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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Inner product modules over $ B\sp{\ast} $-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 $ {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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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0355613-0
PII: S 0002-9947(1973)0355613-0
Article copyright: © Copyright 1973 American Mathematical Society