Schwartz functions.

Universal cycles and homological invariants of locally convex algebras

Six-Term Exact Sequences for Smooth Generalized Crossed Products

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