Galois Objects and Cocycle Deformations

Suppose (W, Δ) is a locally compact quantum group with a left invariant faithful semifinite normal weight x. Let α be a right coaction of (W, Δ) on a von Neumann algebra U that is ergodic, i.e. with fixed point algebra ℂ $${\mathbb C}$$ , and is integrable, meaning that y1 = (ι ⊗ x)α is a semifinite weight on U. Consider the basic construction ℂ ⊂ U ⊂ B ( V y 1 ) $${\mathbb C}\subset U\subset B(V_{y_1})$$ , and the normal ∗-epimorphism f : U ⋊ α W → B ( V y 1 ) $$f\colon U\rtimes _\alpha W\to B(V_{y_1})$$ uniquely determined by fα = ι and f(1 ⊗ (ω ⊗ ι)(N)) = (ι ⊗ ω)(Kr) for ω ∈ W∗, where N = ( J x ̂ ⊗ J x ̂ ) M ̂ ( J x ̂ ⊗ J x ̂ ) $$N=(J_{\hat {x}}\otimes J_{\hat {x}})\hat {M}(J_{\hat {x}}\otimes J_{\hat {x}})$$ and Kr is the unitary implementation of α.