On Local Properties of Hochschild Cohomology of a C- Algebra

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 and show that, for every n ∈ N, bounded local n-cocycles from A into X are n-cocycles. The study of the local properties of Hochschild cohomology of a Banach algebra was initiated by introducing the concept of “local derivations”. Let A be a Banach algebra, and let X be a Banach A-bimodule. An operator D : A → X is a local derivation if for each a ∈ A, there is a derivations Da : A → X such that D(a) = Da(a). This concept was introduced independently by R. V. Kadison [5] and D. R. Larson [6] and it has been interests of studies since then. Kadison’s motivation was based on his and J. R. Ringrose’s earlier investigation of Hochschild cohomology of various operator algebras, whereas Larson’s motivation was to investigate algebraic reflexivity (resp. reflexivity) of the linear space of derivations (resp. bounded derivations) from a Banach algebra. Local derivations have been investigated for various classes of Banach algebras such as operator algebras, Banach operator algebras, group algebras, and Fourier algebras (see [3], [8], [9] and the references therein). In [5], Kadison showed that bounded local derivations from a von Neumann algebra into any of its dual bimodules are derivations. He then raised the question of whether the preceding result can be extended to the local higher cohomology. The purpose of this article is to answer affirmatively to this question in more general setting. We show that if A is a C-algebra and n ∈ N, then bounded local n-cocylces from A(n) into any Banach A-bimodule are n-cocycles. This has already been obtained by B. E. Johnson in [4] for the case n = 1. Our approach is as follow: Let A be a Banach algebra, let X be a Banach A-bimodule, and let n ∈ N. In Section 2, we introduce certain n-linear maps from A(n) into X which are more general than local n-cocycles. We call them n-hyperlocal maps. We show that in order to characterize bounded local n-cocycles from A(n) into X, it suffices to first extend them to A♯(n), where A is the unitization 1991 Mathematics Subject Classification. Primary 47B47, 46L57; Secondary 46J10.