The irreducibility in ordered Banach algebras

Abstract

Let A be an ordered Banach algebra. Put

$$\mathbf{OI}(A)=\{b\in A: 0 \le b\le e, b^2 = b\},$$

where e is a unit of A. An element z ≥ 0 is said to be order continuous if ${b_\alpha\downarrow 0}$ implies ${b_\alpha z \downarrow 0}$ and ${zb_\alpha\downarrow 0}$ for any ${b_\alpha \in \mathbf{OI}(A)}$ . It is shown that if E is a Dedekind complete Banach lattice then the set of all order continuous elements in L(E) coincides with the set of all positive order continuous operators on E. An algebra A is said to have a (strongly) disjunctive product if for any order continuous x and y in A(x, y ≥ 0) with xy = 0 there exists ${b \in \mathbf{OI}(A)}$ such that xb = (e − b)y = 0. We show that the algebra L(E) has the strongly disjunctive product iff E has order continuous norm. An element ${z\in A}$ is said to be irreducible if for every ${b \in \mathbf{OI}(A)}$ the relation (e − b)zb = 0 implies either b = 0 or b = e. We investigate spectral properties of irreducible elements in algebras with a disjunctive product. The spectral radius r(z) is called an f-pole of the resolvent R(·, z) if 0 ≤ x ≤ z implies r(x) ≤ r(z) and if r(x) = r(z) then r(z) is a pole of R(·, x). We show that under some natural assumptions on the Banach lattice E, if ${0\le T \in L(E)}$ then r(T) is an f-pole of R(·,T) iff r(T) is a finite-rank pole of R(·, T). We also present a theorem about the Frobenius normal form of z when r(z) is an f-pole of R(·, z). Some applications to the spectral theory of irreducible operators and the general spectral theory of positive elements are provided. In particular, we show that under some conditions 0 ≤ x < z implies r(x) < r(z).