Inner product modules over -algebras

Author:
William L. Paschke

Journal:
Trans. Amer. Math. Soc. **182** (1973), 443-468

MSC:
Primary 46K05; Secondary 46H25

MathSciNet review:
0355613

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Abstract: This paper is an investigation of right modules over a -algebra *B* which posses a *B*-valued ``inner product'' respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which *B* is a -algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and -algebras. The extension of an inner product module over *B* by a -algebra *A* containing *B* as a -subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0355613-0

Article copyright:
© Copyright 1973
American Mathematical Society