  We consider a Moebius covariant sector, possibly with infinite dimension, of
a local conformal net of von Neumann algebras on the circle. If the sector has
finite index, it has automatically positive energy. In the infinite index case,
we show the spectrum of the energy always to contain the positive real line,
but, as seen by an example, it may contain negative values. We then consider
nets with Haag duality on the real line, or equivalently sectors with
non-solitonic extension to the dual net; we give a criterion for irreducible
sectors to have positive energy, namely this is the case iff there exists an
unbounded Moebius covariant left inverse. As a consequence the class of sectors
with positive energy is stable under composition, conjugation and direct
integral decomposition.
