  Let $A$ be a C$^*$-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 $C^*$-algebra
%for $n$-cocycles from $A^{(n)}$ into $X$ and show that, for every $n\in \N$,
bounded local $n$-cocycles from $A^{(n)}$ into $X$ are $n$-cocycles.
