mail is 'sent" but does not show up in the sent box and does not arrive at destination

I have four devices which sync across apple mail. iMac, iPad Pro, Powerbook and iPhone.

On the iMac any mail sent rom my gmail account or my cloud account seems to be "sent' as it disappears and the whoosh sound signifies it being sent.

It never arrives at destination and does not show up in the sent box.

But it It does receive mail as normal.

The other three devices act as normal.

At iMac start up mail tells tells that it has to rebuild because another device is using mail, so I shut everything down including the iMac...then started it up again, leaving the others shut down.

Mail opened as normal, no rebuild message but the problem remains.

I have sent test messages to my self from all the accounts in the iMac and they all have the same fault so it does not appear to be a problem with just one account.

Any idea where I go from here ???