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.
gollark: Fun fact: getting to sleep is very hard.
gollark: That's what I do!
gollark: No clue, this is hard.
gollark: Anyway. A replay attack could happen if your system encrypts "open the door" as, say, "a" constantly and "close the door" as "b" constantly. While the message is technically secure in that they can't arbitrarily encrypt a value, if someone wants to open the door they can just send "a".
gollark: Yep!
References
The paper by Atiyah is reprinted in volume 3 of his collected works, (Atiyah 1988a, 1988b)
- Connes, Alain (1994), Non-commutative geometry, 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
External link
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.