Representations of semigroups of partial isometries

We present a new proof that the irreducible representations of the von Neumann algebra generated by a strongly continuous semigroup of partial isometries of index 1 are unique up to equivalence, as well as a proof that when such an algebra is a factor, its representations are completely reducible. As an application, we show that the irreducible representations of a strongly continuous semigroup of isometries { U ( α ), α ⩾ 0} such that U ( α ) x → α → ∞ 0 are equivalent.