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)
How to Cite This Entry:
Character of an associative algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_of_an_associative_algebra&oldid=46315
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article