We purchased this home 2 years ago. To get the house up to code for inspection, GFCI outlets had to be installed. A few months after we moved in, I have had a GFCI outlet on one side of my kitchen that keeps tripping. Because of this, my other outlets on this side of the kitchen don't work. I have checked all the outlets with a voltage tester and they all check out fine. I unplugged the refrigerator that is on that same wall and it still trips. There is nothing plugged into any of the other outlets. Any ideas whether this could be a faulty GFCI outlet or faulty wiring?

Could either be a defective unit or the wiring. To prove the GFI works, disconnect the Load wiring from the GFI receptacle, this will isolate the unit for the wiring it serves downstream. Restore power and check the GFI. If it trips, it is defective, if not, there must be a problem with the wiring, or something is still plugged in and causing the trip.

Yes, as TK suggests and if the GFI still trips, use the same general methodology as TK, find the next outlet in line and disconnect, if OK move onto the next... etc. There is no fast way to fix this, move down the circuit one at a time until you find the glitch.