Inner product modules over 𝐵*-algebras

Paschke, William L.

doi:10.1090/S0002-9947-1973-0355613-0

Inner product modules over $B^{\ast }$-algebras
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by William L. Paschke PDF

Trans. Amer. Math. Soc. 182 (1973), 443-468 Request permission

Abstract:

This paper is an investigation of right modules over a ${B^\ast }$-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 ${W^\ast }$-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 ${W^\ast }$-algebras. The extension of an inner product module over B by a ${B^\ast }$-algebra A containing B as a $^\ast$-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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