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