A New Geometric Morita Invariant for Higher Rank Graph $C^*$-algebras

Higher rank graphs, also known as $k$-graphs, are a $k$-dimensional generalization of directed graphs and a rich source of examples of $C^$-algebras. In the present paper, we contribute to the geometric classification program for $k$-graph $C^$-algebras by introducing a new move on $k$-graphs, called LiMaR-split, which is a generalization of outsplit for directed graphs. We show, under one additional assumption, that LiMaR-split preserves the $k$-graph $C^*$-algebras up to Morita equivalence.

💡 Research Summary

This paper contributes to the geometric classification program for higher‑rank graph (k‑graph) C⁎‑algebras by introducing a new graph move called the LiMaR‑split, which generalizes the out‑split operation from ordinary directed graphs to k‑graphs without the restrictive “pairing condition” that previous definitions required. The authors begin by reviewing the historical development of geometric classification for graph C⁎‑algebras, noting the seminal moves (source deletion, insplit, outsplit, reduction, Cuntz splice, and the P‑move) that generate flow‑equivalence and preserve Morita equivalence in the 1‑graph setting. They then discuss earlier extensions of these moves to k‑graphs, especially the work of Eckhardt‑et al., which introduced four moves (insplitting, delay, sink deletion, reduction) but left the out‑split limited by a pairing condition.

The LiMaR‑split is defined on a source‑free, row‑finite k‑graph Λ by first selecting a vertex v and its “neighbourhood” (the collection of edges of all colours incident to v). Instead of splitting a single vertex, the move simultaneously splits the entire neighbourhood: for each colour eᵢ, the set of outgoing edges of colour eᵢ from v is replaced by a family of new vertices, and similarly for incoming edges. The underlying 1‑skeleton G of Λ is altered to a new coloured graph G_L, and the equivalence relation ∼ that encodes the factorisation property of a k‑graph is modified to ∼_L. The authors verify that ∼_L satisfies the four Kumjian‑Pask conditions (KG0)–(KG3), thereby proving (Theorem 3.10) that the quotient G_L⁎/∼_L is again a k‑graph, denoted Γ, provided Λ is sink‑free (the additional mild hypothesis ensures no new sinks are created).

Having constructed Γ, the paper turns to the algebraic consequences. In the Kumjian‑Pask algebra setting (KP_R(Λ) over a commutative ring R), Theorem 4.9 shows that KP_R(Λ) is Zᵏ‑graded ∗‑isomorphic to a corner e KP_R(Γ) e, where e is a projection built from the new vertices. Moreover, this isomorphism carries the diagonal subalgebra D(Λ) onto the corresponding corner of D(Γ). Consequently, KP_R(Λ) and KP_R(Γ) are Zᵏ‑graded Morita equivalent. The proof proceeds by constructing explicit generators for the corner algebra, checking the Kumjian‑Pask relations (KP1)–(KP4) on these generators, and using the graded uniqueness theorem to obtain the isomorphism.

The C⁎‑algebraic result follows from the fact that the Kumjian‑Pask algebra over ℂ is dense in the k‑graph C⁎‑algebra C⁎(Λ). By extending the graded ∗‑isomorphism to the completions, Corollary 4.11 establishes that C⁎(Λ) and C⁎(Γ) are stably ∗‑isomorphic; the isomorphism preserves the canonical diagonal subalgebras and intertwines the natural gauge action of the k‑torus 𝕋ᵏ. As a direct corollary (4.12), the Nᵏ‑actions (the dual actions) associated with Λ and Γ are conjugate. Hence, LiMaR‑split preserves the Morita equivalence class of the associated C⁎‑algebras, and even more strongly, it yields a stable ∗‑isomorphism respecting the full gauge symmetry.

The paper is organized as follows: Section 2 collects necessary preliminaries on k‑graphs, their C⁎‑algebras, and Kumjian‑Pask algebras, emphasizing Theorem 2.1 which identifies k‑graphs with k‑coloured directed graphs modulo an equivalence relation. Section 3 introduces the LiMaR‑split, proves that the resulting structure is a k‑graph, and discusses the role of the sink‑free hypothesis. Section 4 develops the algebraic machinery, proves a series of technical lemmas for both Kumjian‑Pask and C⁎‑algebras, and culminates in the main results (Theorem 4.9, Corollaries 4.11 and 4.12). The authors also remark that the LiMaR‑split can be iterated and combined with previously known moves, suggesting a path toward a finite generating set of moves for the full geometric classification of row‑finite, source‑free k‑graph C⁎‑algebras.

In summary, the LiMaR‑split enriches the toolkit for geometric classification of higher‑rank graph C⁎‑algebras by providing a flexible, neighbourhood‑based out‑splitting operation that avoids earlier technical constraints, and by establishing that this move preserves both the analytic (C⁎‑algebra) and algebraic (Kumjian‑Pask) structures up to the strongest possible equivalences. This work paves the way for future research aiming to achieve a complete geometric classification of k‑graph C⁎‑algebras analogous to the celebrated classification of unital graph C⁎‑algebras.