1 Theorem

Local Proof of Algebraic Characterization of Free Actions

\ArticleName

Local Proof of Algebraic Characterization
of Free ActionsThis paper is a contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel. The full collection is available at http://www.emis.de/journals/SIGMA/Rieffel.html

\Author

Paul F. BAUM  and Piotr M. HAJAC

P.F. Baum and P.M. Hajac

Mathematics Department, McAllister Building, The Pennsylvania State University,
University Park, PA 16802, USA \EmailDbaum@math.psu.edu

Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski,
ul. Hoża 74, 00-682 Warszawa, Poland

\ArticleDates

Received February 13, 2014, in final form May 21, 2014; Published online June 06, 2014

\Abstract

Let  be a compact Hausdorff topological group acting on a compact Hausdorff topological space . Within the -algebra of all continuous complex-valued functions on , there is the Peter–Weyl algebra which is the (purely algebraic) direct sum of the isotypical components for the action of  on . We prove that the action of  on  is free if and only if the canonical map is bijective. Here both tensor products are purely algebraic, and denotes the Hopf algebra of “polynomial” functions on .

\Keywords

compact group; free action; Peter–Weyl–Galois condition

\Classification

22C05; 55R10; 57S05; 57S10

With admiration and affection, to Marc A. Rieffel on the occasion of his 75th birthday

## 1 Theorem

Given a compact Hausdorff topological group , we denote by the dense Hopf -subalgebra of the commutative -algebra spanned by the matrix coefficients of irreducible representations of . Let  be a compact Hausdorff topological space with a given continuous (right) action of . The action map

 X×G∋(x,g)⟼xg∈X

determines a map of -algebras

 δ: C(X)⟶C(X×G).

Moreover, denoting by the purely algebraic tensor product over the field of complex numbers, we define the Peter–Weyl subalgebra [bhms, (3.1.4)] of as

Using the coassociativity of , one can check that is a right -comodule algebra. In particular, . The assignment is functorial with respect to continuous -equivariant maps and comodule algebra homomorphisms. We call it the Peter–Weyl functor. Equivalently [s-pm11, Proposition 2.2], is the (purely algebraic) direct sum of the isotypical components for the action of  on  (see [m-gd61, p. 31] and [l-lh53], cf. [p-p95, Theorem 1.5.1]). In the special case that the action of  on  is free, is the algebra of all continuous sections of the vector bundle  on , where

 E:=X×GO(G). (1)

Note that in forming this vector bundle, is topologized as the direct limit of its finite-dimensional vector subspaces, not by the norm topology.

The theorem of this paper is:

{theorem}

Let  be a compact Hausdorff space equipped with a continuous right action of a compact Hausdorff group . Then the action is free if and only if the canonical map

 can: x⊗y⟼(x⊗1)δ(y), (2)

is bijective. Here both tensor products are purely algebraic. {definition} The action of a compact Hausdorff group  on a compact Hausdorff space  satisfies the Peter–Weyl–Galois PWG condition iff the canonical map (2) 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 homology – noncommutative geometry [bh04].

## 2 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.  (where  is the identity element of ), if and only if

 F: X×G⟶X×X/GX, F: (x,g)⟼(x,xg),

is a homeomorphism. Here is the subset of consisting of pairs such that and are in the same -orbit.

This is equivalent to the assertion that the -homomorphism

 F∗: C(X×X/GX)⟶C(X×G)

obtained from the above map  is an isomorphism. Note that  is always surjective, so that the -homomorphism is always injective. Furthermore, there is the following commutative diagram in which the vertical arrows are the evident maps:

 (3)

Since the right-hand side of the canonical map (2), i.e. , is dense in the -algebra , validity of the PWG-condition combined with the commutativity of the diagram (3) implies that the image of the -homomorphism is dense in . Therefore, as the image of a -homomorphism of -algebras is always closed, PWG implies surjectivity of , which in turn implies that the action of  on  is free.

### 2.2 Reduction to surjectivity and matrix coefficients

{lemma}

Let  be a compact Hausdorff space equipped with a continuous right action of a compact Hausdorff group . Then the canonical map is surjective if and only if for any matrix coefficient  of an irreducible representation of , the element is in the image of the canonical map. Moreover, if the canonical map is surjective, then it is bijective.

###### Proof.

First observe that the canonical map is a homomorphism of left -modules. The first assertion of the lemma follows by combining this observation with the fact that matrix coefficients of irreducible representations span as a vector space.

The Hopf algebra is cosemisimple. Hence, by the result of H.-J. Schneider [s-hj90, Theorem I], if the canonical map is surjective, then it is bijective. ∎

Alternately, assuming that the action of  on  is free, injectivity can be directly proved by using the vector-bundle point of view indicated in (1) (see [bdh]).

### 2.3 Reduction to free actions of compact Lie-groups

Assume that Theorem 1 holds for compact Lie groups. In this section, we prove that this special case implies Theorem 1 in general.

Let be any finite-dimensional representation of . Set

 Xφ:=X×GU(n).

Thus , where  acts on by . The group acts on by , and this action is free. The map

 Φ: X⟶Xφ,x⟼[(x,In)],

where is the identity matrix, has the equivariance property

 Φ(xg)=Φ(x)φ(g). (4)

Hence and induce maps and . The equivariance property (4) implies commutativity of the diagram

 PO \xymatrixPO

Therefore surjectivity of the upper canonical map implies that is in the image of the lower canonical map, where  is any matrix coefficient of . By Lemma 2.2, this implies the PWG-condition.

We recall the theorem of A.M. Gleason: {theorem}[[g-am50]] Let  be a compact Lie group acting freely and continuously on a completely regular space . Then  is a locally trivial -bundle over .

Combining the Gleason theorem with Section 2.3, we infer that 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  on  is free and locally trivial. Since the quotient space is compact Hausdorff, we can choose a finite open cover of such that each is a trivializable principal -bundle over . Here is the quotient map. On , let be a partition of unity subordinate to the cover . Then, for each  there is the canonical map

Assume that for each there exist elements

 pj1,qj1,…,pjn,qjn∈PG(π−1(supp(ψj))such thatcanj(n∑i=1pji⊗qji)=1⊗h.

Let ’s and ’s be functions on  obtained respectively from functions ’s and ’s by zero-value extension. Then for each  and  we take

 ˜pji√ψj∘π,˜qji√ψj∘π∈PG(X),

and for any and , using the commutativity of the diagram (3), we obtain

 can(r∑j=1n∑i=1˜pji√ψj∘π⊗˜qji√ψj∘π)(x,g) =(r∑j=1n∑i=1˜pji√ψj∘π⊗˜qji√ψj∘π)(x,xg) =∑alljs.t.π(x)∈Uj(ψj∘π)(x)n∑i=1pji(x)qji(xg) =∑alljs.t.π(x)∈Uj(ψj∘π)(x)canj(n∑i=1pji⊗qji)(x,g) =r∑j=1(ψj∘π)(x)h(g) =(1⊗h)(x,g).

Hence validity of the PWG-condition in the trivial-bundle case implies that the PWG-condition holds for actions that are free and locally trivial.

### 2.6 The trivial-bundle case

First note that

 PG(Y×G)=C(Y)⊗O(G). (5)

This is implied by two facts: (1) quite generally is the purely algebraic direct sum of the isotypical components for the action of G on  [s-pm11, Proposition 2.2]; and (2) each isotypical component for the action of G on is of the form , where  is an isotypical component for the action of  on .

As is a Hopf algebra, the dual of the homeomorphism

 FG: G×G∋(g1,g2)⟼(g1,g1g2)∈G×G

and the dual of its inverse restrict and corestrict respectively to

 canG: O(G)⊗O(G)∋T⟼T∘FG∈O(G)⊗O(G), can−1G: O(G)⊗O(G)∋T⟼T∘F−1G∈O(G)⊗O(G).

Granted the identification (5), we now obtain the following commutative diagram:

 PO \xymatrixPO

Hence the bijectivity of implies the bijectivity of .

## 3 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} Let be a compact quantum group acting on a unital -algebra . The assignment of the Peter–Weyl algebra to a -algebra yields a functor from the category with objects being unital -algebras with an -coaction and morphisms being equivariant unital -homomorphisms to the category whose objects are -comodule algebras and whose morphisms are colinear algebra homomorphisms. Furthermore, this functor commutes with all equivariant pullbacks, that is,

 PH(A×BC)=PH(A)×PH(B)PH(C).
{lemma}

[Lemma 3.2 in [hkmz11]] Let be a Hopf algebra with bijective antipode, and let

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