On the peripheral spectrum of positive elements

Abstract

Let A be an ordered Banach algebra with a unit e and a cone A+. An element p of A is said to be an order idempotent if p2= p and 0 ≤ p≤ e. An element a∈ A+ is said to be irreducible if the relation (e- p) ap= 0 , where p is an order idempotent, implies p= 0 or p= e. For an arbitrary element a of A the peripheral spectrum σper(a) of a is the set σper(a) = { λ∈ σ(a) : | λ| = r(a) } , where σ(a) is the spectrum of a and r(a) is the spectral radius of a. We investigate properties of the peripheral spectrum of an irreducible element a. Conditions under which σper(a) contains or coincides with r(a) Hm, where Hm is the group of all mth roots of unity, and the spectrum σ(a) is invariant under rotation by the angle 2πm for some m∈ N, are given. The correlation between these results and the existence of a cyclic form of a is considered. The conditions under which a is primitive, i.e., σper(a) = { r(a) } , are studied. The necessary assumptions on the algebra A which imply the validity of these results, are discussed. In particular, the Lotz–Schaefer axiom is introduced and finite-rank elements of A are defined. Other approaches to the notions of irreducibility and primitivity are discussed. Conditions under which the inequalities 0 ≤ b< a imply r(b) < r(a) are studied. The closedness of the center Ae, i.e., of the order ideal generated by e in A, is proved.