Limit algebras and integer-valued cocycles, revisited

A triangular limit algebra A is isometrically isomorphic to the tensor algebra of a C∗-correspondence if and only if its fundamental relation R(A) is a tree admitting a Z+0 -valued continuous and coherent cocycle. For triangular limit algebras which are isomorphic to tensor algebras, we give a very concrete description for their defining C∗-correspondence and we show that it forms a complete invariant for isometric isomorphisms between such algebras. A related class of operator algebras is also classified using a variant of the Aho-Hopcroft-Ullman algorithm from computer aided graph theory.