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If you own Live Suite or you've purchased the Max for Live add-on for Live Standard, you can activate your license of Max for Live by going to Live's Preferences (Live > Settings), clicking on the Licenses & Updates tab, and clicking "Authorize at ableton.com". You will then be able to use Max for Live devices and instruments in Live.
If you own Max for Live but Max says you are not authorized, this is probably because you are using the standalone Max application, rather than Max for Live. Max for Live and Max are two separate products and require separate authorization. For instructions on how to switch back to Max for Live, please see the article Common Questions About Max for Live.
If you are still having problems with the authorization of Max for Live, try reauthorizing the Live application, or contact Ableton support.