JLO cocycle


In noncommutative geometry, the JLO cocycle is a cocycle in entire cyclic cohomology. It is a non-commutative version of the classic Chern character of the conventional differential geometry. In noncommutative geometry, the concept of a manifold is replaced by a noncommutative algebra of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra contains the information about the topology of that noncommutative space, very much as the de Rham cohomology contains the information about the topology of a conventional manifold.
The JLO cocycle is associated with a metric structure of non-commutative differential geometry known as a -summable spectral triple.

\theta-summable spectral triples

A -summable spectral triple consists of the following data:
A Hilbert space such that acts on it as an algebra of bounded operators.
A -grading on,. We assume that the algebra is even under the -grading, i.e., for all.
A self-adjoint operator, called the Dirac operator such that
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.