What is Islamically preferable: registering the building in the name of the husband or the wife, if each paid half the amount and the purpose of the building is to be rented out?

It is permissible for a husband and wife to gift each other whatever they wish from their wealth. If the husband gifts his share to his wife, then the building should be registered in her name. If the wife gifts her share to her husband, then it should be registered in his name. If each of them wishes to retain their share, then the partnership should be registered between them. The intent is for the registration to conform to the reality of the situation, in order to preserve rights. If circumstances necessitate registering the building in the name of one person, then the other may keep documentation proving their right, such as an official document with witnesses.