Remove the primitivity condition from the CB-primitive spectrum

Determine whether the condition that the kernel of a topologically irreducible bounded representation of the quantised function algebra C^\infty(K_q) be primitive can be removed from the definition of the CB-primitive spectrum; equivalently, determine whether the CB-primitive spectrum equals the set of all kernels of topologically irreducible bounded representations of C^\infty(K_q) that admit a completely bounded extension to a Beurling–Fourier algebra A(K_q,w_\beta) for some \beta\geq 1.

Background

The paper defines the CB-primitive spectrum of C\infty(K_q) using only topologically irreducible bounded representations whose kernels are already known to be primitive. Theorem 3.?? identifies this spectrum with the union of the homogeneous Joseph strata.

The authors ask whether the primitivity requirement is redundant. They explain that removing it would require controlling all topologically irreducible bounded representations admitting completely bounded extensions to the relevant Beurling–Fourier algebras. They further show that resolving this question is at least as difficult as the invariant subspace problem for bounded operators on separable infinite-dimensional Hilbert spaces, so the question remains unresolved in the paper.

References

One may wonder whether the condition $Ker(\theta) \in Prim (C\infty(K_q)) $ in the definition of the CB-primitive spectrum $ Prim_{CB}(C\infty(K_q)) $ can be removed, or equivalently, whether $ Prim_{CB}(C\infty(K_q)) $ in Theorem \ref{thm:Spectra of Beurling-Fourier covers Prim} can be replaced by the set of all kernels of topologically irreducible representations $ C\infty(K_q) \rightarrow L(H) $ which CB extend to $ A(K_q, w_\beta) $ for some $ \beta \geq 1 $.

— On the complexification of $ q $-deformations of compact semisimple Lie groups  (2610.23236 - Lee et al., 16 Oct 2026) in Remark 3.??, Section 3, subsection “Beurling-Fourier algebras and primitive spectrum”