-summable spectral triples
A -summable spectral triple consists of the following data:
(a) A Hilbert space such that acts on it as an algebra of bounded operators.
(b) A -grading on , . We assume that the algebra is even under the -grading, i.e. , for all .
(c) A self-adjoint (unbounded) operator , called the Dirac operator such that
- (i) is odd under , i.e. .
- (ii) Each maps the domain of , into itself, and the operator is bounded.
- (iii) , for all .
A classic example of a -summable spectral triple arises as follows. Let be a compact spin manifold, , the algebra of smooth functions on , the Hilbert space of square integrable forms on , and the standard Dirac operator.
The cocycle
The JLO cocycle is a sequence
of functionals on the algebra , where
for . The cohomology class defined by is independent of the value of .