The real property in Italy is controlled by Italian law and not US law regardless of where you live. Even if you are living in NY State, to assure that the house passes to the person that you wish you have to have a Will and renunciations according to Italian Law.
You will likely also need a will that is proper in NY State to control any assets that you will own in the US. US assets would include bank accounts, investment accounts and personal property located in the US. I am happy to follow up with some information regarding the requirements for a handwritten will
I will caveat that if you are an Italian citizen living in the US then your situation is particularly complex. Even if you do not have a tremendous amount of assets you could have an situation where estate tax occurs in both countries on all your assets. You really need to sit down with an experienced estate planning attorney that can review US/Italian Tax Treaties to determine how to best structure your estate to avoid a tremendous amount of double taxation.
I cannot provide you with legal advise. I have provided you with information about the law related to your question. My answer, and any information that you find online, should not take the place of having a consultation with a lawyer in your area to advise you regarding your specific issues.
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