The $H^{\infty }$ algebras of higher rank graphs

Abstract

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.

22 pages, preprint version

References in corpus (5)

• Free Semigroupoid Algebras
• Ideal Structure in Free Semigroupoid Algebras from Directed Graphs
• Factoring in Non-commutative Analytic Toeplitz Algebras
• Non-selfadjoint Operator Algebras generated by Weighted Shifts on Fock Space
• A dual graph construction for higher-rank graphs, and $K$-theory for finite 2-graphs

Cited by in corpus (1)

• Representations of product systems over semigroups and dilations of commuting CP maps