A dynamical proof of the van der Corput inequality

We provide a dynamical proof of the van der Corput inequality for sequences in Hilbert spaces that is based on Furstenberg correspondence principle. This done by reducing to mean ergodic theorem contractions spaces. The key difficulty therein principle is, priori, limited scalar-valued sequences. We, therefore, discuss how interpreting via Gelfand–Naimark–Segal construction C∗-algebras allows study not just scalar but general space-valued terms unitary operators. yields spirit and flexibility this method discussed new proofs different variants inequality.