The spectral theorem for bimodules in higher rank graph $C\sp *$-algebras

Abstract

In this note we extend the spectral theorem for bimodules to the higher rank graph $C^*$-algebra context. Under the assumption that the graph is row finite and has no sources, we show that a bimodule over a natural abelian subalgebra is determined by its spectrum iff it is generated by the Cuntz-Krieger partial isometries which it contains iff the bimodule is invariant under the gauge automorphisms. We also show that the natural abelian subalgebra is a masa iff the higher rank graph satisfies an aperiodicity condition.