If you received a Notice to File a Business Property Statement from the Assessor and your business is no longer in operation, you are still required to file the Business Property Statement. You should also include a note on the Business Property Statement indicating that the business has closed, so the Assessor does not continue to assess the property.
Limited liability companies must pay state fees during the incorporation process. These fees can be deducted from taxes. S-Corps must pay state fees to legally incorporate. These fees can be deducted from taxes. C-Corps must pay state fees to become legally recognized. These fees can be deducted from taxes. Non-Profits pay state fees when they incorporate. These fees can be deducted from taxes. Since Sole Proprietorships aren't incorporated entities, they don't pay formation or compliance fees.
On the other hand, if you are not responsible for its tax, then you should declare the equipment in Part III of the Business Property Statement (Equipment belonging to others). Where equipment is declared in Part III of the Business Property Statement, the Assessor will also send a Notice to File to the person reported as the equipment's actual owner.