K-theory invariance under equality of positive operator norms for unconditional completions

Determine whether two unconditional completions A(G) and B(G) of ℂG that induce equal operator norms on positive elements via the left regular representation necessarily have isomorphic K-theory for their operator closures 𝔄^*_r(G) and 𝔅^*_r(G).

Background

The paper discusses unconditional completions and compares norm closures of the left regular representation on different completions. It is known that amenability yields norm equalities that lead to K-theory coincidences.

The authors ask if equality of operator norms on positive elements alone suffices to guarantee that the associated operator closures have the same K-theory, extending the amenability paradigm.

References

Open question. Let A(G) and B(G) two unconditional completions of C. Consider the algebras 𝔄*_r(G) and 𝔅*_r(G) which are respectively the norm closures of λ(C) in the bounded operators on A(G) and B(G), where λ is the left regular representation. Suppose that for all f∈ℝ+G we have ∥f∥{𝔄r(G)}=∥f∥{𝔅^_r(G)}, does it implies that 𝔄*_r(G) and 𝔅*_r(G) have the same K-theory?

— The rapid decay property for pairs of discrete groups  (2412.07994 - Chatterji et al., 2024) in Open question, Section: K-theoretical questions