Fredholm module

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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[edit]

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[edit]

The paper by Atiyah is reprinted in volume 3 of his collected works, (Atiyah 1988a, 1988b)