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) |
Character of an associative algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_of_an_associative_algebra&oldid=46315