Hilbert C^*-Module

The notion of a Hilbert -module is a generalization of the notion of a Hilbert space. The first use of such objects was made by Kaplansky (1953). The research on Hilbert -modules began in the 1970s in the work of the induced representations of-algebras by Rieffel (1974) and the doctoral dissertation of Paschke (1973). Hilbert -modules are useful tools in -algebra theory, theory of operator algebras, operator -theory, group representation theory, and theory of operator spaces. It is also used to study Morita equivalence of -algebras, -theory of -algebras, -algebra quantum group (Lance 1995, Wegge-Olsen 1993).

A pre-Hilbert module over a -algebra is a complex linear space which is a left -module (and where , , and ) equipped with an -valued inner product satisfying:

1. ,

2. iff ,

3. ,

4. ,

5. .

A pre-Hilbert -module is called a Hilbert -module or Hilbert -module over , if it is complete with respect to the norm . If the closed linear span of the set is dense in then is called full. For example every -algebra is a full Hilbert -module whenever we define .