Can a memory care center Baker Act a dementia patient and still allow the person to live there? What's the purpose?

I have never heard the Baker act used for a resident in a facility. I understood it to be used in the private sector when someone has a mental condition where they could harm themselves or others. But I guess its the same thing in a facility setting.

You posted later about not being told this was being done. I answered on that post that when a resident becomes violent and prescribed medication no longer works, the resident is sent out to a Psychiatric facility to find the correct combination of medication to control his violent behaviour. The resident will be better monitored. If all goes well, he should return to the MC unit. Its very a very common thing.