Spectral triples on irreversible C*-dynamical systems

Given a spectral triple on a C*-algebra A together with a unital injective endomorphism α, the problem of defining a suitable crossed product C*-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and Hawkins, Skalski, White, and Zacharias, and on our previous papers. The embedding of α(A) in A can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on A and α(A).