  We discuss the influence of closed Dirac strings on the photon propagator in
the Landau gauge emerging from a study of the compact U(1) gauge model in 2+1
dimensions. This gauge also minimizes the total length of the Dirac strings.
Closed Dirac strings are stable against local gauge-fixing algorithms only due
to the torus boundary conditions of the lattice. We demonstrate that these
left-over Dirac strings are responsible for the previously observed unphysical
behavior of the propagator of space-like photons (D_T) in the deconfinement
(high temperature) phase. We show how one can monitor the number N_3 of thermal
Dirac strings which allows to separate the propagator measurements into N_3
sectors. The propagator in N_3 \neq 0 sectors is characterized by a non--zero
mass and an anomalous dimension similarly to the confinement phase. Both mass
squared and anomalous dimension are found to be proportional to N_3.
Consequently, in the N_3=0 sector the unphysical behavior of the D_T photon
propagator is cured and the deviation from the free massless propagator
disappears.
