Just bought, installed and nothing, the appdoes not open.
Any clu?
IM SO LAME SORRY
Look into Console.app, there should be some error displayed in the log.
btw. why do you buy software right away when you have a trial period available? 
Had the same problem. At the end of the installation, it told me that the installation failed. This was the only message in the logs:
12/10/13 4:56:31.102 PM _securityagent[1552]: Restarting dock and killing any running TotalSpaces
… so I thought I’d try one more thing - went to spotlight and launched TotalSpaces2 manually and it started up with no problem.
Now I’m trying to get used to the Dashboard in Space #1 - totally messes up my workflow, but that’s a story for another thread.