Reflexivity and Hyperreflexivity of Bounded N-cocycles from Group Algebras

We introduce the concept of reflexivity for bounded n-linear maps and investigate the reflexivity of Zn(L1(G),X), the space of bounded ncocycles from L1(G)(n) into X, where L1(G) is the group algebra of a locally compact group G and X is a Banach L1(G)-bimodule. We show that Zn(L1(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 L1(G)-bimodule, then we show that Z1(L1(G), X) satisfies a stronger property, namely hyperreflexivity. This, in particular, implies that Z1(L1(G), L1(G)) is hyperreflexive. The concept of reflexivity for linear subspaces of bounded operators on Banach spaces has its origin in operator theory. LetX be a Banach space, and let A ⊂ B(X) be an algebra of bounded operators on X. Let LatA denote the set of the closed subspaces of X invariant under A; i.e. for every T ∈ A and I ∈ LatA we have T (I) ⊂ I. We say that A is reflexive if A is the algebra generated by LatA; i.e. every bounded operator satisfying T (I) ⊂ I for every I ∈ LatA belongs to A. This concept is closely related to the well-known invariant subspaces problem: whether a bounded operator T ∈ B(X) has an invariant subspace. In [10], D. R. Larson generalized the concept of reflexivity, both algebraically and topologically, to subspaces of B(X,Y ) for Banach spaces X and Y . One motivation was to study the local behavior of derivations from a Banach algebra A to a Banach A-bimodule X. The main question that one asks is, for which algebras every so-called “local derivation” is a derivation, or equivalently, which algebras have algebraically reflexive derivation spaces? One can also ask the topological version of this question; i.e. when is the linear space of bounded derivations reflexive [10]? In the last two decades, the question of (algebraic) reflexivity of the derivation space has received considerable attention from various researchers, and some very interesting results have been obtained. In [9], R. D. Kadison showed that bounded local derivations from a von Neumann algebra into any of its dual bimodules are derivations. Kadison’s result was generalized later by showing that the space of bounded derivations from a C∗-algebra into any of its modules is both algebraically Received by the editors January 3, 2010 and, in revised form, February 20, 2010. 2010 Mathematics Subject Classification. Primary 47B47, 43A20.