Limit algebras and integer-valued cocycles, revisited

TL;DR

This paper characterizes when triangular limit algebras are isomorphic to tensor algebras via cocycles and provides concrete descriptions and invariants for these algebras, connecting operator algebra theory with graph algorithms.

Contribution

It offers a complete characterization of certain limit algebras using cocycles and introduces a graph-theoretic classification method.

Findings

Triangular limit algebras are isomorphic to tensor algebras if their fundamental relation is a tree with a specific cocycle.

Provides a concrete description of the defining C*-correspondence for these algebras.

Uses a graph algorithm to classify a related class of operator algebras.

Abstract

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.