Eventually, I would hope so. If Italy is occupied relatively bloodlessly (possible, since a fraction of the Goths would be backing Amalasuntha and Belisarius would have a large force form the start) and the previous structures are maintained (and I have no reason to doubt they would not be), then that just leaves Africa, Spain and southern Gaul.As is there an army? Do your empire have enough manpower to occupy all the areas you want to conquer and enforce the law at the same time?
AFAIK there was no major piracy problem coming in from Africa, and if either the carrot or the stick method is used on the Maurs competently (instead of the half-assed measures from OTL), then that's one less head-ache.
Since one of the three main field armies would be stationed in Provence to watch over the Franks (the other 2 are presumably on the Danube and in the East), that just leaves the Spanish coastline, but not that much trade goes through that area anyway, and eventually things would calm down enough for rule of law and prosperity to return.