Christensen–Sinclair converse for the C*-algebraic coefficient module

Determine whether the Christensen–Sinclair converse holds for the C*-algebraic coefficient module S_τ of admissible completely bounded bilinear maps, namely, whether the coboundary property of the cocycle Φ_cs associated with a faithful tracial C*-algebra implies amenability of the trace τ.

Background

For a faithful tracial state τ on a unital C*-algebra A, the paper defines the Banach A-bimodule S_τ of admissible completely bounded bilinear maps on B(H)×B(H), where H=L²(A,τ). The Christensen–Sinclair cocycle Φ_cs is shown to be a coboundary whenever τ is amenable.

The unresolved direction is the converse: the coefficient module S_τ is generally larger than the von Neumann algebra coefficient module used in the original Christensen–Sinclair result, because its maps are required to vanish on JAJ×B(H), rather than on M′×B(H). The paper identifies two technical obstructions: compact implementers need not exist, and even pointwise implementers may not admit a simultaneous linear completely bounded choice.

References

No converse is asserted here. The von Neumann algebra module in \S2 requires vanishing on $M'\times B(H)$, whereas the present, generally larger, module requires vanishing only on $JAJ\times B(H)$. Moreover, the converse proof in Theorem~2.3 uses the fact that, on the continuous von Neumann summand, the relevant compact-valued derivations have unique compact implementers with uniform norm estimates. For the present cyclic C*-representation, Christensen's theorem guarantees only an implementer in $B(H)$. If $D:A\to K(H)$ is written as $D(a)=[T,a]$, Corollary \ref{cor:calkin} identifies the obstruction to a compact implementer with the coset of $q(T)\in q(A)'\cap\mathcal Q(H)$ modulo $q(A')$. Nothing in the present hypotheses forces this coset to vanish. Even if all pointwise obstructions vanish, a converse would still require a simultaneous linear, completely bounded choice of implementers. Thus the Christensen--Sinclair converse does not presently transfer to this C*-algebraic coefficient module.

— Cohomology of Amenable Traces  (2609.18756 - Moradi, 16 Sep 2026) in Remark \ref{rem:cs-converse}, Section 6, “A C*-algebraic Christensen--Sinclair two-cocycle”