In
noncommutative geometry, a Fredholm module is a mathematical structure used to quantize the
differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
. 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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''H'', together with a
self-adjoint operator ''F'', of square 1 and such that the
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
The commutator of two elements, a ...
:
'F'', ''a''
is a
compact operator, for all ''a'' in ''A''.
References
The paper by Atiyah is reprinted in volume 3 of his collected works,
*
*
*{{citation, last= Atiyah, first= Michael, authorlink=Michael Atiyah, title= Collected works. Vol. 3. Index theory: 1 , series=Oxford Science Publications, publisher= The Clarendon Press, Oxford University Press, publication-place=New York, year= 1988a, isbn= 0-19-853277-6 , url=https://books.google.com/books?isbn=0198532776, mr= 0951894
External linkFredholm module
on PlanetMath
Noncommutative geometry
Mathematical quantization