Geodesic
The Double B-Dual Of An Inner Product Module Over a C*-Algebra B
Canadian journal of mathematics, Tome 26 (1974) no. 5, pp. 1272-1280

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Paschke, William L. The Double B-Dual Of An Inner Product Module Over a C*-Algebra B. Canadian journal of mathematics, Tome 26 (1974) no. 5, pp. 1272-1280. doi: 10.4153/CJM-1974-121-0
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