Equality of the minimal and intermediate Klein twisted tensor-product norms

Determine whether the minimal twisted tensor-product norm on the Klein four-group system coincides with the intermediate norm defined using products of invariant states under the subgroup Δ₊^⊥, without assuming nuclearity of the underlying C*-algebra.

Background

For a C*-dynamical system based on the Klein four-group, the paper defines an intermediate C*-norm using product states invariant under the subgroup Δ₊⊥. The authors prove that this norm lies between the minimal and maximal twisted tensor-product norms and that all three coincide under nuclearity. Without nuclearity, the subgroup-invariant states may strictly contain the Klein-group-invariant states, so the equality between the minimal and intermediate norms is left unresolved.

References

Without assuming the nuclearity of $$, it seems unclear whether $|\cdot|{}$ coincides with $|\cdot|{+}$ or not, since in general ${\Delta+\perp}()$ can properly contain $_{K_4}()$.

— Infinite twisted $C^*$-tensor product and symmetric states  (2609.09494 - Fidaleo et al., 8 Sep 2026) in Section 9, concluding remarks, subsection “Symmetric states on the (infinite) noncommutative torus”