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We consider projections of SU(2) lattice link variables onto Z(2) center andU(1) subgroups, with and without gauge-fixing. It is shown that in the absenceof gauge-fixing, and up to an additive constant, the static quark potentialextracted from projected variables agrees exactly with the static quarkpotential taken from the full link variables; this is an extension of recentarguments by Ambjorn and Greensite, and by Ogilvie. Abelian and centerdominance is essentially trivial in this case, and seems of no physicalrelevance. The situation changes drastically upon gauge fixing. In the case ofcenter projection, there are a series of tests one can carry out, to check ifvortices identified in the projected configurations are physical objects. Allthese criteria are satisfied in maximal center gauge, and we show here thatthey all fail in the absence of gauge fixing. The non-triviality of centerprojection is due entirely to the maximal center gauge-fixing, which pumpsinformation about the location of extended physical objects into local Z(2)observables.Comment: 18 pages, 6 figures, Latex2

Center projection with and without gauge fixing