First of all, on all the mute inputs I've seen, they have all been negative. Consult your install manual on this one.
As for getting a low voltage AC audio signal converted to a switched ground, I may have an idea.
If you are comfotable with relays then this could be a snap! I've never done exactly what you are trying to do, but my idea is very similar to what I've done to create a viable amp turn-on signal by using speaker outputs from a head unit.
Anyways, get a hold of a LVT (low voltage trigger). Try to find one that is adjustable if at all possible. Tag this to the audio output of your nav device. Then the LVT should have a positive output that is 12v. If you find one that has a negative output as well then your set. If not then just use the relay to switch to a negative output witch you can connect to the mute wire! If you need any further help, such as how to wire up the relay or LVT, I can help.
Let me know how it turns out!