On local properties of Hochschild cohomology of a C${}^{\ast }$- algebra

Let $A$ be a C${}^{\ast }$-algebra, and let $X$ be a Banach $A$-bimodule. B. E. Johnson showed that local derivations from $A$ into $X$ are derivations. We extend this concept of locality to the higher cohomology of a $C^{\ast }$-algebra %for $n$-cocycles from $A^{(n)}$ into $X$ and show that, for every $n\in \mathbb{N}$, bounded local $n$-cocycles from $A^{(n)}$ into $X$ are $n$-cocycles.