I'm currently looking for a citation to a statement of Imai-Takai duality, specifically for reduced $C^{*}$-algebra crossed products, and I can't seem to find one.
It's clear to me that such a result does hold, but I'm trying to find a source to cite that explicitly states this duality theorem in the context of a reduced crossed product.
I found a reference. For anyone else who may ever need it, it's in "A Categorical Approach to Imprimitivity Theorems for C*-Dynamical Systems", theorem A.68 in the appendix.