Ideal Structure in Free Semigroupoid Algebras from Directed Graphs

A free semigroupoid algebra is the weak operator topology closed algebra generated by the left regular representation of a directed graph. We establish lattice isomorphisms between ideals and invariant subspaces, and this leads to a complete description of the wot-closed ideal structure for these algebras. We prove a distance formula to ideals, and this gives an appropriate version of the Carathéodory interpolation theorem. Our analysis rests on an investigation of predual properties, specifically the An properties for linear functionals, together with a general Wold Decomposition for n-tuples of partial isometries. A number of our proofs unify proofs for subclasses appearing in the literature. In [18] and [19], the second author and Stephen Power began studying a class of operator algebras called free semigroupoid algebras. These are the wot-closed (nonselfadjoint) algebras LG generated by the left regular representations of directed graphs G. Earlier work of Muhly and Solel [24, 25] considered the norm closed algebras generated by these representations in the finite graph case; they called them quiver algebras. In the case of single vertex graphs, the LG obtained include the classical analytic Toeplitz algebra H∞ [13, 14, 29] and the noncommutative analytic Toeplitz algebras Ln studied by Arias, Popescu, Davidson, Pitts, and others [1, 9, 10, 11, 20, 27, 28]. In this paper, we consider algebraic structure-type problems for the algebras LG. In particular, we derive a complete description of the wot-closed ideal structure; for instance, there is a lattice isomorphism between right ideals and invariant subspaces of the commutant LG = RG [11]. Furthermore, we prove a distance formula to ideals in these algebras [2, 12, 23]. This yields a version of the Carathéodory interpolation theorem [2, 12] for LG. A valuable tool in our analysis is a general Wold Decomposition [25, 26], which we establish for n-tuples of partial isometries with initial and final projections satisfying natural 2000 Mathematics Subject Classification. 47L75, 47L55. key words and phrases. Hilbert space, Fock space, directed graph, partial isometry, nonselfadjoint operator algebra, partly free algebra, Wold Decomposition, distance formula, Carathéodory Theorem. 1 2 M.T.JURY AND D.W.KRIBS conditions. This leads to information on predual properties for LG. We prove LG satisfies property A1 [6, 11]; that is, every weak-∗ continuous linear functional may be realized as a vector functional. A number of our proofs for general LG unify the proofs for the special cases of Ln and H∞, which were previously established by different means. The first section contains a brief review of the notation associated with these algebras. Our attention will be focused on the cases when the directed graph G has ‘no sinks’; that is, every vertex in G is the initial vertex for some directed edge. We include a list of some examples generated by simple graphs. We begin the analysis proper in the second section, with a Wold Decomposition for n-tuples of partial isometries. This leads into the topic of the third section; an investigation into the basic properties of linear functionals on LG. In particular, we show the ampliation algebras L (n) G have the factorization property An. In the subsequent section, we prove the subclass of algebras with partly free commutant discovered in [18, 19] are precisely those LG which satisfy the stronger factorization property Aא0 , when an initial restriction is made on the graph. Using property A1 for LG and the Beurling Theorem from [19], we establish complete lattice isomorphisms between ideals and invariant subspaces of LG in the fifth section. This allows us to describe, for example, the wot-closure of the commutator ideal, and precisely when wot-closed ideals are finitely generated. In the penultimate section we prove a completely isometric distance formula to ideals of LG, and we apply this in special cases to obtain a Carathéodory Theorem in the final section. 1. Free Semigroupoid Algebras Let G be a countable (finite or countably infinite) directed graph with edge set E(G) and vertex set V (G). Let F(G) be the free semigroupoid determined by G; that is, F(G) consists of the vertices which act as units, written as {k}k≥1, and allowable finite paths in G, with the natural operations of concatenation of allowable paths. Given a path w = eim · · · ei1 in F(G), an allowable product of edges eij in E(G), we write w = k2wk1 when the initial and final vertices of w are, respectively, k1 and k2. Further, by |w| we mean the number of directed edges which determine the path w. Assumption 1.1. For our purposes, it is natural to restrict attention to directed graphs G with ‘no sinks’; that is, every vertex is the initial vertex for some directed edge. Let HG = `(F(G)) be the Hilbert space with orthonormal basis {ξw : w ∈ F+(G)} indexed by elements of F(G). For each edge e ∈ IDEAL STRUCTURE IN FREE SEMIGROUPOID ALGEBRAS 3 E(G) and vertex k ∈ V (G), define partial isometries and projections on HG by: Leξw = { ξew if ew ∈ F(G) 0 otherwise