The purple star with a number inside is your Reputation level. This increases when you earn more Reputation Points, and allows you to attract richer clients into your bank - this gets you bigger cash deposits!

You earn Reputation Points by upgrading your bank. Make sure you upgrade all areas of your bank:
• Main Hall
• Marketing Office
• Service Area
• Vault Room

Make sure you also complete:
• Missions which appear top right of the game screen