∗-representation of the complex group ring

Invariant subspaces of ∗-representations and unitary representations coïncide

Consider a mutually inducing pair of a Unitary representation and a ∗-representation . Then every invariant subspace under is an invariant subspace of and vice-versa. #m/thm/rep Thus is an irrep iff is irreducible, i.e. has no non-trivial invariant subspace.

Proof

Let be an invariant subspace of . Then is also an invariant subspace of , because for any and

Likewise if is an invariant subspace of then it is also an invariant subspace of , because for any and

as required.

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