Positive energy unitary representation of the circle ($U_{e^{it}}=e^{itL}$, $\inf \sigma (L)>-\infty$)

A strongly continuous unitary representation of the circle has integer spectral weights: $H$ is the orthogonal sum of spaces on which $U_z=z^m$. Positive energy means that these weights are bounded below, equivalently its self-adjoint generator has spectrum bounded below. Multiplication by an integer character shifts the lower bound to zero. A unitary operator with a nonzero rotation weight cannot exist in its operator algebra: repeated application of it or its inverse would move a nonzero vector to arbitrarily low energy.