Many thanks to Stuei for the following:
I don't know about idle actuator, although there is an idle control device, which is part number 4 here http://www.realoem.com/bmw/showparts.do?model=0192&mospid=47977&btnr=13_0897&hg=13&fg=15
This controls the idle. Quite a lot of C1s have this problem, and have to hold the throttle open to start it.
To test this, unscrew the two phillips screws and pull out the unit. You will see a shaft and taper on the underside of the unit. Turn the ignition on, and the shaft should extend (keep your hand over it otherwise it will pop off the end!). When you turn the ignition off, the shaft will retreat and then extend slightly. This is controlled by the ECU. If it does not do this, then it is likely that you have a problem. Try blasting the connector with WD40 and shoving the plug in and out a few times to clean it.
They are extremely difficult to take apart, but if you get inside you will see 4 filament size wires, sometimes these can shake loose of their soldering. These are only insulated by a very thin layer of laquer. A new one costs 54 pounds including the VAT.