Character of an algebra

A character on a complex algebra is a nonzero multiplicative linear functional $\phi :A\to \mathbb{C}$.

Algebra characters form the domain of the Gelfand representation; they are distinct from traces of group representations.

In a commutative complex unital Banach algebra, a maximal ideal is closed: its closure remains proper, since an element within distance less than one of the identity is invertible by the Neumann series. The quotient is a complex division Banach algebra, hence is the complex field by the Gelfand-Mazur theorem. Its quotient map is a character of an algebra. Conversely a nonzero multiplicative complex linear functional has value one at the identity, is onto the complex field, and has maximal kernel. It is automatically continuous because its value at each element belongs to that element's spectrum. The kernel determines the algebra character uniquely since the quotient is one-dimensional.

In a commutative complex unital Banach algebra, a maximal ideal is closed: its closure remains proper, since an element within distance less than one of the identity is invertible by the Neumann series. The quotient is a complex division Banach algebra, hence is the complex field by the Gelfand-Mazur theorem. Its quotient map is a character of an algebra. Conversely a nonzero multiplicative complex linear functional has value one at the identity, is onto the complex field, and has maximal kernel. It is automatically continuous because its value at each element belongs to that element's spectrum. The kernel determines the character uniquely since the quotient is one-dimensional.

For a self-adjoint element $h$, the exponentials $\exp (ith)$ are unitary. A continuous algebra character therefore satisfies $|\exp (it\varphi (h))|\le 1$ for every real $t$, forcing $\varphi (h)$ to be real. Decomposing an arbitrary element into real and imaginary self-adjoint parts proves the displayed identity. Thus every algebra character is a star-preserving scalar homomorphism; commutativity of the whole algebra is not needed for this fact.

For a character of an algebra $\phi$ on a complex unital Banach algebra, $\phi (1)=1$ and $\phi (a)\in \sigma (a)$, so $|\phi (a)|\le \Vert a\Vert$ without any prior continuity assumption. Under the usual normalization $\Vert 1\Vert =1$, this gives $\Vert \phi \Vert =1$; for an unnormalized algebra norm, only the upper bound is automatic.

The character space ${\Phi }_A$ is the set of all characters on $A$.

For a commutative complex unital Banach algebra, maximal ideals correspond bijectively to characters of an algebra by taking kernels. Give the maximal ideal space the Gelfand topology through this correspondence. It is a compact Hausdorff space, since the characters form a weak-star closed subset of the dual unit ball, which is compact by the Banach-Alaoglu theorem.

For a compact set $K\subset \mathbb{C}$, let $\mathcal{R}(K)$ be the uniform closure on $K$ of the rational functions with no poles on $K$. Every character of $\mathcal{R}(K)$ is evaluation at a unique point of $K$, so ${\Phi }_{\mathcal{R}(K)}$ is naturally homeomorphic to $K$.

If $X=Y\oplus Z$ and the Banach spaces $Y$ and $Z$ are isomorphic, then the operator algebra $\mathcal{B}(X)$ has no characters. The two complementary projections have unequal character values, while mutually inverse off-diagonal maps force those values to be equal.

The Gelfand topology on ${\Phi }_A$ is the weak-star topology inherited from $A^{\ast }$, equivalently the coarsest topology making every map $\phi \mapsto \phi (a)$ continuous.

This is the topology on the character space used in the Gelfand representation.

For an algebra of functions on a space, evaluation at $x$ is the character ${\delta }_x(f)=f(x)$.

For a unital commutative complex Banach algebra $A$, the Gelfand transform sends $a\in A$ to the continuous function $\hat{a}(\phi )=\phi (a)$ on its character space of an algebra. It is a contractive unital algebra homomorphism into $C({\Phi }_A)$.

The complex dual numbers $\mathbb{C}[\epsilon ]/({\epsilon }^2)$, with norm $\Vert a+b\epsilon \Vert =|a|+|b|$ and involution $a+b\epsilon \mapsto \bar{a}+\bar{b}\epsilon$, is a commutative unital Banach algebra with an isometric involution. Its sole character of an algebra kills $\epsilon$, so the Gelfand transform is neither injective nor isometric. The C-star identity excludes this example and gives the Commutative Gelfand--Naimark theorem. Even preservation of the involution on characters can fail without that identity: on ${\mathbb{C}}^2$, the involution $(a,b{)}^{\ast }=(\bar{b},\bar{a})$ exchanges the two evaluation characters.

For a commutative complex unital Banach algebra, a character of an algebra annihilates $a-\varphi (a)1$, so this element cannot be invertible. Conversely a noninvertible $a-\lambda 1$ generates a proper ideal, which lies in a maximal ideal. The corresponding character then has value $\lambda$ at $a$. Thus the Gelfand transform determines every element's spectrum even when the algebra is not semisimple.

For a commutative complex unital Banach algebra, its Gelfand transform is a continuous unital algebra homomorphism into the continuous functions on its compact character space. The values of the transform of $a$ are exactly ${\sigma }_A(a)$, so $\Vert \hat{a}{\Vert }_{\infty }=r(a)$. Its kernel is the intersection of the kernels of the algebra characters. For a commutative C-star algebra, spectral radius norm equality for normal elements and the Stone-Weierstrass theorem upgrade this to an isometric C-star homomorphism onto $C(\Delta (A))$.