Naimark's Problem for graph C*-algebras and Leavitt path algebras
Kulumani M. Rangaswamy, Mark Tomforde
TL;DR
The paper proves that Naimark's problem has an affirmative answer for graph $C^*$-algebras and the algebraic analogue for Leavitt path algebras by constructing boundary-path based irreducible representations and exploiting the graph's shift-tail structure. It identifies a precise correspondence between unitary equivalence classes of irreducible representations and shift-tail equivalence classes of boundary paths in the countable-spectrum case, and provides a comprehensive classification via saturated hereditary subsets and graph quotients. A trichotomy for the spectrum of graph $C^*$-algebras is established: when cycles are absent, spectra are countable and align with boundary-path STE classes; when cycles exist, the spectrum becomes uncountable and can vary widely with STE classes. The results unify representation theory, ideal structure, and graph topology, yielding both a positive resolution of Naimark-type questions for these algebras and a detailed map from graph geometry to spectral properties.
Abstract
We describe how boundary paths in a graph can be used to construct irreducible representations of the associated graph C*-algebra and the associated Leavitt path algebra. We use this construction to establish two sets of results: First, we prove that Naimark's Problem has an affirmative answer for graph C*-algebras, we prove that the algebraic analogue of Naimark's Problem has an affirmative answer for Leavitt path algebras, and we give necessary and sufficient conditions on the graphs for the hypotheses of Naimark's Problem to be satisfied. Second, we characterize when a graph C*-algebra has a countable (i.e., finite or countably infinite) spectrum, and prove that in this case the unitary equivalence classes of irreducible representations are in one-to-one correspondence with the shift-tail equivalence classes of the boundary paths of the graph.