Everywhere in the mediterranean supplies oil for olive oil. And calling it "Italian" is virtually meaningless. Either it comes from a very specific region of Italy so you can taste its local flavor, or it's just an oil blend. The back of the bottle will tell you which countries the oil comes from.
I highly suggest buying from Greece instead of California, if only as a form of financial assistance via trade. Greek products are extremely good, but difficult to get here. There's so little money in Greece that they can't afford to build up the export infrastructure they had a century ago.
I am someone who still has all but my immediate family still over there, and I really appreciate you saying this. It really goes a long way. Greece has been at the very short end of the stick on some EU trade deals, austerity measures, increased war games from their millennial long mortal enemies, front lines of a refugee crisis, and more. For a country that has been through so much it truly is a testament to their resolve. And they still take great pride in making the best damn olives they can.
I highly suggest buying from Greece instead of California, if only as a form of financial assistance via trade. Greek products are extremely good, but difficult to get here. There's so little money in Greece that they can't afford to build up the export infrastructure they had a century ago.