arXiv:1007.2896v1 [math.RT] 17 Jul 2010 TOEPLITZ OPERATORS IN HILBERT SPACE OVER INFINITE GRAPHS ILWOO CHO AND PALLE E. T. JORGENSEN Abstract. Associated with a given graph G, typically an infinite tree, and motivated by applications, we introduce two families of operators in a Hilbert space H G induced by G. To realize the Hilbert space, we first develop some representation theory. We obtain the first family of operators on H G by an extension of the more familiar case of groups: free representations of the group- algebra. Because of their classical counter parts, we call the operators in our first family, graph operators; and the second Toeplitz operators. We focus on the interconnections between the two families. We introduce and study graph operators in two steps: first, starting with a fixed graph G, we introduce a groupoid over G; and from this, the groupoid von Neumann algebra M G . Our graph operators will then be finitely supported elements of M G . 1. Introduction Before, starting our problem, we open with a historical comment, and a com- parison between the case of groups and graphs. In a number of recent papers there have been a variety of different approaches to introducing algebras of operators in Hilbert space (for a sample, see the papers cited below). A number of these ideas are motivated by what works for groups, i.e., starting with the group alge- bra, and then build representations of it. Each representation serves some purpose, or is dictated by an application, for example to harmonic analysis or to quantum mechanics. More than half a century ago, von Neumann introduced the ring of operators (now called von Neumann algebras) generated by the free group F n with n-generators, leading to non-hyperfinite factors L(F n ) (See [27]). While the con- struction is simple enough, the questions are difficult. Now, for F n , the natural Hilbert space is l 2 (F n ). Since a group acts on itself, we get operators in l 2 (F n ), i.e., regular representation; and L(F n ) is simply the von Neumann algebra generated by the regular representation. Now, let G be a countable directed graph, i.e., a system of vertices and edges (with direction) subject to simple axioms, details below. It is tempting, in the analysis of graphs, to mimic some of the constructions used for groups. But a glance at the comments above and literature shows that there are difficulties for graphs that do not arise in the case of groups. A key idea we employ is in brief outline this: Starting with a graph G, we introduce first an “enveloping” groupoid Date : Aug., 2010. 1991 Mathematics Subject Classification. 05C62, 05C90, 17A50, 18B40, 47A99. Key words and phrases. Directed Graphs, Graph Groupoids, Graph Operators, Toeplitz Operators. The second named author is supported by the U. S. National Science Foundation. 1