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Settings > apps > all > play store > clear everything > uninstall updates > restart your phone
which notification?
try to clear data and cache of google play services in all apps.
reboot and see if it works.
if nothing work, repair it with SUS
Update Service (SUS)
PC Companion (PCC)
Bridge (for Mac)
Alternatives on How to backup Xperias
http://talk.sonymobile.com/thread/36355