Takesaki duality generalizes to weak* closed L^2-operator crossed products but fails to generalize to L^p-operator crossed products for p ≠ 2.
Larsen, An introduction to the theory of multipliers, Die Grundlehren der math- ematischen Wissenschaften, Band 175 Springer-Verlag, New York-Heidelberg, 1971, xxi+282 pp
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Takesaki duality for weak* closed $L^p$-operator crossed products
Takesaki duality generalizes to weak* closed L^2-operator crossed products but fails to generalize to L^p-operator crossed products for p ≠ 2.