Customer Center > Show Activity on Case > There are no Activities associated on this record

Make sure that the user's Customer Center role has at least View level of access to Track Messages permission. To add the permission:

     1. Navigate to Setup > Users/Roles > Manage Roles.
     
     2. Edit the Customer Center role.
     
     3. Under Permissions tab > Lists subtab, add Track Messages and set Level to at least View.
     
     4. Click Save.

Note: By default, the standard Customer Center role has Track Messages permission level set to View.