Conformal field theory
The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of "c". Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
Beckman–Quarles theorem
Beckman and Quarles also provide a counterexample for Hilbert space, the space of square-summable sequences of real numbers. This example involves the composition of two discontinuous functions: one that maps every point of the Hilbert space onto a nearby point in a countable dense subspace, and a second that maps this dense set into a countable unit simplex (an infinite set of points all at unit distance from each other). These two transformations map any two points at unit distance from each other to two different points in the dense subspace, and from there map them to two different points of the simplex, which are necessarily at unit distance apart. Therefore, their composition preserves unit distances. However, it is not an isometry, because it maps every pair of points, no matter their original distance, either to the same point or to a unit distance.