Hyperreflexivity of bounded $$N$$ N -cocycle spaces of banach algebras

Abstract

Let \(X\) and \(Y\) be Banach spaces, \(n\in \mathbb {N}\), and \(B^n(X,Y)\) the space of bounded \(n\)-linear maps from \(X\times \ldots \times X\) (\(n\)-times) into \(Y\). The concept of hyperreflexivity has already been defined for subspaces of \(B(X,Y)\), where \(X\) and \(Y\) are Banach spaces. We extend this concept to the subspaces of \(B^n(X,Y)\), taking into account its \(n\)-linear structure. We then investigate when \(\mathcal {Z}^n(A,X)\), the space of all bounded \(n\)-cocycles from a Banach algebra \(A\) into a Banach \(A\)-bimodule \(X\), is hyperreflexive. Our approach is based on defining two notions related to a Banach algebra, namely the strong property \((\mathbb {B})\) and bounded local units, and then applying them to find uniform criterions under which \(\mathcal {Z}^n(A,X)\) is hyperreflexive. We also demonstrate that these criterions are satisfied in variety of examples including large classes of C\(^*\)-algebras and group algebras and thereby providing various examples of hyperreflexive \(n\)-cocyle spaces. One advantage of our approach is that not only we obtain the hyperreflexivity for bounded \(n\)-cocycle spaces in different cases but also our results generalize the earlier ones on the hyperreflexivity of bounded derivation spaces, i.e. when \(n=1\), in the literature. Finally, we investigate the hereditary properties of the strong property \((\mathbb {B})\) and b.l.u. This allows us to come with more examples of bounded \(n\)-cocycle spaces which are hyperreflexive.