Hi,
What is the account type for this live.com account in Outlook 2013? Exchange ActiveSync?
You can click on the "Send/Receive All Folders" button and check if there are any error messages in the synchronization Status bar.
Please also check the Account Settings from FILE -> Account Settings -> Account Settings -> Offline Settings, check if you have selected "All" for "Mail to keep offline".
Please also check if you have any anti-virus or anti-spam integration with Outlook. If you do have this kind of integration, disable it to check the result.
Regards,
Melon Chen
TechNet Community Support
It's recommended to download and install
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