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 $\tau$ is faithful, $H_{\tau }=L^2(A,\tau )$, and $J$ is the canonical conjugation, the cocycle has the particularly simple form $${\delta }_{\tau }(a)(x)=J{\pi }_{\tau }(a^{\ast })Jx-xJ{\pi }_{\tau }(a^{\ast })J\mathrm{.}$$ It is inner in a natural Banach $A$-bimodule if and only if $\tau$ is amenable. We also give the corresponding reformulation of embeddability into an ultrapower of the hyperfinite ${\mathrm{I}\mathrm{I}}_1$ 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.