Character of an associative algebra

over a field

A non-zero homomorphism of into . A character of the algebra is sometimes also called a multiplicative functional on . Every character is surjective and has the property . The kernel is a maximal ideal in .

If is a finitely generated commutative algebra and if the field is algebraically closed, then any maximal ideal in is the kernel of a unique character, so that the correspondence between characters and maximal ideals is bijective. The collection of all characters of a commutative algebra , its so-called maximal spectrum, has the natural structure of an affine variety. Every element determines a function on , given by the formula , and the functions form the algebra of regular functions on . Conversely, if is an affine variety and is the algebra of regular functions on , then can be identified with : To every point corresponds the character defined by the formula .

The characters of a commutative Banach algebra over have similar properties. Every character is continuous and has norm . Every maximal ideal in is the kernel of a unique character of . The set of all characters, regarded as a subset of the unit ball in endowed with the weak topology, is compact and is called the spectrum of the algebra , and there is a natural homomorphism of into the algebra of continuous functions on . For example, if is the algebra of all complex-valued continuous functions on a compact set , equipped with the norm , then can be identified with : To every element corresponds the character defined by the formula , . A character of a symmetric commutative Banach algebra is called Hermitian if (); is Hermitian if and only if is a symmetric maximal ideal.

References

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