Editing Banach algebra (section)

==Ideals and characters==

Let <math>A</math> be a unital ''commutative'' Banach algebra over <math>\Complex.</math> Since <math>A</math> is then a commutative ring with unit, every non-invertible element of <math>A</math> belongs to some [[maximal ideal]] of <math>A.</math> Since a maximal ideal <math>\mathfrak m</math> in <math>A</math> is closed, <math>A / \mathfrak m</math> is a Banach algebra that is a field, and it follows from the Gelfand–Mazur theorem that there is a bijection between the set of all maximal ideals of <math>A</math> and the set <math>\Delta(A)</math> of all nonzero homomorphisms from <math>A</math> to <math>\Complex.</math> The set <math>\Delta(A)</math> is called the "[[structure space]]" or "character space" of <math>A,</math> and its members "characters".

A character <math>\chi</math> is a linear functional on <math>A</math> that is at the same time multiplicative, <math>\chi(a b) = \chi(a) \chi(b),</math> and satisfies <math>\chi(\mathbf{1}) = 1.</math>  Every character is automatically continuous from <math>A</math> to <math>\Complex,</math> since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (that is, operator norm) of a character is one. Equipped with the topology of pointwise convergence on <math>A</math> (that is, the topology induced by the weak-* topology of <math>A^*</math>), the character space, <math>\Delta(A),</math> is a Hausdorff compact space.

For any <math>x \in A,</math>
<math display=block>\sigma(x) = \sigma(\hat x)</math>
where <math>\hat x</math> is the [[Gelfand representation]] of <math>x</math> defined as follows: <math>\hat x</math> is the continuous function from <math>\Delta(A)</math> to <math>\Complex</math> given by <math>\hat x(\chi) = \chi(x).</math>  The spectrum of <math>\hat x,</math> in the formula above, is the spectrum as element of the algebra <math>C(\Delta(A))</math> of complex continuous functions on the compact space <math>\Delta(A).</math>  Explicitly,
<math display=block>\sigma(\hat x) = \{\chi(x) : \chi \in \Delta(A)\}.</math>

As an algebra, a unital commutative Banach algebra is [[semisimple algebra|semisimple]] (that is, its [[Jacobson radical]] is zero) if and only if its Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when <math>A</math> is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between <math>A</math> and <math>C(\Delta(A)).</math>{{efn-la|Proof: Since every element of a commutative C*-algebra is normal, the Gelfand representation is isometric; in particular, it is injective and its image is closed. But the image of the Gelfand representation is dense by the [[Stone–Weierstrass theorem]].}}