On outer automorphisms of certain graph $C^{*}$-algebras

Given a countable abelian group $A$, we construct a row finite directed graph $Γ(A)$ such that the $K_{0}$-group of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(Γ(A))$ is canonically isomorphic to $A$. Moreover, each element of $\textrm {Aut}(A)$ is a lift of an automorphism of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(Γ(A))$.

💡 Research Summary

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The paper addresses the problem of realizing every automorphism of a countable abelian group A as an automorphism of a C*‑algebra whose K₀‑group is exactly A. The authors work within the framework of graph C*‑algebras, exploiting the fact that for a row‑finite directed graph G the K₀‑group of C*(G) can be described as the cokernel of a simple linear map derived from the adjacency structure.

After recalling basic notions—inner automorphisms, approximately inner automorphisms, and the K‑outer automorphism group Kout(C)=Aut(C)/Inn(C)—the authors note that while for AF‑algebras Kout(C) coincides with the positive automorphism group of K₀(C), in general many K₀‑automorphisms fail to lift to genuine C*‑automorphisms (the classic example being Mₙ(ℂ), whose K₀ is ℤ but all automorphisms are inner).

The core construction proceeds as follows. For any countable abelian group A, consider its canonical presentation: generators {a} for each element a∈A and relations {a}+{b}-{c}=0 whenever a+b=c. The authors translate this presentation into a row‑finite directed graph Γ(A). Each generator a yields a vertex vₐ. For each relation a+b=c they introduce auxiliary vertices u_{ab} and u_c together with edges arranged so that in K₀(C*(Γ(A))) the relation u_{ab}=vₐ+v_b−v_c holds. To force u_{ab}=0 they add an infinite chain of vertices uⁿ_{ab} (n≥1) linked in a way that each yields the equation uⁿ_{ab}=uⁿ⁻¹_{ab}+uⁿ_{ab}, which forces uⁿ_{ab}=0 in K₀. When a=b, a special pair of intermediate vertices wₐ, w_b is inserted to avoid multiple edges, ensuring the construction remains row‑finite.

With this design, the K₀‑group of C*(Γ(A)) is generated by the classes