There are no fewer than three distinct notions of the term local C^*-algebra used throughout functional analysis. A normed algebra A = (A, left bracketing bar · right bracketing bar _A) is said to be a local C^*-algebra provided that it is a local Banach algebra and that the norm left bracketing bar · right bracketing bar _A is a pre-C^*-norm. An alternative definition most in the spirit of the above identifies a local C^*-algebra to be a pre-C^*-algebra A, each of whose positive elements is contained in a complete C^*-subalgebra A' of A. An algebra satisfying this property is said to admit a functional calculus on its positive elements.

algebra | analytic function | Banach algebra | completion | involution | local Banach algebra | normed space | positive element | subalgebra