Injective Envelopes of a Hilbert C^*Module

As in homology, the notion of injectivity is introduced in the category whose objects are Hilbert C module over a C algebra and whose morphism are bounded module operators. The definition of injective envelopes of an extension of a Hilbert C modules over a C algebra is introduced, and is characterized in terms of the injectivity and essence. It is shown that every Hilbert C module has a unique (up to H isometrics) injective envelope if it exists. It is also shown that an extension of a Hilbert C module is an injective envelope if and only if it is an injective and essential extension. Moreover, every Hilbert C module over a W algebra has a unique (up to H isometrics) injective envelope and the injective envelope of a Hilbert C module H is maximal essential extension of H .