Put Captain Solo in the Cargo Hold said:
I'm always under the assumption that the general public (I mean the people in the Star Wars universe) doesn't know Vader and Anakin are one and the same. That it's a dark secret known to only to Obi-Wan, Yoda and the Emperor. Though in order for this to work, you have to disregard the prequels and EU.
Even if you do regard the prequels then you only really have to add Bail Organa, a battalion of Clone Troopers, and perhaps that blue dude with the horns.
If you follow the line of Anakin from the moment he falls to the dark side to him going into the suit, then there are only a few stops along there way for people to see him doing evil stuff out of the suit and pretty much all of those people are dead.
He leave the Senate, hooded mind you, directly to the temple with the 501st, who actually probably picked him up from the Senate and flew him straight there. After killing EVERYONE in the temple, he then leaves for Mustafar making only a pit stop at Padme's apartment alone the way. He's in a Jedi starfighter so he presumably came straight from the temple to Padme's and wouldn't have had to make any stops between there and Mustafar. He kills all the separatists on Mustafar and other than a few aliens doing some mining who have no idea what the hell is going on, nobody sees Anakin from that point until he goes into the suit except for the Emperor and the Clones.
Anakin's actions where not great public spectacles . They happened in the shadows. So the only way to assume that the general public of the galaxy knew that Anakin fell, is if you assume that one of the few people who did know went announcing the fact to everyone. Which you can't assume.