electrical switch controlled remotely via mobile bluetooth arduino
Are you simply asking if it is possible to switch an electrical circuit using Bluetooth? Or do you specifically have to use an Arduino for some reason?

If so the answer is yes. I'm not sure from your question if you must use an Arduino or you're asking how to do it without one.

Without the Arduino, you can use an RN-41 from Roving Networks. The RN41 has some general purpose I/O pins. After the RN41 is paired with the host Bluetooth controller, you can send commands to it to control outputs. One of these outputs can drive a relay. The question of how to control a relay from a logic level output has been answered here many times, so you can search for that solution. There are other Bluetooth devices that have built in I/O, but that one is easily available and cheap.

With the Arduino, you have more flexibility, since any pair of Bluetooth devices supporting SPP (Serial Port Protocol) will allow you to set up a wireless serial connection between devices. Now you can send control messages wirelessly from your PC, etc to the Arduino which is then responsible for controlling the relay.