[section] [section] EXPLICIT CONSTRUCTION AND UNIQUENESS FOR UNIVERSAL OPERATOR ALGEBRAS OF DIRECTED GRAPHS
نویسنده
چکیده
Given a directed graph, there exists a universal operator algebra and universal C-algebra associated to the directed graph. In this paper we give intrinsic constructions of these objects. We provide an explicit construction for the maximal C-algebra of an operator algebra. We also discuss uniqueness of the universal algebras for finite graphs, showing that for finite graphs the graph is an isomorphism invariant for the universal operator algebra of a directed graph. We show that the underlying undirected graph is a Banach algebra isomorphism invariant for the universal C-algebra of a directed graph. There has been significant work in the study of operator algebras associated to combinatorial objects (e.g. groups, semigroups, and graphs). We have continued this study in [3] where the universal operator algebra of a directed graph and the universal C-algebra of a directed graph were introduced and described. The aim of this paper is twofold: first we refine the construction of the universal operator algebras of directed graphs, then we discuss invariants of the universal algebras of finite directed graphs. First we use ideas from [2] to define intrinsic norms on OA(Q) the universal operator algebra of a directed graph. This allows a more concrete construction than was given in [3]. We also describe a construction of the maximal C-envelope of an operator algebra, see [1]. This construction is defined instrinisically using the free product operator algebra construction of Blecher and Paulsen [2]. This suggests that the maximal C-envelope is not as mysterious as is presumed. In fact having a canonical construction should allow a more detailed study of the maximal operator algebra of a directed graph in particular cases. Kribs and Power show in [7] that the graph is a complete unitary invariant for the Toeplitz quiver algebra of a directed graph. Recent work on these Toeplitz quiver algebras by Katsoulis and Kribs, [6] and by Solel [12], has demonstrated that the graph is a complete isomorphism invariant for these algebras. In this paper we extend the techniques of [6] to show that for finite graphs the graph is a complete isomorphism invariant for OA(Q). This fact is perhaps not surprising, although the technique requires more subtlety than in [6] and [12]. For the universal Calgebra of a finite directed graph we are able to show that the underlying directed graph is an isomorphism invariant for the algebra. This is very surprising since the Cuntz-Krieger algebra of a directed graph is not classified by the graph. Before proceeding, we would like to emphasize a difference between the operator algebras in the present paper and those defined in [9]. When we construct the 2000 Mathematics Subject Classification. 47L40, 47L55, 47L75, 46L80.
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