Dear New York Tenant:
If you sublet the apartment as furnished, then until you are able to end the subtenant's right to possession, you may not enter the apartment and may not take back your furniture.
Since you are no longer on the lease, you may not even possess the right to bring an eviction proceeding based on the expiration of the sublease and the failure of the subtenant to move.
Moreover, as you state you have no formal contract then it is likely the sublease violated the tenant's lease with the landlord, and is actually an "illegal sublease."
New York State strictly controls a tenant's right to sublease an apartment (unless the lease itself provides a less restricted right) by a statute. The law requires both a written sublease agreement made by the tenant and subtenant, but also that the sublease must be acknowledged in the same manner a deed in a real property transaction. Assuming for a moment the emails may somehow rise to the level of a contract, the agreement is not acknowledged and is not a sublease as required by the statute.
The statute also requires the landlord's advance consent to the sublease.
Your subtenant may have an attorney and may know that you cannot legally enter the apartment or take back your furniture. You may have created the bane of all subletting tenants: The Illusory Tenancy, in which case, you may need an attorney to negotiate your way back to your furniture.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.
You can let him know you will call the police if he doesn't give you your stuff. Assuming the police won't get involved you can sue for conversion. I don't recomnend entering with the key and taking it. You could be arrested for trespass and maybe even burglary if he claims to own the stuff.
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You have a mess. Suing the guy will cause additional animosity, even destruction of your property. If you enter the premises, he will attempt to have cops arrest you. You can file a holdover proceeding in housing court against him ad a subtenant, but that will create even more animosity.
I wish you had a concrete lease and kept your property separate from his.