Reddit mentions of The Cambridge Companion to Arabic Philosophy (Cambridge Companions to Philosophy)

Do you mean like rational theology?

1. Cambridge Companion to Islamic Theology
   
   
2. Cambridge Companion to Arabic Philosophy
   
   
3. Moderation in Belief by Al-Ghazali. This is the famous Ash'arite (orthodox Islamic theology) thinker's defense of the school's doctrine. You will also be interested in his Incoherence of the Philosophers, the famous systematic refutation of avicennan philosophy (again if you're interested in these things).
   
   If, however, you want something of a more basic "introduction" to Islamic thought, which I'm guessing is what you actually want. You need to read the Qur'an. With a good commentary. I recommend nothing (and I mean nothing else- because I've read this one among others and this is the best intro for westerners) other than Muhammad Asad's "Message of the Qur'an". After this, you aught to read "Hadith: Muhammad's Legacy to the Medieval and Modern World". This is a great introduction to our "second scripture".
   
   Also, if you want a good ground-up introduction to Islamic theology you should check out this:
   
   https://islamtheologyscience.files.wordpress.com/2011/08/islamictheologyandsciencedraft.pdf
   
   Its written by one of our mods. Really good. But if rational theology/kalaam was not what you were looking for, then please read Muhammad Asad's quran commentary and Jonathan brown's book on Hadith that I listen above.
   
   

Check out BBC Radio 4's In Our Time podcast episode on Avicenna, too. I read that it recommends Peter Adamson's (the OP) The Cambridge Companion to Arabic Philosophy as further reading.

This looks like a good place to start. I would read up on Islamic philosophy on the side. Good intro