The short answer is no. Assuming your webmail admin has direct access to the web server which is running Roundcube then they would also have the ability to cover their tracks, removing logs etc. If your mail administrator (the people who run your IMAP server) are different to the webmail admin then they might be able to give you some info from the IMAP server logs. IMO if you cannot trust your admin the best thing to do is to move providers.