The app on the Sony reader which says "Google Books" and looks like it would be the official Google Books app is not the actual Google Books app. It takes you to a section of the Sony Reader store where you can search and download the *free* Google Books.
If you want to download Google Books you've purchased, you will need to download them to the Sony PC/Mac software on your computer and then sync them to your Reader (if they are DRM-enabled). Non-DRM ePubs you can download directly to the reader using the books.google.com site.
So that means you won't be able to use some of the nice features from Google Books, such as automatic page synchronization between devices (allowing you to read on multiple devices and then continue where you left off using another).
If you're willing to root your device, it's quite likely you could then install the official Google Books app and get this functionality back. But it's not included in Sony's stock firmware.
Scott
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