The term local 
-algebra
has no fewer than three distinct definitions 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-C*-algebra 
, each of whose positive
elements is contained in a complete subalgebra 
of 
that is a C*-algebra
(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 C*-algebras with
the given norm and involution 
(Behncke and Cuntz 1977). Here, maximality
of a 
-subalgebra 
of 
is defined to mean that 
is not properly contained in any larger commutative 
-subalgebra
of 
.
-Algebras." J. Funct. Anal. 45, 297-340,
1982.