Cohomology of Amenable Traces

Abstract: 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 [ δτ(a)(x) =Jπτ(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.