An Analogue of Peetre's Theorem in Non‐Commutative Topology

A linear mapping ∇: E1 → E2 between non‐commutative vector bundles E1, E2 over a C^*‐algebra A is said to be local if, for all a, b ∈ A, ξ ∈ E1, aξ = ξb = 0 implies that a(∇ξ)b = 0. We show that a bounded linear mapping ∇ is local if and only if there is a derivation δ: A → LA(E1, E2) such that ∇(aξ) = (δa)ξ + a∇ξ for all a ∈ A, ξ ∈ E1.For a not necessarily bounded, linear mapping ∇: A → A on a unital C^*‐algebra A, a variant of locality implies the existence of a derivation δ: A → A and an element h ∈ A such that ∇(x) = δ(x) + hx, x ∈ A.