To deal with technical issues in noncommuntative geometry for nonunital algebras, we introduce a useful class of algebras and their modules. Thes algebras and modules allo us to extend all of the smoothness results for spectral triples to the nonunital case. In addition, we show that smooth spectral tiples are closed under the C- functional calculus of self-adjoint elements. In the final section we show that our algebras allow the formulation of Poincare Duality and that the algebras of smooth spectral triples are H-unital.
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Rennie, A. C. (2003). Smoothness and locality for nonunital spectral triples. K-Theory: interdisciplinary journal for the development, application and influence of K-theory in the mathematical sciences, 28 (2), 127-165.