Publisher Summary This chapter presents a theorem that provides sufficient conditions that a non-zero element of Banach algebra should generate a strict locally compact semi-algebra. A theorem is presented in the chapter that states that if the peripheral spectrum of the generator is a set of poles and if the semi-algebra it generates is strict then the spectral radius is a pole of the generator of maximal order in the peripheral spectrum. The sufficient conditions for an element in semi-algebra have its spectral radius in its spectrum. These conditions are phrased in terms of ideals of the semi-algebra. The chapter provides methods that yield the generalizations of the Krein–Rutman theorem. The theorem states that if a bounded irreducible positive linear operator on a Banach space has a peripheral spectrum consisting of poles of finite rank then the spectral radius is an eigenvalue and the associated eigenspace is one dimensional.