How can I pour a concrete patio under my existing deck without surrounding the deck posts in concrete?
Support either side of each post, cut it off, insert a steel stub base or pour a new concrete base to a height slightly above the planned slab, pour the new slab.

Or, just make sure the "expansion joint" around each post is actually around each footing, boxing out the whole area, and fill with pea stone or marble chips after pouring the slab.

Or skip pouring a slab and just put in a base for pavers and build the patio from pavers.