Spectral Triples and Generalized Crossed Products

Abstract

We give a construction for lifting 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 module. To prove that the lifted spectral triple is the above-mentioned Kasparov product, we rely on operator- *algebras and connexions.

The extension class and KMS states for Cuntz–Pimsner algebras of some bi-Hilbertian bimodules

“…The bounded Kasparov module representing the extension in this case is implicit in Pimsner [35], and numerous similar constructions of Kasparov modules associated to circle actions have appeared in [2,5,30,34] amongst others. Similar results for the unbounded Kasparov module were obtained in [15]. The Fock module associated to C * -bimodules E over a C *algebra A is defined as…”

Pimsner Algebras and Circle Bundles

“…For commutative algebras this connection was already in [13,Prop. 5.8] with the following result: Proposition 4.6.…”

“…Pimsner algebras, which were introduced in the seminal work [22], provide a unifying framework for a range of important C * -algebras including crossed products by the integers, Cuntz-Krieger algebras [8,9], and C * -algebras associated to partial automorphisms [11]. Due to their flexibility and wide range of applicability, there has been recently an increasing interest in these algebras (see for instance [13,24]). A related class of algebras, known as generalized crossed products, was independently invented in [1].…”

Lifting spectral triples to noncommutative principal bundles

“…In [26] Gabriel and Grensing proposed a construction of spectral triples for a class of crossed product-like algebras that relies on a noncommutative version of the geometric notion of a connection. It gives an unbounded representative of the Kasparov product of the original spectral triple and the Pimsner-Toeplitz extension associated with the crossed product by a Hilbert module.…”

“…This and the fact that the cocycle of the factor system takes values in C • 1 A 2 θ entail that the hypotheses of Theorem 5.9 are fulfilled. Furthermore, coming back to Remark 5.2, it is an easy task to show that Equation (26) implies that Ω 1 (A 2 θ ) naturally embeds into Ω 1 (A 4 θ ).…”