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 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, (Atiyah 1988a, 1988b)

• Connes, Alain (1994), Non-commutative geometry (PDF), Boston, MA: Academic Press, ISBN 978-0-12-185860-5
• Atiyah, M. F. (1970), "Global Theory of Elliptic Operators", Proc. Int. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), University of Tokio, Zbl 0193.43601
• Atiyah, Michael (1988a), Collected works. Vol. 3. Index theory: 1, Oxford Science Publications, New York: The Clarendon Press, Oxford University Press, ISBN 0-19-853277-6, MR 0951894