characters of a $C^*$-algebra

Question

I have read that a state $\rho$ on a unital $C^*$-algebra $A$ is a character (i.e. multiplicative) if and only if, for all unitary $u\in A$, $|\rho(u)| = 1$. Is there an easy proof, or can someone direct me to one in the literature?

Answer 1

One way of proving this is using the idea of multiplicative domain. It works on more generality than states. Namely, if $\rho:A\to B$ is a unital completely positive map (when $B=\mathbb C$ this reduces to $\rho$ being a state), you have the Schwarz inequality $$ \rho(a)^*\rho(a)\leq\rho(a^*a),\ \ \ \ a\in A. $$ The set $\{a\in A:\ \rho(a)^*\rho(a)=\rho(a^*a)\}$ is called the multiplicative domain of $\rho$ and has the property that, for any $x\in A$, $\rho(xa)=\rho(x)\rho(a)$ (you can find details on this in Paulsen's book or in Ozawa's QWEP paper among other places).

If all the unitaries in $A$ satisfy $\rho(u)^*\rho(u)=I=\rho(u^*u)$, then all unitaries are in the multiplicative domain, so $\rho(xu)=\rho(x)\rho(u)$ for all $x\in A$. As every element in $A$ is a linear combination of unitaries, we get that $\rho$ is multiplicative.