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Beeper vs Apple battle intensifies: Lawmakers demand DOJ investigation - Android Authority
(www.androidauthority.com)
This is a most excellent place for technology news and articles.
If they function identically to a normal client though what's the issue? As an example Google indexes pages all over the web without the explicit permission of those websites, that requires them to read the page and make requests to someone else's infrastructure.
What part exactly here is illegal?
The websites in question getting crawled and indexed are generally open and available for anyone to browse. There are parts of the web that are gated off and require authentication and authorization to access. Imagine now that Google found a way to authenticate as you with your bank's website and index your online banking portal. (It's not a perfect analogy to what's happening with Beeper, but I'm just using the one you laid out.)
In a similar way, iMessage as a service requires authentication and authorization to use. It is not open for anyone to use. Beeper is doing something to spoof or otherwise fool Apple into giving the client access. This is the part that's illegal. And potentially not just "file a lawsuit" illegal but criminally so.
It doesn't really matter why Apple doesn't want Beeper or anyone else to use it. The fact that they simply don't is all that matters.