Character of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. A character of $A$ is a nonzero $*$ -homomorphism $\chi:A\to\mathbb C$ . Equivalently, it is a nonzero multiplicative complex linear functional on $A$ ; the $C^*$ -structure then forces continuity and $\chi(a^*)=\overline{\chi(a)}$ . Every character is positive and has norm one, so it is a state. When $A$ is unital, a character automatically satisfies $\chi(1)=1$ . Its kernel is a maximal closed two-sided ideal, and $A/\ker\chi\cong\mathbb C$ . The zero homomorphism is excluded.

Existence and examples

For $A=C_0(X)$ , evaluation at $x\in X$ , $\chi_x(f)=f(x)$ , is a character, and every character has this form. The matrix algebra $M_n(\mathbb C)$ has no characters for $n>1$ , because it is simple and cannot have a one-dimensional quotient. More generally, characters exist precisely when $A$ has a quotient isomorphic to $\mathbb C$ .

Multiplicativity and purity

Every character is a pure state. Conversely, a pure state need not be a character: vector states on $M_n(\mathbb C)$ are pure, but no nonzero state there is multiplicative when $n>1$ . On a commutative $C^*$ -algebra, however, the pure states are exactly the characters. These facts underlie the construction of the character space Murphy, chapter on commutative $C^*$ -algebras.

Terminology

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: the chapter on characters, pure states, and commutative $C^*$ -algebras.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: the introductory discussion of states and multiplicative functionals.