Cohomology of Amenable Traces

We construct a bounded Hochschild one-cocycle which detects amenability of tracial states on unital C*-algebras. For an arbitrary trace the construction uses a faithful representation containing the GNS representation as a direct summand. When \(τ\) is faithful, \(H_τ=L^2(A,τ)\), and \(J\) is the canonical conjugation, the cocycle has the particularly simple form \[ δ_τ(a)(x) =Jπ_τ(a^*)Jx-xJπ_τ(a^*)J. \] It is inner in a natural Banach \(A\)-bimodule if and only if \(τ\) is amenable. We also give the corresponding reformulation of embeddability into an ultrapower of the hyperfinite \(\mathrm{II}_1\) factor. Finally, for faithful traces we study a C*-algebraic variant of the Christensen--Sinclair two-cocycle and prove, with all admissibility details, that amenability makes this cocycle a coboundary.