Free actions of compact quantum groups on unital \(C^\ast\)-algebras
The theorem of this paper is stated as follows. Let \(A\) be a unital \(C^*\)-algebra with an action of a compact quantum group \(H_q\) with co-multiplication \(\Delta: H_q\rightarrow H_q \otimes H_q\), with the coaction \(\delta\) as an injective unital \(\ast\)-homomorphism from \(A\) to \(A \otimes H_q\). Then there are equivalences among: (1) the action of \(H_q\) on \(A\) is free; (2) the action of \(H_q\) on \(A\) satisfies the Peter-Weyl-Galois (PWG) condition; (3) the action of \(H_q\) on \(A\) is strongly monoidal. Also provided by the theorem above is the following characterization. Let \(G\) be a compact Hausdorff group acting continuously on a compact Hausdorff space \(X\). Then the action of \(G\) on \(X\) is free if and only if the canonical map, defined below, is an isomorphism. Recall several definitions concerning the first statement as follows. By definition, the coaction \(\delta\) satisfies both the co-associativity related to \(\Delta\) and the co-unitality (or density) in \(A\otimes H_q\) related to \(A\) and \(H_q\). By definition, the coaction is free if the density in \(A\otimes H_q\) related to only \(A\) holds. Moreover, for a compact quantum group \(H_q\), there is its dense Hopf \(\ast\)-subalgebra \(H_f\) spanned by the matrix coefficients of its irreducible unitary representations. Then the Peter-Weyl subalgebra \(PW_{H_q}(A)\) of \(A\) is defined as the inverse image of \(\delta\) in \(A\otimes H_f\). By definition, the coaction satisfies the PWG condition if the canonical map from \(PW_{H_q}(A)\otimes_B PW_{H_q}(A)\) to \(PW_{H_q}(A) \otimes H_f\) involving \(\delta\) is bijective, where \(B\) is the unital (fixed point) \(C^*\)-subalgebra of \(A\) of coaction invariants \(a\in A\) so that \(\delta(a)= a\otimes 1\). By definition, the coaction \(\delta\) is strongly monoidal if the map extended from the canonical map above by involving co-tensor products with any left \(H_f\)-comodules \(V\) and \(W\) as well as \(V \otimes W\), respectively on the left and right sides, is bijective.
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