A Banach algebra is an algebra B over a field F endowed with a norm left double bracketing bar · right double bracketing bar such that B is a Banach space under the norm left double bracketing bar · right double bracketing bar and left double bracketing bar x y right double bracketing bar <= left double bracketing bar x right double bracketing bar left double bracketing bar y right double bracketing bar . F is frequently taken to be the complex numbers in order to ensure that the operator spectrum fully characterizes an operator (i.e., the spectral theorems for normal or compact normal operators do not, in general, hold in the operator spectrum over the real numbers). If B is commutative and has a unit, then x element B is invertible iff x^^(ϕ)!=0 for all ϕ, where x↦x^^ is the Gelfand transform.
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