We continue our development of spectral theory for positive elements in an ordered Banach algebra. In particular we provide a suitable version of the Krein-Rutman theorem, obtain some results concerning the peripheral spectrum of a positive element and provide a characterisation of positive quasi inessential elements, in the context of an ordered Banach algebra.

More Spectral Theory in Ordered Banach Algebras

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).

The irreducibility in ordered Banach algebras

We investigate some properties of the set S∂(a) = {λ ∈ ℂ: λ – a ∈ ∂S} (which we call the boundary spectrum of a) where ∂S denotes the topological boundary of the set S of all non-invertible elements of a Banach algebra A, and where a is an element of A.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/E04809FF59D1FD36A326378C74005E83/S0004972700035681a.pdf/div-class-title-on-the-boundary-spectrum-in-banach-algebras-div.pdf

On the boundary spectrum in banach algebras

Convergence Properties of Positive Elements in Banach Algebras

We recall from [9] the definition and properties of an algebra cone C of a real or complex Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A. The Banach algebra A is then called an ordered Banach algebra. An important property that the algebra cone C may have is that of normality. If C is normal, then the order structure and the topology of A are reconciled in a certain way. Ordered Banach algebras have interesting spectral properties. If A is an ordered Banach algebra with a normal algebra cone C, then an important problem is that of providing conditions under which certain spectral properties of a positive element b will be inherited by positive elements dominated by b. We are particularly interested in the property of b being an element of the radical of A. Some interesting answers can be obtained by the use of subharmonic analysis and Cartan’s theorem.

/pdf/domination-properties-in-ordered-banach-algebras-180j4y21bn.pdf

Domination properties in ordered Banach algebras

Cones in Banach algebras

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Cones in Banach algebras