摘要
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 .
采用同调理论的观点探讨了C 代数上HilbertC 模作为对象和有界模算子作为态射构成的范畴 .研究C 代数上HilbertC 模扩张的内射性和内射包络 ,通过内射性和本性给出内射包络的特征描述 .证明了如果一个C 代数的HilbertC 模的内射包络存在 ,则在H等距意义下是唯一的 .其次给出了HilbertC 模的扩张是内射包络 ,当且仅当此扩张是内射的和本性的 .进一步得到在H等距意义下W 代数上的任何HilbertC 模都有唯一的一个内射包络而且HilbertC
