A Dynamical Interpretation of K Theory for Graph C Star Algebras
DOI:
https://doi.org/10.62051/27qj2b25Keywords:
K theory, graph C star algebra, etale groupoid, symbolic dynamics, shift space, Smith normal form, essential skeleton, move equivalence.Abstract
This paper develops a dynamical interpretation of the K theory of graph C star algebras through a systematic analysis of five canonical graph families, selected to isolate distinct dynamical phenomena: transient flow, periodic recurrence, mixing, and component interaction. For each family, the groups and are computed and the resulting invariants are correlated with explicit features of the associated graph groupoid and shift dynamics. The free part of is shown to count irreducible sink components, torsion in records the period of cyclic subsystems, and the rank of measures the number of independent cycles in the essential skeleton of the graph. A conjectural dictionary is proposed, linking these algebraic invariants to structural and dynamical properties of directed graphs, and pathways toward rigorous proofs via move equivalence and groupoid homology are indicated. Connections to network robustness, Markov chains, and higher rank graphs are briefly explored.
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