Is Allah the same as the God of the Christians and Jews?

It is my understading that the Christian God and the Jewish God are the same, except Christians accept Jesus to be Messiah and God. What about the Muslims? The bible says in Genesis that even though God establishes his covenant with Isaac, God was with Ishmael as he grew up. So if the God of Ishmael was the same God as Isaac, then is Allah the same God as the Christians and Jews? If not, when and how did things change since the time of Ishmael and Jacob?