A condition for spectral continuity of positive elements

Abstract

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