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A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].

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Hilbert $C^*$-modules and approximate units

I haven't properly thought about it, but it seems to me that for countably generated modules a positive answer follows from proposition 1.6 in http://arxiv.org/abs/math/0301259
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