Smoothness and Locality for Nonunital Spectral Triples

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The Noncommutative Geometry of k-graph C*-Algebras (2005, background): “ quasi-local algebra (when A c is understood.) If A c is a dense ideal with local units, we call A c ⊂ A local. Quasi-local algebras have an approximate unit {φ n} n≥1 ⊂ A c such that φ n+1φ n = φ n, [R1]. Example For a k-graph C∗-algebra A= C∗(Λ), Equation (1) shows that A c = span{S µS ∗ ν : µ,ν∈ ”
Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars (2014, background): “However, the terminology ‘regular spectral triple’ is often used in the literature with a different meaning, related to ‘smoothness’ properties (see for instance [48]).”
The noncommutative geometry of graph C*-algebras I: The index theorem (2005, background): “Quasi-local algebras have an approximate unit {φn}n≥1 ⊂ Ac such that for all n, φn+1φn = φn, [30]; we call this a local approximate unit.”
What is geometry in quantum theory (2004, background): “Moreover, this isomorphism is a categorial equivalence [92], and its variant takes place if X is locally compact [93].”
Moyal Planes are Spectral Triples (2003, background): “The algebras Aθ are not known to have bilateral ideals; it is very likely that they are simple, and if so, they would not be local in the sense of [73], either (thus, it is not clear if Rennie’s device can carry the full weight of noncommutative spin geometry).”