Are you saying that you booked a place through Airbnb and you were somehow able to deduce that the listing was a fake, @Nini27 ? Did you report it to Airbnb? Did you pay through the platform or send the listing host money directly?
A refund is depending on the cancellation policy. When you cancelled you were informed about the refund (if any), as it is calculated automatically. But seen the reason for your cancellation: did you report the fake profile or listing (upfront) to Airbnb ? As they could investigate and cancel for you with full refund anyway.