The stable uniqueness theorem for unitary tensor category equivariant KK-theory

We introduce the Cuntz-Thomsen picture of $\mathcal{C}$-equivariant Kasparov theory, denoted $\mathrm{KK}^\mathcal{C}$, for a unitary tensor category $\mathcal{C}$ with countably many isomorphism classes of simple objects. We use this description of $\mathrm{KK}^\mathcal{C}$ to prove the stable uniqueness theorem in this setting.

💡 Research Summary

The paper develops a Cuntz‑Thomsen picture of equivariant Kasparov theory for actions of a unitary tensor category 𝒞 on C∗‑algebras, denoted KK^𝒞, and uses this framework to prove a stable uniqueness theorem in the categorical setting. After recalling the necessary background on Hilbert C∗‑modules, C∗‑correspondences, and the structure of unitary tensor categories (including dual objects, standard solutions of the conjugate equations, and the pivotal structure), the authors define a 𝒞‑co‑cycle representation as a family of linear maps {ϕ_X : α(X) → L(B,β(X))}_X∈𝒞 satisfying compatibility conditions that encode the categorical action. Proposition 1.11 shows that such a representation can be viewed as a single object, which allows the construction of a “𝒞‑Cuntz pair” (ϕ,ψ) analogous to the classical Cuntz pair in KK‑theory.

Section 2 introduces the Cuntz‑Thomsen picture for KK^𝒞. The authors prove that homotopy classes of 𝒞‑Cuntz pairs correspond bijectively to elements of KK^𝒞((α,u),(β,v)). The key technical tool is a notion of “stable operator homotopy,” which stabilizes both representations by tensoring with the compact operators 𝒦 and then connects them via a norm‑continuous path of unitaries in the multiplier algebra of B⊗𝒦.

Section 3 develops the theory of stable operator homotopy in detail, establishing that the homotopy class is invariant under addition of a third representation and under unitary conjugation. This prepares the ground for the absorption arguments that follow.

In Section 4 the authors define absorbing 𝒞‑co‑cycle representations. Lemma 4.14 provides sufficient conditions for a representation to absorb another, extending the Elliott‑Kucerovsky absorption theorem to the categorical context. Theorem 4.16 shows that, under mild separability and σ‑unitality hypotheses, absorbing representations always exist. The proof relies on constructing countable direct sums of copies of a given correspondence and using the duality in 𝒞 to control the left and right actions.

Section 5 establishes asymptotic unitary equivalence. Given two 𝒞‑Cuntz pairs (ϕ,ψ) that represent the same KK^𝒞 class, the authors construct a third absorbing representation θ and a norm‑continuous path of unitaries u_t in the unitization of 1+B⊗𝒦 such that for every object X∈𝒞 and every ξ∈α(X) one has  (ψ_X⊕θ_X)(ξ) = lim_{t→∞} u_t·(ϕ_X⊕θ_X)(ξ)·u_t^*. The construction uses the categorical dual objects X̄, the standard solutions R_X, \bar R_X, and the pivotal isomorphisms μ_X to replace the group‑theoretic invertibility used in earlier work. The path u_t is built by intertwining the representations through a sequence of partial isometries that become asymptotically unitary after stabilization.

Finally, Section 6 states the main result, Theorem 6.2 (also called Theorem A). It asserts that for separable 𝒞‑C∗‑algebras (A,α,u) and σ‑unital (B,β,v), two 𝒞‑co‑cycle representations ϕ,ψ : (A,α,u) → (B⊗𝒦,β⊗id_𝒦,v⊗1) form a 𝒞‑Cuntz pair representing the zero element in KK^𝒞 if and only if there exists an absorbing representation θ and a norm‑continuous unitary path u_t with u_0=1 such that the asymptotic unitary equivalence above holds. This theorem generalizes the classical stable uniqueness theorem of Dadarlat‑Eilers and Lin, as well as its group‑equivariant version by Gabe‑Szabó, to the setting of quantum symmetries described by unitary tensor categories.

The paper concludes with acknowledgments and a bibliography that includes foundational works on KK‑theory, Cuntz‑Thomsen pictures, subfactor theory, and recent developments in tensor‑category actions on C∗‑algebras. The results provide a crucial tool for extending the Kirchberg–Phillips classification program to C∗‑algebras equipped with quantum symmetries, opening new avenues for classification results involving actions of fusion categories, quantum groups, and related structures.