Reflexivity and hyperreflexivity of bounded $N$-cocycles from group algebras

Abstract:

We introduce the concept of reflexivity for bounded $n$-linear maps and investigate the reflexivity of $\mathcal {Z}^n(L^1(G),X)$, the space of bounded $n$-cocycles from $L^1(G)^{(n)}$ into $X$, where $L^1(G)$ is the group algebra of a locally compact group $G$ and $X$ is a Banach $L^1(G)$-bimodule. We show that $\mathcal {Z}^n(L^1(G),X)$ is reflexive for a large class of groups including groups with polynomial growth, IN-groups, maximally almost periodic groups, and totally disconnected groups. If, in addition, $G$ is amenable and $X$ is the dual of an essential Banach $L^1(G)$-bimodule, then we show that $\mathcal {Z}^1(L^1(G),X)$ satisfies a stronger property, namely hyperreflexivity. This, in particular, implies that $\mathcal {Z}^1(L^1(G),L^1(G))$ is hyperreflexive.