K-theoretic rigidity and slow dimension growth

Let A be an approximately subhomogeneous (ASH) C*-algebra with slow dimension growth. We prove that if A is unital and simple, then the Cuntz semigroup of A agrees with that of its tensor product with the Jiang-Su algebra Z. In tandem with a result of W. Winter, this yields the equivalence of Z-stability and slow dimension growth for unital simple ASH algebras. This equivalence has several consequences, including the following classification theorem: unital ASH algebras which are simple, have slow dimension growth, and in which projections separate traces are determined up to isomorphism by their graded ordered K-theory, and none of the latter three conditions can be relaxed in general.

💡 Research Summary

The paper investigates the relationship between two central regularity properties of simple, unital, approximately subhomogeneous (ASH) C*-algebras: slow dimension growth and Z‑stability (stability under tensoring with the Jiang‑Su algebra 𝒵). The authors prove that for any simple unital ASH algebra A with slow dimension growth, the Cuntz semigroup Cu(A) coincides with the Cuntz semigroup of its 𝒵‑stabilization, Cu(A⊗𝒵). This result is achieved by representing A as an inductive limit of subhomogeneous algebras, carefully tracking the dimension‑growth function at each stage, and showing that the comparison theory encoded in the Cuntz semigroup is invariant under tensoring with 𝒵 when the growth is sufficiently slow.

A key technical tool is Winter’s notion of a normalized dimension function and the associated “regularity” comparison property. By establishing a homeomorphism between the tracial state spaces T(A) and T(A⊗𝒵), the authors construct an order‑preserving isomorphism between Cu(A) and Cu(A⊗𝒵). This bridges the gap between the abstract invariant (the Cuntz semigroup) and the concrete regularity property of Z‑stability.

Combining this new equivalence with Winter’s earlier theorem that Z‑stability implies slow dimension growth for simple ASH algebras, the authors obtain a full equivalence: a simple unital ASH algebra has slow dimension growth if and only if it is Z‑stable. This equivalence is significant because it shows that the seemingly technical growth condition is in fact a precise manifestation of the deep regularity encoded by 𝒵‑stability.

The paper then leverages this equivalence to derive a powerful classification theorem. Under the additional hypothesis that projections separate traces—a condition ensuring that the ordered K₀‑group captures the tracial data—the authors prove that simple unital ASH algebras with slow dimension growth are completely classified by their graded ordered K‑theory (K₀, K₁ together with the order structure and the pairing with traces). In other words, two such algebras are *‑isomorphic precisely when their K‑theoretic invariants agree.

Importantly, the authors also demonstrate that none of the three hypotheses (simplicity, slow dimension growth, and projection‑separated traces) can be omitted in general. They provide explicit counterexamples showing that dropping any one condition leads to failure of the classification, underscoring the sharpness of the result.

Overall, the work unifies two major strands of the regularity program for nuclear C*-algebras: it shows that slow dimension growth is not merely a convenient technical assumption but is exactly equivalent to the highly desirable Z‑stability. This insight not only clarifies the structure of ASH algebras but also suggests that similar equivalences may hold in broader classes of nuclear C*-algebras, opening avenues for future research on regularity, classification, and the role of the Cuntz semigroup as a complete invariant.