JLO Cocycle - -summable Fredholm Modules

-summable Fredholm Modules

A -summable Fredholm module 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 Fredholm module 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.