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, Hτ=L2(A,τ), and J is the canonical conjugation, the cocycle has the particularly simple form δτ(a)(x) =Jπτ(a*)Jx−xJπτ(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 II1 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.
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BibTeX
@misc{moradi2026cohomology,
title = {{Cohomology of Amenable Traces}},
author = {Mehdi Moradi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18756},
eprint = {2609.18756},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18756},
}
RIS
TY - UNPB
TI - Cohomology of Amenable Traces
AU - Mehdi Moradi
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18756
UR - https://arxiv.org/abs/2609.18756
AB - 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 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.
ER -