Fredholm module
In noncommutative geometry, a Fredholm module is a mathematical structure used to quantize the differential calculus. Such a module is, up to trivial changes, the same as the abstract elliptic operator introduced by {{harvtxt|Atiyah|1970}}.
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 H, together with a self-adjoint operator F, of square 1 and such that the commutator
:[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, {{harvs|last=Atiyah|year1=1988a|year2=1988b}}
- {{Citation | last1=Connes | first1=Alain | author1-link=Alain Connes | title=Non-commutative geometry | url=https://archive.org/details/noncommutativege0000conn | publisher=Academic Press | location=Boston, MA | isbn=978-0-12-185860-5 | year=1994 | url-access=registration }}
- {{citation|last=Atiyah|first= M. F.|authorlink=Michael Atiyah
|chapter=Global Theory of Elliptic Operators|title= Proc. Int. Conf. on Functional Analysis and Related Topics (Tokyo, 1969)|year= 1970|publisher=University of Tokio|zbl=0193.43601}}
- {{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 links
- [https://planetmath.org/fredholmmodule Fredholm module], on PlanetMath