A condition for spectral continuity of positive elements

Let $a$ be an element of a Banach algebra $A$. We introduce a compact subset $T(a)$ of the complex plane, show that the function which maps $a$ onto $T(a)$ is upper semicontinuous and use this fact to provide a condition on $a$ which ensures that if $(a_n)$ is a sequence of positive elements converging to $a$, then the sequence of the spectral radii of the terms $a_n$ converges to the spectral radius of $a$ in the case that $A$ is partially ordered by a closed and normal algebra cone and $a$ is a positive element.