On the peripheral spectrum of positive elements

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 p 2 = p and 0 ≤ p ≤ e . An element a ∈ A + is said to be irreducible if the relation ( e - p ) a p = 0 , where p is an order idempotent, implies p = 0 or p = e . For an arbitrary element a of A the peripheral spectrum σ per ( a ) of a is the set σ per ( 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 σ per ( a ) contains or coincides with r ( a ) H m , where H m is the group of all m th roots of unity, and the spectrum σ ( a ) is invariant under rotation by the angle 2 π m 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., σ per ( 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 A e , i.e., of the order ideal generated by e in A , is proved.