Character of an associative algebra

$ A $ over a field $ k $

A non-zero homomorphism of $ A $ into $ k $. A character of the algebra $ A $ is sometimes also called a multiplicative functional on $ A $. Every character $ \chi : A \rightarrow k $ is surjective and has the property $ \chi ( 1) = 1 $. The kernel $ \mathop{\rm Ker} \chi $ is a maximal ideal in $ A $.

If $ A $ is a finitely generated commutative algebra and if the field $ k $ is algebraically closed, then any maximal ideal in $ A $ is the kernel of a unique character, so that the correspondence between characters and maximal ideals is bijective. The collection $ \mathop{\rm Specm} A $ of all characters of a commutative algebra $ A $, its so-called maximal spectrum, has the natural structure of an affine variety. Every element $ a \in A $ determines a function $ \widetilde{a} $ on $ \mathop{\rm Specm} A $, given by the formula $ \widetilde{a} ( \chi ) = \chi ( a) $, and the functions $ \widetilde{a} $ form the algebra of regular functions on $ \mathop{\rm Specm} A $. Conversely, if $ X $ is an affine variety and $ A $ is the algebra of regular functions on $ X $, then $ \mathop{\rm Specm} A $ can be identified with $ X $: To every point $ x \in X $ corresponds the character $ \chi _ {x} $ defined by the formula $ \chi _ {x} ( a) = a ( x) $.

The characters of a commutative Banach algebra $ A $ over $ \mathbf C $ have similar properties. Every character $ \chi : A \rightarrow \mathbf C $ is continuous and has norm $ \| \chi \| \leq 1 $. Every maximal ideal in $ A $ is the kernel of a unique character of $ A $. The set $ \Phi ( A) $ of all characters, regarded as a subset of the unit ball in $ A ^ {*} $ endowed with the weak topology, is compact and is called the spectrum of the algebra $ A $, and there is a natural homomorphism of $ A $ into the algebra of continuous functions on $ \Phi ( A) $. For example, if $ A $ is the algebra of all complex-valued continuous functions on a compact set $ X $, equipped with the norm $ \| f \| = \max _ {X} | f | $, then $ \Phi ( A) $ can be identified with $ X $: To every element $ x \in X $ corresponds the character $ \chi _ {x} $ defined by the formula $ \chi _ {x} ( f) = f ( x) $, $ f \in A $. A character $ \chi $ of a symmetric commutative Banach algebra $ A $ is called Hermitian if $ \chi ( a ^ {*} ) = \chi ( a) $( $ a \in A $); $ \chi $ is Hermitian if and only if $ \mathop{\rm Ker} \chi $ is a symmetric maximal ideal.

References

[1]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)