Read your employee handbook and IT regulations thoroughly; personal use of company equipment is almost always forbidden and doing so for profit doubly so. Whether looking at emails from your startup counts as the former or latter is subject to interpretation by HR and the legal department.
In practice, many (most?) people do check personal email and more on company equipment and nobody much cares ... unless, one day, the company has a reason to want to know what you've been doing (for example, because you had a side startup you haven't cleared with them first (check you employee handbook) or your manager randomly decides he doesn't like your face anymore) and then the rules apply full force and it's game over.