1) Rabbi Abraham Maimonides, the son of the Rambam, as head of the Jewish community in 13th Century Egypt, was a self professed Jewish Sufi. He included Sufi (Islamic) Practices into Jewish Worship including prostrations during prayer, making proper ablutions (wudu) before prayer, being in a state of ritual purity after marital relations (ghusl) before prayer, doing Zikr (Remembrance of God), Khalwa (Spiritual Retreats), wearing the Sufi cloak, following a Jewish Rebbe (Shaykh) in learning the discipline of the Purification of the Heart (Tasawuf). They were the original Chasidim (Pious ones).
2) One can follow these Islamic/Sufi practices while not converting from Judaism to Islam. First, Rabbi Abraham and all of his followers were devoutly Orthodox Jews while they practiced these customs. He kept Shabbat, Kosher, wore Tzitzit, Tefillin, all of the Holidays, Brit Millah, etc.
3) If one does indeed accept that Prophet Muhammad is the last Prophet of God, it still does not constitute a conversion or abandonment of Judaism as long as they still accept the legitimacy and validity of the Torah, the Tanakh, the Mishnah, believes in the 13 articles of faith, and respects and loves the Sages of Israel (the Rabbis of the Talmud). As long as they still love the Jewish people, love the land of Israel, and still believes that God has made a special covenant with the Children of Israel that he will not abandon or abolish, he would still remain Jewish. It is when someone starts hating the Jewish people or believes that the Torah is corrupt or that the covenant with the Bnei Yisrael has been replaced with a newer covenant, that that person should then be seen as an apostate or someone who has left Judaism and converted to another religion.
4) Other Medieval Jewish Sufis included Rabbis Bahya ibn Paquda, Yehuda HaLevi, Isaac of Acre, Solomon Ibn Gabirol, Abraham Abulafia, Obadyah Maimonides, David Maimonides, etc.
5) Rabbi Abraham’s defense of following Islamic/Sufi practices was that these practices listed above all originated from the Prophets of Israel. That these are not foreign practices to the Nation of Israel but rather due to the destruction of the Temple, the absence of the Sanhedrin and true Semicha (Rabbinical Ordination), exile, persecution, etc., that these practices had been abandoned, but that they needed to be reinstituted as truly authentic Jewish practices as had been practiced by the Hebrew Prophets.
6) Other very prominent Rabbis including Moses Maimonides and Rabbi Ben Abrahamson teach that Prophet Muhammad was indeed a true Prophet of God, a messenger to the Gentiles, to bring them to true Monotheism (no trinity, no God incarnating into the flesh of man, etc.) and that his purpose was to bring the Gentiles to Monotheism in preparation for the coming of the Messiah and the Messianic era when all nations will recognize that the God of Abraham is the true God and that Jerusalem, specifically the Temple Mount, is the gateway to the heavens. This is in fact, exactly what the Prophet Muhammad accomplished! Throughout the Qur’an, it repeatedly denounces the belief in the Trinity and the divinity of Jesus and also that he was the unique son of God. It also rejects the belief that he died on the cross for everyone’s sins. In addition, it repeatedly firmly establishes that Islam (meaning simply Submission to God) was the original religion of Abraham and that the God of the Qur’an is the God of Abraham. What this means is that Judaism is defined by the covenant of the Torah and of course Abraham predates this covenant. Islam contains within it the Covenant of Abraham (circumcision) and the 7 laws of Noah amongst others. Additionally, the Qur’an also establishes that Jerusalem is the gateway to heaven (as was seen in Jacob’s vision) by narrating that God sent the Angel Gabriel to get the Prophet Muhammad from Mecca and bring him to the Temple Mount in Jerusalem to ascend through the 7 heavens and meet many of the Hebrew Prophets (including Abraham, Moses, David, etc.) Prior to Islam, the Byzantine Christians used the Temple Mount as a Dung Hill. When the Caliph Omar capture Jerusalem in 638, he cleansed the Temple Mount due to its sacredness in Islam, and later the Dome of the Rock and Masjid Al Aqsa were built there to honor its sanctity.
Isaiah Chapter 2 יְשַׁעְיָהוּ
2 And it shall come to pass in the end of days, that the mountain of the LORD'S house shall be established as the top of the mountains, and shall be exalted above the hills; and all nations shall flow unto it.
3 And many peoples shall go and say: 'Come ye, and let us go up to the mountain of the LORD, to the house of the God of Jacob; and He will teach us of His ways, and we will walk in His paths.' For out of Zion shall go forth the law, and the word of the LORD from Jerusalem.
4 And He shall judge between the nations, and shall decide for many peoples; and they shall beat their swords into plowshares, and their spears into pruninghooks; nation shall not lift up sword against nation, neither shall they learn war any more.
5 O house of Jacob, come ye, and let us walk in the light of the LORD.