Let M be a Hilbert C*-module over a C*-algebra A. Suppose that K(M) is the space of compact operators on M and the bounded antihomomorphism p: A-->B(M) defined by p(a)(m) = ma for all a in Aand m in M. In this paper, we first provide some characterizations of module maps on Banach modules over Banach algebras by several local conditions (some of our results are a generalization of previous results) and then apply them to characterize the reflexive closure of K(M) and p(A) , i.e., Alg LatK(M) and Alg Lat p(A), where we think of K(M) and p(A) as operator algebras acting on M. As an application of our results on reflexive closure of K(M) and p(A), a characterization of commutativity for C*-algebras is given.
