In this chapter we shall prove Theorems 3.1 and 3.15, and the various lemmas required in the proofs. In 4.1 we sketch the ideas behind the proofs, and after proving all the lemmas we prove 3.1 in 4.7 and 3.15 in 4.8. Throughout this chapter A will denote a Banach algebra with a countable bounded approximate identity bounded by d(≥1), X will denote a left Banach A-module satisfying ∥a.x∥ ≤ ∥a∥.∥x∥ for all a ∈ A and x ∈ X, and [A·x]− will denote the closed linear span of the set {a·x : a ∈ A, x ∈ X}. Taking d = 1 simplifies the calculations slightly. We assume that A does not have an identity.

SKETCH OF THE PROOF

The proof is a variation of Cohen's factorization, theorem (Cohen [1959]) with the analytic semigroup obtained as a limit of exponential semigroups in the unital Banach algebra A#. The variation is influenced by the proof of the Hille-Yoshida Theorem. If the algebra A had an identity, then the factorization results would be trivial as we could take at = 1 for all t ∈ H. Though our algebra does not have an identity, we shall use the case when there is an identity and an approximation to prove 3.1. We work in the algebra A# = A ⊕ ℂl obtained by adjoining an identity to A, and we regard X and Y as left and right Banach A#- modules by defining 1.w = w and u.1 = u for all w ∈ X and u ∈ Y.