Stroumsa’s portrait is most compelling in the chapter on Maimonides and Almohadism. During his formative years, Maimonides not only lived under Almohad rule, but according to contemporary Muslim historians (whose testimony the author accepts), converted to Islam and practiced outwardly its tenets. Stroumsa argues that Maimonides’ law code, innovative in the Jewish legal tradition, follows Almohad legal
methodology, which is to present a conclusion concisely based on legal principles, without following the circuitous history of the argumentation. Moreover, his penchant for including theological principles in his legal works, unprecedented and seldom followed in
the Jewish legal tradition, also has Almohad precedent in Ibn Tumart’s law code, a point made earlier by I. Heinemann. In theology Maimonides follows the Almohads’ rigid rejection of anthropomorophism and makes it a key principle of his thought, classifying this rejection as a foundation of the Torah that even the multitude must accept. Unlike Averroes, who is very strict about revealing philosophical allegorization to non-philosophers, Maimonides finds it occasionally an obligation. Stroumsa also suggests that Maimonides’ military conception of the messiah who spreads monotheism by the sword may be influenced by the frequent military campaigns of the Almohads. Heinemann already connected Maimonides’ ruling that
obligates Jews to force idolaters to choose between accepting the Noahide laws or death with contemporary Muslim notions of jihad; Stroumsa does not consider this particular question, or in what way
the Jewish warrior king emulates Almohad models. She does mention in an earlier chapter that the Almohad rulers restricted the dwelling of Christian merchants to the port cities; this is reminiscent of Maimonides’ banning idolatrous merchants from even temporary
residence within lands over which Jews possess control (A.Z. 3:8). In any event, the “Almohad connection” continues to be an interestingavenue of scholarly exploration, though highly speculative.

Two ironies emerge from Kraemer’s book. First, that the great architect of medieval and modern Judaism seems to have lived for a time, at least outwardly, as a Muslim; whether this was a feigned or true conversion, he was an insider in Muslim culture. And second, that what is often considered original in Maimonides is not very original at all. Throughout the book, Kraemer shows how many of Maimonides’ contributions are derivative, not just of Aristotle and Plato, but also of Muslim thinkers. He notes that Maimonides’s discussion of the five types of speech in Jewish law employs the same five categories contained in Islamic jurisprudence. He shows that Maimonides’s prohibition of using sacred poems for mundane purposes (such as setting them to music at communal gatherings) is taken directly from a commentary on Plato’s Republic by the Muslim philosopher Averroes.
…
Kraemer’s subtitle, One of Civilization’s Greatest Minds, is unfortunate, because the book undermines this claim throughout. Kraemer shows that for Jews and Judaism, Maimonides was certainly an innovator, and the depth of his knowledge and compassion was truly astounding. But as a contributor to the ideas of Western (including medieval Islamic) civilization, he did not have much new to offer.

See here for the full article. For more on Maimonides’s conversion see here and here.

Update: I see that Magid’s review is also discussed at Failed Messiah, although I think that the title of the post is misleading. I have yet to read Kraemer’s book, but my impression from what has been written about it is that he discusses the possible Islamic influences on Maimonides’s philosophy, yet the title of the post at Failed Messiah is “Maimonides The Muslim? Could What We Call Jewish Law Have A Muslim Source?” The possible influence of Islamic law on Maimonides’s legal writings is a very interesting question, although separate, at least in the micro- sense, than influences on his philosophy. On the possible influences of Islamic law on the halakhah during the Geonic period and the time of the Rambam see the writings of Gideon Libson.