Noncommutative topology is a recent program having important and deep applications in several branches of mathematics and mathematical physics. Because every commutative
-algebra
is 
-isomorphic
to 
where 
is the space of maximal ideals of 
(this is the so-called Gelfand theorem) and because an algebraic
isomorphism between 
and 
induces a homeomorphism between 
and 
, 
-algebraic theory may be regarded as a noncommutative analogue
of the algebra of continuous functions vanishing at infinity on a locally compact
T2-space. In other words, every property of a locally
compact T2-space 
can be formulated in terms of a "Gelfand dual" property
of 
and then it will probably be true for any noncommutative 
-algebra.
The following is a list of some such Gelfand dualities (Wegge-Olsen 1993, Moslehian 2002):
| topological language |  -algebraic language |
| locally compact T2-space | C-*-algebra |
| triangulation or the structure of an affine algebraic variety or manifold | system of generators and relations |
| Stone space |  -algebra |
| sub-Stonean space |  -algebra |
| second countable space | projectionless |
| Tychonoff product topology | spatial tensor product |
| proper map |  -homomorphism |
| homeomorphism | automorphism |
| Radon measure | positive linear functional |
| compact space | unital |
 -compact |  -unital |
| compactification | unitization |
| one-point compactification | minimal unitization |
| Stone-Cech compactification | maximal unitization (multiplier algebra) |
| open subset | ideal |
| closed subset | quotient |
| clopen | ideal with unit |
| open dense | essential ideal |
| complement of singleton | maximal ideal (prime ideal) |
| discrete | maximal ideals being principal |
| isolated point | minimal ideal |
| retract closed subspace | complemented ideal |
."
Inter. Math. J. 2, 1055-1059, 2002.Wegge-Olsen, N. E.