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If A is an algebra of operators on a C-space H, then the action of A induces a faithful module AH whose submodules are the A-invariant subspaces of H, and if A is generated by a commuting family of nilpotent operators, then A is a split local commutative C-algebra in the sense that, as C-spaces, A = C⊕ J with J = J(A) the unique maximal ideal of A. Let K be an arbitrary field. A module RM over a K-algebra R is called reflexive (see [6] or [3], for example) if the only K-linear transformations of M that preserve the submodule lattice of RM are multiplications by elements of R. Thus, if λ : R → End(KM) is the ring homomorphism induced by R-scalar multiplication, then λ(R) ∼= R/ann(M) and RM is reflexive if and only if λ(R) is a reflexive algebra of K-operators on M . In the terminology of [3], the set of K-linear transformations of M that preserve the submodule lattice of RM is alglat(M) = {α ∈ End(KM) | αm ∈ Rm for all m ∈M},

On reflexivity of representations of local Commutative Algebras