There are no fewer than three distinct notions of the term local 
-algebra used throughout functional analysis.
A normed algebra 
is said to be a local
-algebra
provided that it is a local Banach algebra
and that the norm 
is a pre-
-norm (Blackadar 1998).
An alternative definition most in the spirit of the above identifies a local 
-algebra to be a pre-
-algebra 
, each of whose positive elements
is contained in a complete 
-subalgebra 
of 
(Blackadar and Handelman 1982). An algebra satisfying this
property is said to admit a functional calculus on its positive elements.
Elsewhere in the literature, one finds that a complex normed *-algebra 
is called a local 
-algebra if all its maximal commutative 
-subalgebras are themselves 
-algebras with the given norm and involution 
(Behncke and Cuntz 1977). Here, maximality
of a 
-subalgebra
of 
is defined to mean that 
is a closed subalgebra of
.
-Algebras." J. Functional Anal. 45, 297-340,
1982.