Editing Banach algebra (section)

==Spectral theory==
{{Main|Spectral theory}}

Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The ''spectrum'' of an element <math>x \in A,</math> denoted by <math>\sigma(x)</math>, consists of all those complex [[scalar (mathematics)|scalar]]s <math>\lambda</math> such that <math>x - \lambda \mathbf{1}</math> is not invertible in <math>A.</math>  The spectrum of any element <math>x</math> is a closed subset of the closed disc in <math>\Complex</math> with radius <math>\|x\|</math> and center <math>0,</math> and thus is  [[Compact space|compact]]. Moreover, the spectrum <math>\sigma(x)</math> of an element <math>x</math> is [[non-empty]] and satisfies the [[spectral radius]] formula:
<math display=block>\sup \{|\lambda| : \lambda \in \sigma(x)\} = \lim_{n \to \infty} \|x^n\|^{1/n}.</math>

Given <math>x \in A,</math> the [[holomorphic functional calculus]] allows to define <math>f(x) \in A</math> for any function <math>f</math> [[holomorphic function|holomorphic]] in a neighborhood of <math>\sigma(x).</math>  Furthermore, the spectral mapping theorem holds:<ref>{{harvnb|Takesaki|1979|loc=Proposition 2.8.}}</ref>
<math display=block>\sigma(f(x)) = f(\sigma(x)).</math>

When the Banach algebra <math>A</math> is the algebra <math>L(X)</math> of bounded linear operators on a complex Banach space <math>X</math> (for example, the algebra of square matrices), the notion of the spectrum in <math>A</math> coincides with the usual one in [[operator theory]]. For <math>f \in C(X)</math> (with a compact Hausdorff space <math>X</math>), one sees that:
<math display=block>\sigma(f) = \{f(t) : t \in X\}.</math>

The norm of a normal element <math>x</math> of a C*-algebra coincides with its spectral radius.  This generalizes an analogous fact for normal operators.

Let <math>A</math> be a complex unital Banach algebra in which every non-zero element <math>x</math> is invertible (a division algebra).  For every <math>a \in A,</math> there is <math>\lambda \in \Complex</math> such that
<math>a - \lambda \mathbf{1}</math> is not invertible (because the spectrum of <math>a</math> is not empty) hence <math>a = \lambda \mathbf{1}:</math> this algebra <math>A</math> is naturally isomorphic to <math>\Complex</math> (the complex case of the Gelfand–Mazur theorem).