Weakly almost periodic functionals, representations, and operator spaces

Details Weakly almost periodic functionals, representations, and operator spaces Weakly almost periodic functionals, representations, and operator spaces

2007-01-31

arxiv.org/abs/math/0701935v2

Mathematics

Functional Analysis

Latex, 13 pages; minor corrections; added reference to work of Uygul which shows the same result

Scientific paper

A theorem of Davis, Figiel, Johnson and Pe{\l}czy\'nski tells us that weakly-compact operators between Banach spaces factor through reflexive Banach spaces. The machinery underlying this result is that of the real interpolation method, which has been adapted to the category of operator spaces by Xu, showing the this factorisation result also holds for completely bounded weakly-compact maps. In this note, we show that Xu's ideas can be adapted to give an intrinsic characterisation of when a completely contractive Banach algebra arises as a closed subalgebra of the algebra of completely bounded operators on a reflexive operator space. This result was shown by Young for Banach algebras, and our characterisation is a direct analogue of Young's, involving weakly almost periodic functionals.

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