Gelfand-Naimark theorem
Let be the category of locally compact Hausdorff spaces
with continuous proper maps as morphisms.
And, let be the category of commutative
-algebras
with proper *-homomorphisms
(send approximate units
into approximate units)
as morphisms.
There is a contravariant functor
which sends each locally compact Hausdorff space to the commutative -algebra ( if is compact
).
Conversely, there is a contravariant functor which sends each commutative -algebra to the space of characters
on (with the Gelfand topology
).
The functors and are an equivalence of categories.
Title | Gelfand-Naimark theorem![]() |
---|---|
Canonical name | GelfandNaimarkTheorem |
Date of creation | 2013-03-22 13:29:28 |
Last modified on | 2013-03-22 13:29:28 |
Owner | mhale (572) |
Last modified by | mhale (572) |
Numerical id | 5 |
Author | mhale (572) |
Entry type | Theorem |
Classification | msc 46L85 |