An Example on Ordered Banach Algebras

Let B be a complex unital Banach algebra. We consider the Banach algebra A = B × C ordered by the algebra cone K = {(a, ξ) ∈ A : ‖a‖ ≤ ξ}, and investigate the connection between results on ordered Banach algebras and the right bound of the numerical range of elements in B. 1. Ordered Banach algebras The aim of this paper is to stress the aspect of the applicability of the ordered Banach algebra theory, within a wider scope of general Banach algebras. To this end we will embed a given (non-ordered) Banach algebra B into a certain ordered Banach algebra A; see section 2. In particular Theorems 6 and 7 in section 5 are results strictly in terms of the originally given Banach algebra B, while the proofs involve results on ordered Banach algebras applied to A. We start with some notations and results on general ordered Banach algebras. Let A be a complex unital Banach algebra with unit 1, and assume that A is ordered by an algebra cone K, that is, K is a closed convex subset of A with λK ⊆ K (λ ≥ 0), K ∩ (−K) = {0}, 1 ∈ K, K ·K ⊆ K, and a ≤ b : ⇐⇒ b−a ∈ K. For a general survey on ordered Banach algebras we refer to [4], [9], and the references given there. The spectrum of a ∈ A is denoted by σ(a), and r(a) denotes its spectral radius. Moreover let τ (a) denote the right spectral bound of a, that is, τ (a) := max{Re λ : λ ∈ σ(a)}. Next, we consider the sets Q+ := {a ∈ A : exp(ta) ≥ 0 (t ≥ 0)}, Q± = Q+ ∩ (−Q+). Amongst other things the following properties of these sets are known. Note that ∅ = W ⊆ A is called a wedge if W is closed, convex and λW ⊆ W (λ ≥ 0). Theorem 1. Under the assumptions above 1. Q+ is a wedge, and K + R1 = Q+; 2. Q± is a closed real subspace of A; 3. Q± is a real Lie algebra; 4. a ∈ Q± =⇒ a ∈ Q+; Received by the editors September 22, 2006 and, in revised form, November 6, 2006. 2000 Mathematics Subject Classification. Primary 47H05, 47A12, 47B60.