My android app starts a service that opens a websocket to communicate to a remote server. The service spawns a thread whose run method looks like this.
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If you want this function to be guaranteed to continue to run while your app's UI is in the background, you will need to make that service run as a foreground service. There are some restrictions/guidelines on the use of foreground services, see the documentation at https://developer.android.com/guide/components/services.html.
Alternatively, if this is work that needs to occur on a periodic recurring basis and does not need to run continuously, you may be able to utilize JobScheduler; see https://developer.android.com/topic/performance/scheduling.html.