The following strengthening of a result of B. A. Barnes is proved: If ϕ \phi is a topologically irreducible representation of a C ∗ {C^ \ast } -algebra A \mathfrak {A} on a Banach space such that ϕ ( A ) \phi (\mathfrak {A}) contains a nonzero finite-rank operator, then ϕ \phi is similar to an irreducible ∗ ^ \ast -representation of A \mathfrak {A} (and is thus automatically continuous).