Why I cannot explain, but, yes, it is needed. See the following website for how: http://italian.about.com/library/fare/blfare132a.htm
Here is a table showing how the possessive adjectives are used in Italian. Apparently they always have the definite article before them. http://italian.about.com/library/fare/blfare132a.htm
Roxanne839598: I recommend modern Greek to see another language that uses the definite article in much the same way. It makes sense. But yes, it is different from English and some other languages, including Spanish long form possessive adjectives (often), e.g., la silla tuya.
Guy.s: I read the "ha" in the sentence as in possession that doesn't imply that he is consuming the [my] lunch. I read the sentence literally: "He has my lunch." Io sono un bambino in understanding Italian but I think actually that "having lunch" may be an American English idiom for eating lunch. I think the Italians "prendere," or "take lunch."