What is the halachic status of the person who converts to Islam, does he remain a Jew in spite of conversion? I am specifically asking the Muslim case since they don't deny the Torah and believe it to be in the six mandatory articles of Islam, nor are Muslims Avodah Zara like majority Christianity which is clearly Avodah Zarah due to diefication of Jesus. Hence is there a special status for them?
Maimonides makes it clear in Laws of Repentance, chapter 3 that a Jew who denies the irrevocable validity of the Torah as Judaism understands it is deemed a sinner but still a Jew. Hence, a Jew who converts to Islam would be in this category.
There are opinions that upon returning to the fold of mainstream Judaism it would be best for the individual to re-immerse in a mikva (ritual bath), as was done by a group of Karaites who returned to mainstream Judaism under the supervision of a famed Rabbi Abraham the Prince (believed to be Maimonides' son).
It's a more interesting question just how far the elastic can bend before snapping; if an entire population abandoned Judaism several generations ago, how long do we still consider them Jews? But at the individual level, yes he would still be considered a Jew.
Some 800+ years ago, Rabbi Abraham ibn Ezra's son actually served as an imam at one point, there was a great deal of discussion as to whether he actually intended to abandon Judaism at that point, or this may have simply been a job he took as he was starving.