Let 
be a 
-algebra having no unit. Then 
as a vector spaces together with
1. 
.
2. 
.
3. 
.
4. 
.
5. 
.
is a 
-algebra with the identity 
and also 
is an isometrically 
-isomorphism from 
into 
. The algebra 
is called the unitization of 
.
For example, the minimal unitization of 
-algebra 
of continuous complex-valued functions on 
vanishing at infinity is the 
-algebra 
of continuous complex-valued functions on the compact
space 
,
where 
is the one point compactification of 
(Wegge-Olsen 1993).
