Character and maximal ideal space

An associative complex algebra is a complex vector space $A$ (Vector space over a field) equipped with a multiplication $A\times A\to A$, $(a,b)\mapsto ab$, which is complex-bilinear and associative: $(\lambda a+\mu b)c=\lambda ac+\mu bc$, $a(\lambda b+\mu c)=\lambda ab+\mu ac$, and $(ab)c=a(bc)$ for all $a,b,c\in A$ and $\lambda ,\mu \in \mathbb{C}$. No multiplicative identity is assumed, and no scalar-algebra or real-algebra convention is imported here; all algebras in this page are complex and the only structure used below is the one just displayed.

A character on $A$ is a map $\chi :A\to \mathbb{C}$ which is nonzero, complex-linear (Linear map between vector spaces over the same field) and multiplicative:

$\chi (\lambda a+\mu b)=\lambda \chi (a)+\mu \chi (b),\chi (ab)=\chi (a)\chi (b)(a,b\in A,\lambda ,\mu \in \mathbb{C})\mathrm{.}$

Two things are deliberately not part of the definition:

• continuity is not assumed; for a unital Banach algebra it is a theorem below, and for a commutative Banach algebra it then follows for free;
• preservation of a unit is not assumed either. If $A$ happens to have an identity $1$, a character is not required to satisfy $\chi (1)=1$ by definition; for a unital Banach algebra this too is proved later on this page.

The character space of $A$ is

$\mathrm{\Delta }(A):=\{\chi :\chi \text{ is a character on }A\},$

a set by Separation, since every character is a subset of $A\times \mathbb{C}$ and $A\times \mathbb{C}$ is a set. Initially $\mathrm{\Delta }(A)$ carries the topology of pointwise evaluation: the coarsest topology for which all the evaluation maps $e_a:\mathrm{\Delta }(A)\to \mathbb{C}$, $e_a(\chi ):=\chi (a)$, are continuous. Equivalently, a basic neighbourhood of ${\chi }_0$ is $\{\chi :\mathrm{\mid }\chi (a_i)-{\chi }_0(a_i)\mathrm{\mid }<\epsilon ,i\le k\}$ for finitely many $a_i\in A$ and $\epsilon >0$. Once characters are known to be bounded linear functionals they are points of the dual $A^{\ast }$, and this topology is exactly the subspace topology induced by the weak-star topology $\sigma (A^{\ast },A)$ (as proved later on this page); no duality theory is used before that point. For a nonzero commutative unital Banach algebra, under the Axiom of Choice, $\mathrm{\Delta }(A)$ is also called the maximal ideal space: only in that setting does the later maximal-ideal correspondence identify its points with all maximal ideals.

• Why "nonzero" is part of the definition. The zero map is linear and multiplicative and would otherwise be a character of every algebra; excluding it is what makes characters the algebraic counterparts of points, and it is used already in the first unitality computation.
• The empty character space is allowed here. For an algebra with no characters at all $\mathrm{\Delta }(A)=\mathrm{\varnothing }$ is a perfectly good value of the definition; compact Hausdorffness and nonemptiness of $\mathrm{\Delta }(A)$ are theorems requiring a nonzero commutative unital Banach algebra.
• A character need not exist. For a general associative complex algebra nothing in this definition produces a character, and the existence statements below spend the Axiom of Choice precisely there.