Do you see Illustrator there?
When you open the Creative Cloud app, does Illustrator have an Open button to the right of it?
If you are still having trouble, you need to contact Adobe Customer Care:
Try the link to auto-open the Chat:
If your browser blocks the Chat window, try the troubleshooting tips here:
You can also send a tweet to @AdobeCare.
Do not reply to PMs, which are likely to be scammers.
Log-out of Creative Cloud Desktop app and log-in again.
1. Open CC Desktop App.
2. Click your Avatar (top right image) > Preferences > General tab > Settings. Select "Always keep Creative Cloud up to date."
3. Click on your Avatar again.
4. Sign-out of Creative Cloud.
5. Close all apps.
6. Restart your computer.
7. Open CC Desktop app.
8. Click your Avatar.
9. Sign-in with your paid ID and password.
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In addition to the suggestions from @jane-e and @Nancy OShea you might want to open a support ticket through the Admin Console if you're still facing difficulties. I see that you have a team membership to Illustrator. Support would be able to help if you're still having difficulties. https://adminconsole.adobe.com/support?promoid=6NCS75BX&mv=other