Your statement above is a bit unclear, but if you meant that Ashkenazi Jews are not genetically much assimilated to Europe at all, that is not exactly correct. The Fst data calculated in this study suggests that the genetic distance between Ashkenazim and European host populations is roughly equal to the distance between Ashkenazim and Levantines. The Atzmon paper discussed on this blog earlier actually gave an admixture estimate of 30-60%, although in his interview with Ostrer in the NYT, Nicholas Wade quotes Prof. Ostrer offering the 30% number, possibly for simplicity sake (and the quote did not specify Ashkenazim, IRC).

There was a Jewish community in North Africa for 2000 or 2500 years. More Jews came from Spain later. There is no source, no documentation, no traces of any Berber tribe converting to Judaism ever. And now we have genetic studies who just show that the North African Jews are closer to Russian Jews than to Berbers and Arabs. The surprise being that Ashkenazi Jews are not more assimilated to the European environment were they used to live.
Regarding names, well I have a 100% Jewish aunt whose name is “Christiane” – the least Jewish name possible. My grand mother had a Arabic name (Nedjma), my parents and uncles had French names, my generation was a French-Hebrew mix, my children and nefew generation is 100% Hebrew names. Jews have adopted the names and languages of the society where they live during all history. Jews had and still have Persian names (Esther and Mordekhai), Greek names, Roman names, Arabic names, French names, American names.

I think it would be more correct and unambiguous to use “besides” or “in addition to” here instead of “among”.

It would be better if I used “the” here instead of “a”, as I was exclusively talking about Iberia.

But eventually Iberian Arabic became the dominant language among Iberian Muslims, as it was the prestige language or rather a colloquial form (but nevertheless prestigious) of the real prestige language (Classical Arabic).

Abdallah means God’s servant in Arabic, and as Allah has been the name Christians and Jews have used for their God when speaking Arabic, Persian or Turkish, it is very normal for them to take it as a name. After all, being a servant of God is fully in accordance with Christian and Jewish theologies.

Of course, I was referring to Turks of Turkey (who are genetically much more Anatolian + south Balkanic than anything else), not Central Asian Turkic populations.

And I haven’t seen a study comparing populations on both sides of the Straits

There is a genetic study specifically investigating the degree to which the Mediterranean Sea has acted as a barrier to gene flow, and it includes Spaniards, Moroccans and Turks among other Mediterranean populations:

http://www.biomedcentral.com/1471-2148/10/84

I suggest you to especially look at the PCA plots.

but apparently at some time the two sides were closer together (they were connected millions of years ago)

In those times there were no humans (not even archaic humans) living in either of the two sides of the Mediterranean Sea.

and there were migrations

Of course, there were migrations across the Mediterranean Sea, but they only started beginning from the Neolithic times with the developments in seafaring technologies, but even since then vast majority of migrations in the Mediterranean Basin have been by land down to the and including the modern era, as evident in genetic studies and history.

(some people swim accross the Straits even now!)

That, of course, is not a logical way of crossing the Mediterranean Sea.

Which language do you mean here by “special” and “invented”? Iberian Arabic?

Not so fast. Iberians are genetically much closer to Turks, who live in the other extreme of the north Mediterranean Basin, than to Moroccans, who live just across the Mediterranean Sea. The Mediterranean Sea has been a great barrier to gene flow throughout most of human history in the Mediterranean Basin. Humans have mostly moved by land rather than by sea in their migrations.

What is so surprising about that? Arabic names have usually been frequent among Christians and Jews living in Islamic countries. Here in Anatolia, it was very normal for Greeks, Armenians and Jews to take Muslim names (Arabic, Persian or Turkish) in Ottoman times, and still many of them have such names.