NONCOMMUTATIVE WOLD DECOMPOSITIONS FOR SEMIGROUPS OF ISOMETRIES

The structure of the isometric representations of unital discrete semi groups with the left cancellation property and no divisors of the iden tity is studied. We obtain Wold decompositions and prove that the left regular representation of such semigroups is irreducible. Contractive representations of the free product (n)(i=1)G(i)(+) of discrete subs emigroups G(i)(+) of R+ are considered in connection with their minima l isometric dilations. Using the Wold decomposition, we characterize t he class C-0 of all contractive representations of (n)(i=1)G(i)(+) fo r which the minimal isometric dilation is unitarily equivalent to the left regular representation of (n)(i=1)G(i)(+) on l(2)(*(n)(i=1)G(i)( +)) x H. A Rota-Foias model theorem is obtained for contractive repres entations of class C-0. This is used to provide a noncommutative analo gue of the Nagy-Foias H-infinity-functional calculus for C-0 contracti ons.