JLO cocycle
In noncommutative geometry, the JLO cocycle is a cocycle (and thus defines a cohomology class) 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 (also known as a -summable Fredholm module). -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 .