On C∗-algebra of a semigroup of partial isometries

A C∗-algebra associated to strongly continuous one-parameter semigroups of partial isometries is introduced as a groupoid C∗-algebra C∗(G). A one-to-one correspondence between non-degenerate representations of C∗(G) and the semigroups is established. It is proved also that C∗(G) contains an ideal J isomorphic to C0((0,+∞],K) and C∗(G)/J≃C0(R), where K is the algebra of compact operators. The C∗-algebra C∗(G) is shown to be generated by one unbounded affiliated element whose image under each non-degenerate representation is the infinitesimal generator of the corresponding semigroup.