Suppose the Muslims take control of, say, Naples (other cities in Southern Italy could do just as well), but, due to the large Christian population, establish a tolerant state where both Christians and Mulisms are free to live and trade.
Then, this port becomes a key trading center, as Mulism traders from the Middle East and Christian traders from Europe are both free to do business there. This trading activity generates the wealth, and the presence of Christians and Muslims in close proximity generates the cultural exchange.
Although I'm not sure if the Venetians, for example, would still prefer to trade for Eastern goods through Constantinople rather than trading with Muslim merchants in Naples. Maybe Constantinople needs to fall sooner so that trade route gets cut off....