A local Banach algebra is a normed algebra 
which satisfies the following
properties:
1. If 
and 
is an analytic function on a neighborhood of
the spectrum of 
in the completion of 
, with 
if 
is non-unital, then 
.
2. All matrix algebras over 
satisfy property (1) above.
Here, if 
is a *-algebra, then it will be called a local Banach
-algebra;
similarly, if 
is a pre-
-norm,
then 
is called a local C-*-algebra (though different
literature uses the term "local 
-algebra" to refer to different structures altogether).
An algebra satisfying condition (1) above is said to be closed under holomorphic functional calculus.
