ON STRONG SHIFT EQUIVALENCE FOR ROW-FINITE GRAPHS AND C*-ALGEBRAS

Kevin Aguyar Brix, Pete Gautam

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Theorem 3.4 proves that if E1 and E2 are row-finite graphs that are elementary SSE via a bipartite graph E3, then there are complementary full projections P1,P2 in M(C*(E3)), an action α of T, and diagonal-preserving, gauge-equivariant isomorphisms onto the corners P1C*(E3)P1 and P2C*(E3)P2; hence the triples (C*(E1),γE1,D(E1)) and (C*(E2),γE2,D(E2)) are Morita equivalent (, ). The candidate solution establishes the same conclusion via the squared-graph viewpoint and a slightly different, and actually cleaner, choice of circle action. The only substantive issue in the paper is a small presentation slip in the definition of α (they write z^{1/2} on edges), which does not define a T-action; replacing it by integer weights (exactly as in the model’s construction) fixes the matter without affecting the argument (, cf. weighted gauge actions in ).

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

This note strengthens Bates’ SSE-to-Morita equivalence by incorporating graphs with sources and by preserving the diagonal and the gauge action. The construction through a bipartite mediator and θ-bijections is concise and effective. A small but important correction is needed in the definition of the circle action α (replace half-powers by integer-weighted exponents), and a brief explicit verification of the fullness of P1 and P2 under the source condition would improve completeness. With these minor changes, the paper is correct, clear, and a useful contribution to the graph C*-algebra literature.