A local Banach algebra is a normed algebra A = (A, left bracketing bar · right bracketing bar _A) which satisfies the following properties: 1. If x element A and f is an analytic function on a neighborhood of the spectrum of x in the completion of A, with f(0) = 0 if A is non-unital, then f(x) element A. 2. All matrix algebras over A satisfy property (1) above. Here, if A is a *-algebra, then it will be called a local Banach *-algebra; similarly, if left bracketing bar · right bracketing bar _A is a pre-C^*-norm, then A is called a local C^*-algebra (though different literature uses the term "local C^*-algebra" to refer to different structures altogether).
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