Final answer:
The Ottoman state did not control southern Italy after the collapse of the Byzantine Empire. While it controlled many of the former Byzantine territories, including Anatolia, the Balkans, and Constantinople, certain areas remained outside Ottoman influence.
Step-by-step explanation:
The area that the Ottoman state did not control after the collapse of the Byzantine Empire was southern Italy. After the fall of Constantinople in 1453, the Ottomans under Mehmed II took control of what had been the Byzantine territories, including Anatolia, the Balkans, and the city of Constantinople itself, which was renamed Istanbul. However, certain areas remained outside of Ottoman control, such as most of southern Greece, Genoese trading posts on the Black Sea, and parts of the Balkans like Bosnia and Albania.