The irreducibility in ordered Banach algebras

Abstract

Let A be an ordered Banach algebra. Put OI(A) = {b ∈where e is a unit of A. An element z ≥ 0 is said to be order continuous if b α ↓ 0 implies b α z ↓ 0 and zb α ↓ 0 for any b α ∈ 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 x y = 0 there exists b ∈ 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 ∈ A is said to be irreducible if for every b ∈ 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 ≤ T ∈ 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).

Ordered Banach algebras and multi-norms: some open problems

“…Problem 26 The concept of irreducible elements in ordered Banach algebras was established in [1]. Several aspects of these elements, in particular the peripheral spectrum, still have to be investigated.…”

Fredholm theory in ordered Banach algebras

“…Recent work done by Alekhno in [6] shows that, under natural conditions, the spectrum of a positive element is determined by the spectra of irreducible elements. The notion of an irreducible OBA element is also introduced in [6], where it is established that these elements have useful spectral properties.…”

“…Recent work done by Alekhno in [6] shows that, under natural conditions, the spectrum of a positive element is determined by the spectra of irreducible elements. The notion of an irreducible OBA element is also introduced in [6], where it is established that these elements have useful spectral properties. The ideas drawn from Alekhno's work open doors to the study of whether positive elements of arbitrary Dedekind complete semisimple OBAs with disjunctive products have the upper Browder spectrum property.…”

“…The theory of irreducibility in an OBA was developed by Alekhno in [6]. In this section we record some concepts and important results from [6] which will serve as the background needed to study spectral properties of elements in OBAs which have disjunctive products.…”

“…We remark that, if either ( From this point onward we recall concepts and results from [6] that will be useful in the sequel.…”