What is the ruling on working in a shop owned by the state, when it is said that its origin was the property of a Jew or another person, despite the current ownership of the seller being established?

If the state seized the shop from its Jewish owner without a legitimate Islamic justification, then it is not permissible to work in it, as this would be cooperating in falsehood. However, if the seizure was based on a legitimate Islamic justification, then there is no objection to working in it. Regarding your purchase of the shop through what is called "khulū al-rijl" (premium for vacating property), it is permissible if the paid amount is considered part of the rent, but it is not permissible if it is an additional amount not accounted for within the rent.