A character on a discrete group $\Gamma$ is a conjugation invariant function $\tau$ which is of positive type, and is normalized so that $\tau(e) = 1$, where $e$ is the identuty element of $\Gamma$. A character $\tau$ is irreducible if it cannot be represented as $\tau=a\tau_1+b\tau_2$ for some $a,b>0$ and some characters $\tau_1\ne \tau_2$.

Is it true that irreducible characters of a discrete group form a closed set with respect to the ponitwise convergence?

A character on a discrete group $\Gamma$ is a conjugation-invariant function $\tau$ which is of positive type, and is normalized so that $\tau(e) = 1$, where $e$ is the identity element of $\Gamma$. A character $\tau$ is irreducible if it cannot be represented as $\tau=a\tau_1+b\tau_2$ for some $a,b>0$ and some characters $\tau_1\ne \tau_2$.

Is it true that irreducible characters of a discrete group form a closed set with respect to pointwise convergence?