There is a issue with the desktop app approach: if we change app from the interal available option, to access lets say, the Calendar, then the mail notifications and unread mail counter wont work anymore. Whats worse, the user will eventually minimize the application to resume work expecting mail notifications that wont ever arrive.
I think I more solid approach for the desktop app would be for it to have a -minimalistic- sidebar with the options for each application, acting as tabs.
This has seveal benefits:
• Mail application would be always open and available, and thus, always desktop wrapper app always receiving messages (notifications, unread counter, etc).
• Other applications could also benefit from the same, like Calendar having its own set of notifications, like event reminders, alerts, etc.