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.
