Naimark's Problem for graph C*-algebras and Leavitt path algebras

TL;DR

This paper proves that Naimark's Problem has an affirmative answer for both graph C*-algebras and Leavitt path algebras, using boundary paths to classify irreducible representations and spectra.

Contribution

It establishes the affirmative solutions to Naimark's Problem for these algebras and characterizes their spectra based on boundary path equivalence classes.

Findings

Naimark's Problem is affirmatively answered for graph C*-algebras.

The algebraic analogue of Naimark's Problem is affirmatively answered for Leavitt path algebras.

Characterization of when a graph C*-algebra has a countable spectrum.

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.