Local 2-cocycles.
Let \(R\) be a von Neumann algebra and \(S\) be a unital dual Banach \(R\)-bimodule. A bounded bilinear mapping \(\Phi\colon R\times R\to S\) is called a local \(2\)-cocycle if, for each pair \((a,b)\in R\times R\), there exists a bounded bilinear mapping \(\Phi_{a,b}\colon R\times R\to S\) such that \(\Phi(a,b)=\Phi_{a,b}(a,b)\) and \(\Phi_{a,b}\) satisfies the cocycle identity \[ x\Phi_{a,b}(y,z)-\Phi_{a,b}(xy,z)+\Phi_{a,b}(x,yz)-\Phi_{a,b}(x,y)z=0 \qquad(x,y,z\in R).\tag{1} \] The author shows that every local \(2\)-cocycle is in fact a global \(2\)-cocycle, that is, it satisfies identity~(1). The proof proceeds in tedious but straightforward computations.
scientific article; zbMATH DE number 1836079