While looking at Arveson's "An invitation to C* algebras", at the moment of defining type I representations (p. 47), he says that a (non degenerate) representation is type I if every central subrepresentation contains a multiplicity free representation. One line further, he says this is equivalent to the fact that every central subrepresentation contains an irreducible subrepresentation.
My question is why does the fact that there's a multiplicity free subrepresentation implies that there's an irreducible subrepresentation?
Thanks for any help