  We consider projections of SU(2) lattice link variables onto Z(2) center and
U(1) subgroups, with and without gauge-fixing. It is shown that in the absence
of gauge-fixing, and up to an additive constant, the static quark potential
extracted from projected variables agrees exactly with the static quark
potential taken from the full link variables; this is an extension of recent
arguments by Ambjorn and Greensite, and by Ogilvie. Abelian and center
dominance is essentially trivial in this case, and seems of no physical
relevance. The situation changes drastically upon gauge fixing. In the case of
center projection, there are a series of tests one can carry out, to check if
vortices identified in the projected configurations are physical objects. All
these criteria are satisfied in maximal center gauge, and we show here that
they all fail in the absence of gauge fixing. The non-triviality of center
projection is due entirely to the maximal center gauge-fixing, which pumps
information about the location of extended physical objects into local Z(2)
observables.
