In noncommutative geometry

Noncommutative geometry

Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...

, a Fredholm module is a mathematical structure used to quantize the differential calculus

Differential calculus

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

. Such a module is, up to trivial changes, the same as the abstract elliptic operator introduced by .

Definition

If A is an involutive algebra over the complex numbers C, then a Fredholm module over A consists of
an involutive representation of A on a Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

 H, together with a self-adjoint operator

Self-adjoint operator

In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

 F, of square 1 and such that the commutator

Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...

[F, a]

is a compact operator

Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...

, for all a in A.