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You can provide notifications on the web. Support maybe limited to some browsers only but it is possible.

This works in Chrome: http://jsfiddle.net/dandv/wT26x/1/

I think Safari has a system for this too: https://developer.apple.com/notifications/safari-push-notifi...

Firefox: https://developer.mozilla.org/en/docs/Web/API/notification

CanIUse stats: http://caniuse.com/notifications

Although it's not quite the same as a native app, we already have built in support for desktop notifications on all browsers that support them, including some older WebKits.

so how about a tray icon? AFAIK there is no way to create one using HTML5.

What do you need the tray icon for...?

There are still a great many organizations that are using XP. It's a reliable and battle tested OS. Ignoring these users wouldn't be a good idea, I don't think.