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The Double B-Dual Of An Inner Product Module Over a C*-Algebra B

Published online by Cambridge University Press:  20 November 2018

William L. Paschke
William L. Paschke*
Affiliation:
University of Kansas, Lawrence, Kansas
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The principal result of this paper states that if X is a pre-Hilbert B-module over an arbitrary C*-algebra B, then the B-valued inner product on X can be lifted to a B-valued inner product on X″ (the B-dual of the B-dual X′ of X). Appropriate identifications allow us to regard X as a submodule of X″ and the latter in turn as a submodule of X′. In this sense, the inner product on X″ is an extension of that on X. As an example (and application) of this result, we consider the special case in which X is a right ideal of B and give a topological description of X″ when in addition B is commutative.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 5 , October 1974 , pp. 1272 - 1280
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Akemann, Charles A., The general Stone- Weierstrass problem, J. Functional Analysis 4 (1969), 277–294.Google Scholar
2. Douglas, Ronald G. and Pearcy, Carl, On the spectral theorem for normal operators, Proc. Cambridge Philos. Soc. 68 (1970), 393–400.Google Scholar
3. Paschke, William L., Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468.Google Scholar