The H Algebras of Higher Rank Graphs

We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hilbert space and creation operators which are partial isometries acting on the space. We call the weak operator topology closed algebra generated by these operators a higher rank semigroupoid algebra. A number of examples are discussed in detail, including the single vertex case and higher rank cycle graphs. In particular the cycle graph algebras are identified as matricial multivariable function algebras. We obtain reflexivity for a wide class of graphs and characterize semisimplicity in terms of the underlying graph. In [22] Kumjian and Pask introduced k-graphs as an abstraction of the combinatorial structure underlying the higher rank graph Calgebras of Robertson and Steger [31, 32]. A k-graph generalizes the set of finite paths of a countable directed graph when viewed as a partly defined multiplicative semigroup with vertices considered as degenerate paths. The C-algebras associated with k-graphs include k-fold tensor products of graph C-algebras, and much more [2, 21, 26, 27, 30]. On the other hand, as a generalization of the nonselfadjoint free semigroup algebras Ln [3, 5, 6, 7, 20, 28, 29], the authors [17, 18] have recently studied free semigroupoid algebras LG associated with directed countable graphs G. In particular it was shown that these algebras are reflexive. (See also [12, 13, 14, 15, 16, 23, 24, 25, 34] for related recent work.) As it turns out, these algebras arise from the left regular representation of the 1-graph of the directed graph G. In the present paper we consider the higher rank versions of these algebras, the k-graph algebras L(Λ,d) associated with the k-graph (Λ, d), as well as their norm closed subalgebras. To our knowledge such nonselfadjoint higher rank graph algebras have not been considered previously. However, from the perspective of contemporary operator algebra theory they evidently form a natural class and one which may play an 2000 Mathematics Subject Classification. 47L55, 47L75, 47L80.