Exact sequences for locally convex subalgebras of Pimsner algebras with an application to Quantum Heisenberg Manifolds

TL;DR

This paper establishes six-term exact sequences for specific subalgebras of Pimsner algebras, enabling computations of cyclic cohomology generators for Quantum Heisenberg Manifolds and generalizing known results for smooth crossed products.

Contribution

It introduces new six-term exact sequences for subalgebras of Pimsner-Voiculescu type, extending their application to Quantum Heisenberg Manifolds and related structures.

Findings

Derived six-term exact sequences for subalgebras of Pimsner algebras

Applied sequences to compute cyclic cohomology generators of Quantum Heisenberg Manifolds

Unified approach encompassing known results for smooth crossed products

Abstract

We prove six-term exact sequences of Pimsner-Voiculescu type for certain subalgebras of the Cuntz-Pimsner algebras. 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. Further, our results include the known results for smooth crossed products. Our proof is based on a combination of arguments from the setting of (Cuntz-)Pimsner algebras and the Toeplitz proof of Bott-periodicity.