The co-option of another person's personal information (e.g., name, Social Security number, credit card number, passport) without that person's knowledge and the fraudulent use of such knowledge WordNet 3.0 & Open English WordNet

Identity theft, identity piracy or identity infringement occurs when someone uses another's personal identifying information, like their name, identifying number, or credit card number, without their permission, to commit fraud or other crimes. Wikipedia

Deliberate use of someone else's identity, usually as a method to gain a financial advantage or obtain credit and other benefits in the other person's name, and perhaps to the other person's disadvantage or loss Wikidata

A fraud that involves stealing money or getting other benefits by pretending to be someone else. OmegaWiki

The deliberate assumption of another person's identity, usually to gain access to that person's finances or to frame a person for a crime. Wiktionary

Deliberate assumption of another person's identity. Wiktionary (translation)