Les:
I don't know what model of BMW Navigator you are talking about - whether it is a Nav I, II, II+, or III.
If it is a II, II+, or III - in other words, any of the BMW Navigators that have the same physical shape as the Garmin 2xxx series of GPSRs - I can help you with pinout information. I don't know what the actual wire colours correspond to, but you can match the unknown purple wire up to the pinout diagram below using a continuity tester.
Here is how the pins are assigned for the 8 pin connector on the back of the GPSR (or, the 8 pin cable - same thing). There are two vertical rows of pins, one row contains five pins, and the other row contains 3 pins.
For the row containing 5 pins, the assignments go like this (pin 1 is at the bottom, pin 8 at the top):
8 - Speaker positive
6 - RS 232 transmit
4 - Reverse signal (from car backup light, or moto sidestand)
2 - DC Ground (negative)
1 - 12 Volt DC power (positive)
For the row containing 3 pins, the assignments go like this (again, looking at it top to bottom, with 7 being at the top and 3 at the bottom):
7 - Speaker negative
5 - RS 232 receive
3 - Speedsensor signal input
Be aware that not all functions are supported by hardware (i.e. "used") on all models of 2xxx GPSR. For example, only the 2650 and 2660, which have dead reckoning capability, use the signals from pins 3 and 4. I don't know which (if any) support the RS 232 data communication function, and I have no idea at all what the RS 232 data communication function could be used for*, or if Garmin even implemented this feature. Maybe they just reserved a pair of pins for possible future use.
As you can see, the speaker wires consist of a pair of wires from pins 7 and 8, and the power wires consist of a pair from pins 1 and 2.
Let us know what pin you find that the purple wire connects to.
Michael
* But I can guess - lemme see - when the GPSR detects a speed camera, it sends a data signal to automatically disconnect the cruise control on the Gold Wing being driven by Fred & Ethel, an elderly retired couple from Florida...