The Gelfand transform of a commutative complex Banach algebra 
is the map 
defined by 
for each character 
on 
. Here a character is
a nonzero complex-linear
functional satisfying
 |
(1)
|
and
 |
(2)
|
where 
and 
.
The characters form the maximal ideal space 
. Each 
is a continuous function
on 
that is vanishing at infinity. With the supremum norm, the Gelfand transform is a bounded
operator from 
to 
that preserves multiplication, and
 |
(3)
|
If 
has a multiplicative identity, then 
is compact
and 
(Taylor, pp. 6-8).
For example, if 
with the usual norm, then 
is a Banach algebra under
convolution and the Gelfand transform is the Fourier
transform. In fact, 
may be replaced by any locally
compact Abelian group, using a Haar
measure to define 
. This Banach
algebra has a multiplicative identity iff the group is a discrete
group.
