Left Quotients of a C * - algebra , I : Representation via vector sections ∗

Let A be a C*-algebra, L a closed left ideal of A and p the closed projection related to L. We show that for an xp in A∗∗p (∼= A∗∗/L∗∗) if pAxp ⊂ pAp and px∗xp ∈ pAp then xp ∈ Ap (∼= A/L). The proof goes by interpreting elements of A∗∗p (resp. Ap) as admissible (resp. continuous admissible) vector sections over the base space F (p) = {φ ∈ A∗ : φ ≥ 0, φ(p) = ‖φ‖ ≤ 1} in the notions developed by Diximier and Douady, Fell, and Tomita. We consider that our results complement both Kadison function representation and Takesaki duality theorem.