Why is it called *spectral* triple? I know the definition a spectral triple and that it is some kind of non-commutative generalisation of (the ring of functions on) a compact spin manifold.
But, why is it called spectral triple?
 A: Well, it uses the spectral properties of the Dirac operator $D$ in the spectral triple quite extensively.
Also, in the article where he (essentially) introduces the notion of spectral triples ( http://www.alainconnes.org/docs/reality.pdf ) Alain Connes writes about the naming:
"We shall need for that purpose to adapt the tools of the differential and integral calculus
to our new framework. This will be done by building a long dictionary which relates the usual
calculus (done with local differentiation of functions) with the new calculus which will be done
with operators in Hilbert space and spectral analysis, commutators.... The first two lines of the
dictionary give the usual interpretation of variable quantities in quantum mechanics as operators in
Hilbert space. For this reason and many others (which include integrality results) the new calculus
can be called the quantized calculus’ but the reader who has seen the word “quantized” overused
so many times may as well drop it and use “spectral calculus” instead."
