From the outside, with its high minarets and bulbous Mughal-style dome, the Wilmersdorf mosque, located on Brienner Street in southwest Berlin, looks much the same as it did when it was built in the 1920s. But the institution, just like the city around it, has changed. Today, the mosque is a quiet place. It mainly serves as an information center: School children sometimes visit on field trips; it hosts interfaith brunches. A small community of Muslims regularly show up for Friday prayer. It’s all a far cry from the days when the Wilmersdorf mosque was the lively center of a...