Applications of the Wold decomposition to the study of row contractions associated with directed graphs

Abstract

Abstract. Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of operators on Hilbert space which are related to the graph in a natural way. This yields a functional calculus determined by quiver algebras and free semigroupoid algebras. We establish a generalization of Coburn's theorem for the C * -algebra of a CKT family, and prove a universality theorem for C * -algebras generated by these families. In both cases, the C * -algebras generated by quiver algebras play the universal role.

The C*-envelope of the Tensor Algebra of a Directed Graph

“…when restricted to the operators with symbols in G, has range equal to the universal Cuntz-Krieger algebra for G. The next lemma shows that actually (1) is a complete isometry when restricted to λ Gs (T + (G s )), thus providing a shorter proof for the main result of this section, provided that G has no sources. We note that the 'unampliated' version of this lemma has appeared as Proposition 7.3 in [12].…”

Structure of free semigroupoid algebras

“…Here we lay out some of the theory of algebras associated to directed graphs. We recommend [49,34,36,39] for further material and proofs for some standard results mentioned here.…”

“…L G can be thought of as the analogue of the non-commutative analytic Toeplitz algebra for arbitrary directed graph, where L n corresponds to the graph on a single vertex with n loops. In [39,34,33,36] many of the results of Popescu and Arias, and of Davidson and Pitts on L n were extended and expanded to L G for arbitrary graphs. While in these papers L G is called a free semigroupoid algebra, we will expand that definition here so that it extends the original definition of free semigroup algebras.…”

All finite transitive graphs admit a self-adjoint free semigroupoid algebra

“…Towards establishing a non-self-adjoint theory for distinguishing representations of directed graphs, and by building on work of many authors [21,10,14,15,16,17,18], the first author together with Davidson and B. Li extended the theory of free semigroup algebras to classify representations of directed graphs via non-self-adjoint techniques [8]. Definition 1.1.…”