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Inner Product Modules Over B*-Algebras

William L. Paschke
Transactions of the American Mathematical Society
Vol. 182 (Aug., 1973), pp. 443-468
DOI: 10.2307/1996542
Stable URL: http://www.jstor.org/stable/1996542
Page Count: 26
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Inner Product Modules Over B*-Algebras
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Abstract

This paper is an investigation of right modules over a B*-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 W*-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 W*-algebras. The extension of an inner product module over B by a B*-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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