I am fairly sure it is one of the carpenter bees, of which there are many, in the subfamily Xylocopinae.
Yours is probably in the Xylocopini tribe. That also means it is probably in the genus Xylocopa.
Probably the most well-known is Xylocopa violacea (which has no yellow).
I don't have a good list of bees and wasps for Cyprus, so doing some research...
Some of the Cyprus fauna is shared with Turkey, and Turkey has ten species of Xylocopa. The most common are Xx. violacea, valga, iris but none of those have a yellow thorax; ditto X. fenestrata. X. olivieri does occur in Cyprus but it has a brown-buff thorax and striped abdomen, so that's ruled out. Xx. parviceps, armeniaca, rufa are ruled out by distribution.
There are quite a lot of species of Xylocopa (http://en.wikipedia.org/wiki/Carpenter_bee), and several of them have a yellow thorax, eg. Xx. appendiculata circumvolans, sulcatipes, but those do not occur in Cyprus.
(...much more research...)
My best guess is Xylocopa pubescens. It occurs in the eastern Mediterranean (including Syria and Israel) but I have no actual records for it in Cyprus.
https://www.youtube.com/watch?v=M2OCpWl2Pso (good video; just like your specimen)
I wish I had access to G.A. Mavromoustakis' 'The bees (Hymenoptera, Apoidea) of Cyprus', in multiple parts, eg. Journal of Natural History: Series 12, Volume 10, Issue 113, 1957, pp.321-337
Thank you very much for your research and response. I've been in Cyprus for 5 years and this is the first time I'm seeing this kind of bee. This is definitely a male xylocopa pubescens. I released it last night after the video and this post : ) Even though it's harmless, it's very aggressive. I hope I won't see him around again : ))
Would you mind sending me an email, please?
I, too, live in Cyprus.
I'm trying to pursue this issue of X. pubescens in Cyprus with some experts, and I might want to get in touch with you.
Please use <email address removed by Mike Hardman>, temporarily.