The Gelfand transform 
is defined as follows. If 
is linear and multiplicative in the senses
 |
and
 |
where 
is a commutative Banach algebra, then write 
. The Gelfand transform
is automatically bounded.
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, and
then 
has a unit if and only if the group is discrete.)
