Hi There. I am Bing and I look forward to helping you with your issue today.
Please bear with me for a few mins as I need to check Microsoft KB against your issue.
Please the steps to rebuild the identity from the following MS official document. Ignore the "Entourage 2008 for Mac and Entourage for Mac 2004" in the title, it doesn't matter as a reference for your case. I am not asking you to do the given steps, just need to review and let me know if you have done any the same step there.
> If I then choose rebuilt, I lose all messages.
How does your Outlook access the mail server, via IMAP or not?
> I have not rebuilt the database.
Please clarify, you mean you have not "rebuilt the identity" or "repaired the Outlook database"?
> I use IMAP but all the 'sent items' were on my computer, no on a server
That's interesting. Commonly the Sent fold should be also synced. You did that on purpose or not?
FYI - See below how Microsoft says about it. Please read it for your practice in the futhre.
About Time Machine backups and Outlook
> I have not rebuilt the database. I only checked it with microsoft database daemon.
I just successful restored previous previous Outlook data including emails in all folders on my Mac. Basically all your previous Office files were located at:
~/Documents/Microsoft User Data/Office 20xx Identities/your identity
You may copy your previous Identity folder and subfolders here but do NOT overwrite the existing Identity folder, otherwise it will cause inconsistant data across the whole settings.
After switching your Outlook identity, you can export all previous Outlook files to an OLM file then import back to to the existing Identity if required.
You may choose to sync Sent folder to keep a copy on the server as well, the setting is located at Preferences > Accounts > (your account) > Advanced > Folders.