The co-op owns the building, and the residents own the co-op. Yes, the co-op is shares of stock. There are UCC filings and guidelines for documenting the share ownership, you need to get financial documentation from the co-op, etc.
There are co-ops authorized in many different states, and like most regulations, they are slightly different by state. You may want to check with an attorney familiar with co-ops in the state you're going to be dealing with.
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