On local properties of Hochschild cohomology of a C$^*$- algebra

TL;DR

This paper extends the concept of local derivations to higher Hochschild cohomology of C*-algebras, proving that bounded local n-cocycles are indeed n-cocycles, thus generalizing Johnson's result.

Contribution

It introduces the notion of locality to higher cohomology of C*-algebras and proves that bounded local n-cocycles are genuine n-cocycles, expanding previous results on derivations.

Findings

Bounded local n-cocycles are n-cocycles for all n.

Extension of locality concept to higher cohomology.

Generalization of Johnson's theorem to n-cocycles.

Abstract

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.