Smooth approximation of Lipschitz projections

We show that any Lipschitz projection-valued function p on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions q with Lipschitz constant close to that of p. This answers a question of Rieffel.

💡 Research Summary

The paper addresses a problem that lies at the intersection of metric geometry, functional analysis, and non‑commutative geometry: given a Lipschitz projection‑valued function (p) on a closed, connected Riemannian manifold (M), can one find a smooth projection‑valued function (q) that approximates (p) uniformly while keeping the Lipschitz constant essentially unchanged? This question was posed by Marc Rieffel in the context of his work on quantum metric spaces, where projection‑valued functions play the role of “coordinate functions” for vector bundles and are used to define distances in non‑commutative settings.

The authors answer affirmatively. Their construction proceeds in two main stages. First, they smooth the original projection (p) without destroying its self‑adjointness. Using the heat kernel (or, equivalently, a mollifier adapted to the Riemannian structure), they define \