The Spectral Theorem for Bimodules in Higher Rank Graph C*-algebras

TL;DR

This paper extends the spectral theorem for bimodules to higher rank graph C*-algebras, characterizing bimodule spectra and invariance properties under gauge automorphisms.

Contribution

It introduces a spectral theorem for bimodules in higher rank graph C*-algebras, linking spectra, gauge invariance, and aperiodicity conditions.

Findings

Bimodules are determined by their spectrum if generated by contained partial isometries.

Bimodules are invariant under gauge automorphisms iff they are determined by their spectrum.

The abelian subalgebra is a masa iff the graph satisfies an aperiodicity condition.

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.