[ffgjgf1]

Constantinople was the center of the empire long before Italy and Rome were lost (and the Byzantine empire controlled the city of Rome itself until the 750s).

It’s a bit like saying that Angles/English stopped being “English” after they moved from northern Germany/Denmark to the modern territory of Britain.

At least for several centuries the “Byzantine” Empire was the Roman Empire and was undoubtedly recognized as such both in the west and east.