Local Proof of Algebraic Characterization of Free Actions

Let $G$ be a compact Hausdorff topological group acting on a compact Hausdorff topological space $X$. Within the $C^{*}$-algebra $C(X)$ of all continuous complex-valued functions on $X$, there is the Peter-Weyl algebra $\mathcal{P}_G(X)$ which is the (purely algebraic) direct sum of the isotypical components for the action of $G$ on $C(X)$. We prove that the action of $G$ on $X$ is free if and only if the canonical map $\mathcal{P}_G(X)\otimes_{C(X/G)}\mathcal{P}_G(X)\to \mathcal{P}_G(X)\otimes\mathcal{O}(G)$ is bijective. Here both tensor products are purely algebraic, and $\mathcal{O}(G)$ denotes the Hopf algebra of"polynomial"functions on $G$.


Theorem
Given a compact Hausdorff topological group G, we denote by O(G) the dense Hopf *subalgebra of the commutative C*-algebra C(G) spanned by the matrix coefficients of irreducible representations of G. Let X be a compact Hausdorff topological space with a given continuous (right) action of G. The action map X × G ∋ (x, g) −→ xg ∈ X determines a map of C*-algebras δ : C(X) −→ C(X × G) .
Using the coassociativity of δ, one can check that P G (X) is a right O(G)-comodule algebra. In particular, P G (G) = O(G). The assignment X → P G (X) is functorial with respect to continuous G-equivariant maps and comodule algebra homomorphisms. We call it the Peter-Weyl functor. Equivalently [S-PM11, Proposition 2.2], P G (X) is the (purely algebraic) direct sum of the isotypical components for the action of G on C(X) (see [ is bijective. (Here both tensor products are purely algebraic.) Definition 1.2. The action of a compact Hausdorff group G on a compact Hausdorff space X satisfies the Peter-Weyl-Galois (PWG) condition iff the canonical map (1.1) is bijective.
Our result states that the usual formulation of free action is equivalent to the algebraic PWG-condition. In particular, our result provides a framework for extending Chern-Weil theory beyond the setting of differentiable manifolds and into the context of cyclic homologynoncommutative geometry [BH04].

Proof
The proof of the equivalence of freeness and the PWG-condition consists of six steps. The first step takes care of the easy implication of the equivalence, and the remaining five steps prove the difficult implication of the equivalence.
2.1. PWG-condition ⇒ freeness. It is immediate that the action is free, i.e. xg = x =⇒ g = e (where e is the identity element of G), if and only if is a homeomorphism. Here X × X/G X is the subset of X × X consisting of pairs (x 1 , x 2 ) such that x 1 and x 2 are in the same G-orbit. This is equivalent to the assertion that the * -homomorphism obtained from the above map F is an isomorphism. Note that F is always surjective, so that the * -homomorphism F * is always injective. Furthermore, there is the following commutative diagram in which the vertical arrows are the evident maps: Since the right-hand side of the canonical map (1.1), i.e. P G (X) ⊗ O(G), is dense in the C*-algebra C(X × G), validity of the PWG-condition combined with the commutativity of the diagram (2.1) implies that the image of the * -homomorphism F * is dense in C(X × G). Therefore, as the image of a * -homomorphism of C*-algebras is always closed, PWG implies surjectivity of F * , which in turn implies that the action of G on X is free.
2.2. Reduction to surjectivity and matrix coefficients. 2.3. Reduction to free actions of compact Lie groups. Assume that Theorem 1.1 holds for compact Lie groups. In this section we prove that this special case implies Theorem 1.1 in general.
Let ϕ : G → U(n) be any finite-dimensional representation of G. Set , and this action is free. The map where I n ∈ U(n) is the identity matrix, has the equivariance property (2.2) Φ(xg) = Φ(x)ϕ(g) .
Hence Φ and ϕ induce maps Φ * : P U (n) (X ϕ ) → P G (X) and ϕ * : O(U(n)) → O(G). The equivariance property (2.2) implies commutativity of the diagram Therefore surjectivity of the upper canonical map implies that 1 ⊗ h is in the image of the lower canonical map, where h is any matrix coefficient of ϕ. By Lemma 2.1, this implies the PWG-condition.
2.4. Local triviality for free actions of compact Lie groups. We recall the theorem of A. M. Gleason: Theorem 2.2 ([G-AM50]). Let G be a compact Lie group acting freely and continuously on a completely regular space X. Then X is a locally trivial G-bundle over X/G.
Combining the Gleason theorem with Section 2.3, the PWG-condition is valid for free actions if it is valid for locally trivial free actions.

2.5.
Reduction to the trivial-bundle case. Assume that the action of G on X is free and locally trivial. Since the quotient space X/G is compact Hausdorff, we can choose a finite open cover U 1 , . . . , U r of X/G such that each π −1 (U j ) is a trivializable principal G-bundle over U j .
Here π : X → X/G is the quotient map. On X/G, let ψ 1 , . . . , ψ r be a partition of unity subordinate to the cover U 1 , . . . , U r . Then, for each j there is the canonical map If for each j ∈ {1, . . . , r} there exist elements p j1 , q j1 , . . . , p jn , q jn ∈ P G π −1 (supp(ψ j ) such that can j n i=1 p ji ⊗ q ji = 1 ⊗ h , then for each i and j we take p ji ψ j • π, q ji ψ j • π ∈ P G (X), and for any x ∈ X and g ∈ G, using the commutativity of the diagram (2.1), we obtain Here p ji 's and q ji 's are functions on X obtained respectively from functions p ji 's and q ji 's by zero-value extension. Hence validity of the PWG-condition in the trvial-bundle case implies that the PWG-condition holds for actions that are free and locally trivial.
2.6. The trivial-bundle case. Consider Y × G, where Y is a compact Hausdorff topological space and G acts on Y × G by (y, g 1 )g 2 = (y, g 1 g 2 ). First note that This is implied by two facts: (1) quite generally P G (X) ⊆ C(X) is the purely algebraic direct sum of the isotypical components for the action of G on C(X) [S-PM11, Proposition 2.2]; and (2) each isotypical component for the action of G on C(Y × G) is of the form C(Y ) ⊗ V , where V is an isotypical component for the action of G on C(G).

5
As O(G) is a Hopf algebra, the dual of the homeomorphism g 1 g 2 ) ∈ G × G and the dual of its inverse F −1 G restrict and corestrict respectively to . Granted the identification (2.3), we now obtain the following commutative diagram: Hence the bijectivity of can G implies the bijectivity of can.

Appendix
In this appendix, we observe that Step 2.5, i.e. reduction to the trivial-bundle case, is implied by the following general results: Lemma 3.1. Let (H, ∆) be a compact quantum group acting on a unital C*-algebra A. The assignment A → P H (A) of the Peter-Weyl algebra to a C*-algbra yields a functor from the category with objects being unital C * -algebras with an H-coaction (and morphisms being equivariant unital * -homomorphisms) to the category whose objects are O(H)-comodule algebras (and whose morphisms are colinear algebra homomorphisms). Furthermore, this functor commutes with all (equivariant) pullbacks, that is, Lemma 3.2 (Lemma 3.2 in [HKMZ11]). Let H be a Hopf algebra with bijective antipode, and let P be the pullback diagram of surjective right H-comodule algebra homomorphisms. Then P is principal if and only if P 1 and P 2 are principal. (For the definition of "principal" see [BH04] and [HKMZ11, Definition 2.3].) Proving the first lemma is straightforward, and the second lemma is the highlight of [HKMZ11]. For the case considered in this paper (a compact Hausdorff group G acting continuously on a compact Hausdorff space X), we have A = C(X), H = C(G), O(H) = O(G), P H (A) = P G (X), and the condition of being "principal" is equivalent to the PWG-condition. Thus these two lemmas combined with standard induction yield an alternative proof of Step 2.5.
Finally, the theorem of this paper is a special case of a much more general theorem about compact quantum groups acting on unital C*-algebras [BDH]. However, the proof of the general theorem is nonlocal.