Symmetric states for infinite noncommutative tori with non-Hermitian bicharacters

Determine how to define and characterize symmetric states directly on the infinite noncommutative torus when the rotation parameter is irrational and the associated bicharacter is non-Hermitian, despite the absence of a natural finitary permutation-group action.

Background

For irrational rotation parameters, the bicharacters defining the infinite noncommutative torus are non-Hermitian, so the natural action of the finitary symmetric group does not extend to the twisted tensor product itself. The paper instead defines symmetric states indirectly as states induced from the infinite free product. The authors explicitly leave unresolved how symmetric states should be handled directly on the noncommutative torus.

References

It is then unclear how to manage the symmetric states directly on this infinite tensor product $_$.

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

It is a very delicate question to understand whether an analogous result holds true in the case of the one-sided twisted chains constructed by using any, possibly compatible, $C*$-norm.

— 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 various completions of the twisted tensor product”