The armistice line drawn at the end of the 1948 war divided Jerusalem into two. Between 1949 and 1967, the Israelis controlled the western part of Jerusalem, while Jordan took the eastern part, including the old walled city containing important Jewish, Muslim and Christian religious sites.
The Israelis captured and seized East Jerusalem from Jordan during the 1967 war.
East Jerusalem has been occupied by the Israelis since then and has been effectively annexed, in an act internationally condemned, by the Israelis in 1980.
According to Israeli annexation law, therefore, not only does Israeli law apply to the territory of East Jerusalem, the Israelis also considers East Jerusalem an integral part of its territory.
Now the question: who owns East Jerusalem?
The answer depends on how one wish to view, which is heavily politicized nowadays.
But, if you have the guts to start there were we all should have started ... from the beginning .... you will find out that the real owners do no longer live..
You then need to start there were it is still visible today. There's nothing visible that reveals early presence of Judaism in the Old City thus East Jerusalem.
You find about early Islam, the Umayyad period (661–750), like the Dome Of The Rock on the Al Aqsa compound.