On conditional expectations arising from group actions

M. Frank, V. M. Manuilov and E. V. Troitsky

Abstract. Induced conditional expectations of finite index on crossed product C*-algebras are considered which are non-algebraically of finite index. The characteristics of actions of (amenable) topological groups on compact Hausdorff spaces X are investigated ensuring the appearance of a well-defined induced conditional expectation on the corresponding commutative C*-algebra C(X) and its property to be of finite index. For this aim Hilbert C*-module and topological techniques are used. Special emphasis is put on discrete group actions.

Keywords: group actions on compact spaces, induced conditional expectations of finite index,

crossed product C*-algebras, commutative C*-algebras

AMS subject classification: Primary 46 L 99, secondary 46 H 25

In the light of recent results of E. Kirchberg and the first author [6] we generalize some interesting results of Mahmood Khoshkam [10] concerning special conditional expectations on crossed product C*-algebras which are of finite index in the sense of Y. Watatani’s algebraic definition [20] to the more general setting of similar conditional expectations E on them which are of finite index in the sense of M. Baillet, Y. Denizeau and J.-F. Havet [3], i.e. those for which there exists a finite real number K ≥ 1 so that the mapping (K · E − id) is positive.

In section two we consider actions of amenable infinite discrete and topological groups on locally compact Hausdorff spaces which give rise to conditional expectations of finite index. In these cases the action of an amenable topological group G always can be replaced by a suitable action of the completely disconnected group arising as the factor group of G by its connected component of the identity. The influence of the maximal length of orbit in X under the group action on the characteristic constant K(E_{G}) of the derived conditional expectation of finite index on C_{0}(X) is described.

In addition we analyze the constructions of conditional expectations of finite index on commutative C*-algebras C(X) given by Y. Watatani [3: Prop. 2.8.1, 2.8.2], which arise from actions of finite groups on compact Hausdorff spaces X. Special attention is paid to the close interrelation between non-maximal orbits in X under the group action and of the non-algebraic character of the corresponding conditional expectation of finite index. For these investigations we use group action and Hilbert C*-module techniques. Examples illustrate the phenomena appearing for discrete group actions.

M. Frank: University of Leipzig, Inst. Math., D – 04109 Leipzig / V. M. Manuilov and E. V.

Troitsky, Moscow State University, Dept. Mech. Math., 119899 Moscow, Russia

We make use of some basic notions and results from the theory of Hilbert C*- modules during our considerations to make the explanations smooth. For an introduction to this theory we refer the reader to the papers of W. L. Paschke [12] and M. A. Rieffel [17].

1. On liftings of conditional expectations of finite index to crossed products

Throughout the present section we deal with discrete groups G only. If a group G acts on a C*-algebra A as a group of ∗-automorphisms we denote the full and the reduced crossed product C*-algebras by A × G and A ×_{r} G, respectively, cf. [4,14,19]. We consider conditional expectations E : A → B ⊆ A on C*-algebras A, i.e. projections of norm one with range C*-subalgebra B ⊆ A, see [14,19] for detailed results. The C*- algebra A can always be assumed to be unital without loss of generality since E can be continued to a normal conditional expectation E^{∗∗} on the bidual W*-algebra A^{∗∗} of A with range C*-algebra E^{∗∗}(A^{∗∗}) ⊆ A^{∗∗} which is ∗-isomorphic to the bidual W*-algebra B^{∗∗} of B, whereby the reduction of E^{∗∗} to the canonical embedding A ↪ A^{∗∗} recovers E. A conditional expectation E is said to be faithful if E(x^{∗}x) = 0 implies x = 0 for x ∈ A. In that case both A and E(A) = B share a common identity. To consider both conditional expectations E : A → B ⊆ A and actions α of groups G on the C*-algebras involved we suppose that the group action α of G by ∗-automorphisms on A commutes with the conditional expectation E.

We shall investigate conditional expectations of finite index both in the sense of Y. Watatani [20] and in the sense of M. Baillet, Y. Denizeau and J.-F. Havet [3]. If we consider W*-factors and normal conditional expectations on them with factor image W*- algebra both notions reduce to the classical index notion of V. Jones, M. Pimsner and S. Popa [9,15,3]. A conditional expectation E : A → B ⊆ A is said to be algebraically of finite index if there exist elements {u_{1}, ..., u_{n}} ⊂ A, a quasi-basis, such that the equality x = ∑_{i=1}^{n} u_{i}E(u_{i}^{∗}x) is valid for every x ∈ A. In this case the index is defined by Ind(E) = ∑_{i=1}^{n} u_{i}u_{i}^{∗}, which is a positive invertible element in the center of A such that Ind(E) ≥ 1_{A}, and it does not depend on the choice of the quasi-basis {u_{1}, ..., u_{n}} inside A and in particular not on the number of its elements, [20]. From another point of view the existence of a finite quasi-basis for some conditional expectation E : A → B ⊆ A is equivalent to the algebraic statement that A is a finitely generated projective (or, equiv., Hilbert) B-module. There is another interesting class of conditional expectations E : A → B ⊆ A – those for which there exists a finite real number K ≥ 1 such that the mapping (K · E − id_{A}) is a positive map on A. They have the following remarkable property:

Proposition 1.1. Let A be a C*-algebra and E : A → B ⊆ A be a conditional expectation with fixed point set B. Then there exists a finite real number K ≥ 1 such that the mapping (K · E − id_{A}) is positive if and only if E is faithful and the (right) pre-Hilbert B-module {A, E(⟨·, ·⟩_{A})} is complete with respect to the norm ∥E(⟨·, ·⟩_{A})∥_{B}^{1/2} (where ⟨a, b⟩_{A} = a^{∗}b for a, b ∈ A).

Proof. If A is complete with respect to the norm ∥E(⟨·, ·⟩_{A})∥_{B}^{1/2}, then by the general theory of Banach spaces there exists a number K such that K∥E(x^{∗}x)∥ ≥ ∥x^{∗}x∥ holds for every x ∈ A. Set x = a(ε + E(a^{∗}a))^{−1/2} for a ∈ A, ε > 0 and observe that

(ε + E(a^{∗}a))^{−1/2} · E(a^{∗}a) · (ε + E(a^{∗}a))^{−1/2} ≤ 1_{A} .

The inequality K ·1_{A} ≥ (ε+E(a^{∗}a))^{−1/2} ·a^{∗}a·(ε+E(a^{∗}a))^{−1/2} holds. Multiplying with (ε + E(a^{∗}a))^{1/2} from both sides we arrive at K · (ε + E(a^{∗}a)) ≥ a^{∗}a for every a ∈ A, ε > 0. This yields the other statement. The converse is obvious by spectral theory and by the inequality ∥E(x)∥ ≤ ∥x∥ which is valid for every x ∈ A

The difference between the two types of conditional expectations to be of finite index arises in the way the Hilbert B-module {A, E(⟨·, ·⟩_{A})} is generated, finitely or infinitely, see [20: Prop. 2.8.1, 2.8.2], [6] and the third section. By the results of joint work of E. Kirchberg and the first author ([6]) conditional expectations E : A → B ⊆ A of finite index have been identified with those satisfying the following equivalent conditions: for a conditional expectation E on a (unital) C*-algebra A there exists a real number K ≥ 1 such that the mapping (K · E − id_{A}) is positive if and only if there exists a real number L ≥ 1 such that the mapping (L · E − id_{A}) is completely positive, if and only if E is faithful and A is complete with respect to the norm ∥E(⟨·, ·⟩_{A})∥^{1/2} (where ⟨a, a⟩_{A} = a^{∗}a for a ∈ A). Setting

K(E) = inf{K : (K · E − id_{A}) is positive on A} ,
L(E) = inf{L : (L · E − id_{A}) is completely positive on A}

the estimate K(E) ≤ L(E) ≤ K(E) · [K(E)] can be shown, where [·] denotes the entire part of a real number. Moreover, beside the properties of the class of conditional expectations obtained in [1], normal conditional expectations of finite index on W*- algebras commute with the (abstract) projections of W*-algebras to their finite, infinite, discrete and continuous type I, type II_{1}, type II_{∞} and type III parts, i.e. they respect and preserve these W*-decompositions in full. Additionally, the index value and the Jones’ tower always exist in this case. In the general C*-case the index value of E can be obtained only as an element of the center of the bidual W*-algebra A^{∗∗} of A extending E to its canonical normal conditional expectation E^{∗∗} : A^{∗∗} → B^{∗∗} ⊆ A^{∗∗} (see [19,4] and the third section). However, if the index value belongs to the image C*-algebra B, the Jones’ tower can be shown to exist. These observations of [6] extend and complete results of M. Pimsner, S. Popa [15], M. Baillet, Y. Denizeau and J.-F. Havet [3], Y. Watatani [20], P. Jolissaint [8], E. Andruchow and D. Stojanoff [1,2] and others and close the gap of knowledge separating Proposition 3.3 and Theorem 3.5 of [3]. There are examples of conditional expectations of finite index which are not of the purely algebraic type investigated by Y. Watatani, cf. [20: Prop. 2.8.2.] and the third section. A large class of these examples is closely related to group actions on compact Hausdorff spaces and the related C*-algebras as well as to crossed products.

In the special situation of a conditional expectation E : A → B ⊆ A and a discrete group G acting on A, where the action α of G and the mapping E are supposed to commute, M. Khoshkam showed that there exist canonically induced conditional expectations Ẽ (resp., Ẽ_{r}) from A × G (resp., A ×_{r} G) to B × G ⊆ A × G (resp., B ×_{r} G ⊆ A ×_{r} G), [10: Rem. 2.6]. They arise from the projection of l^{1}(G, A) onto l^{1}(G, B) defined by f → E ∘ f.

Now we extend the result of M. Khoshkam [10: Th. 2.8] that the additional property of one of the three interrelated conditional expectations E, Ẽ_{r} and Ẽ to be algebraically of finite index implies the same property for the other two mappings. We would like to consider the more general case of conditional expectations of finite index (cf. [3]) possessing a finite real number K ≥ 1 such that the mapping (K · E − id) is positive.

Proposition 1.2. Let E : A → B ⊆ A be a conditional expectation of finite index with characteristic number K(E). Then E is of finite index if and only if Ẽ_{r} is of finite index, if and only if Ẽ is of finite index. The equality K(Ẽ_{r}) = K(Ẽ) = K(E) holds for their characteristic numbers. Especially, Ẽ is faithful in this case. If, moreover, one of E, Ẽ_{r} or Ẽ is algebraically of finite index in the sense of Y. Watatani [20], then both the other conditional expectations possess the same property.

Proof. If E is faithful, then Ẽ_{r} is faithful, too, (and vice versa) since the conditional expectations E_{0} : A ×_{r} G → A and E′_{0} : B ×_{r} G → B given by the evaluation at the identity of G are faithful and the equality E′_{0} ∘ Ẽ_{r} = E ∘ E_{0} holds. Because of the normdensity of A in the pre-Hilbert B-module {A, E(⟨., .⟩)} the action of G extends to its norm-completion E_{E} for every faithful conditional expectation E. Similarly, E_{E} ×_{r} G can be formed. By [10: Prop. 2.7] both the Hilbert B ×_{r} G-modules EẼ_{r} and E_{E} ×_{r} G are isometrically isomorphic. Furthermore, suppose E is of finite index in the sense of [3]. By Proposition 1.1 the Hilbert B-module E_{E} coincides with A. Consequently, the Hilbert B ×_{r} G-module EẼ_{r} coincides with A ×_{r} G, and Proposition 1.1 shows again the existence of a finite real number K(Ẽ_{r}) ≥ 1 such that the mapping (K(Ẽ_{r}) · Ẽ_{r} − id_{A ×_{r} G}) is positive, i.e. Ẽ_{r} is of finite index. The conclusions work equally well in the converse direction. If Ẽ is of finite index it obviously has to be faithful. Hence [20: Prop. 2.7] and the assertions above apply again, and E turns out to be of finite index, too. Conversely, suppose E to be of finite index. This means the validity of the inequality

K(E) · E(a^{∗}a) ≥ a^{∗}a

for every a ∈ A. Denoting by λ_{g,a} ∈ l^{1}(G, A) the element giving a ∈ A at g ∈ G and zero elsewhere we obtain

\begin{aligned}Ẽ ((∑_{i=1}^{n} λ_{g_{i},a_{i}}) ∗ (∑_{i=1}^{n} λ_{g_{i},a_{i}})^{∗}) (e) & = Ẽ (∑_{i=1}^{n} λ_{g_{i},a_{i}} ∗ λ_{g_{i},a_{i}}^{∗}) (e) = ∑_{i=1}^{n} E(a_{i}a_{i}^{∗}) \\ & = E (∑_{i=1}^{n} a_{i}a_{i}^{∗}) ≥ K(E) · ∑_{i=1}^{n} a_{i}a_{i}^{∗} ≥ 0 , \end{aligned}

which implies the faithfulness of Ẽ on l^{1}(G, A) since the inequality above is preserved for all elements of l^{1}(G, A) by continuity. Consequently, Ẽ is faithful on A × G. To finish, we apply again Proposition 1.1 and the assertions above to Ẽ, which proves Ẽ to be of finite index.

The remaining part of the proof is concerned with the estimates of the characteristic numbers K(E) for the three conditional expectations E, Ẽ_{r} and Ẽ which are now known to be of finite index (and hence, faithful) if and only if one of them is. Again, let E_{0}, E′_{0} be those conditional expectations E_{0} : A ×_{(r)} G → A, E′_{0} : B ×_{(r)} G → B which arise as evaluation at the identity of G. Since both Ẽ and Ẽ_{r} are already shown to be faithful in case E is of finite index the equality

E′_{0} ∘ Ẽ_{(r)} ≡ E ∘ E_{0}  (1)

is valid. Note that E_{0} and E′_{0} are faithful by [5], and that the restriction of E_{0} to B ×_{(r)} G yields E′_{0}. By (1) we obtain the identity

E′_{0} ∘ (K(E) · Ẽ_{(r)} − id_{A ×_{(r)} G}) ≡ (K(E) · E − id_{A}) ∘ E_{0}

valid on A ×_{(r)} G. Suppose E is of finite index with characteristic number K(E) > 1. (K(E) = 1 implies E = id_{A}.) By assumption

((K(E) · E − id_{A}) ∘ E_{0})(a) = (E′_{0} ∘ (K(E) · Ẽ_{(r)} − id_{A ×_{(r)} G}))(a) ≥ 0

for every a ∈ (A ×_{(r)} G)_{h}^{+}. Since E′_{0} is faithful and has a special mapping rule the value (K(E) · Ẽ_{(r)} − id_{A ×_{(r)} G})(a) belongs to (A ×_{(r)} G)_{h}^{+} for every a ∈ (A ×_{(r)} G)_{h}^{+}. This implies K(Ẽ_{(r)}) ≤ K(E). Conversely, suppose Ẽ_{(r)} to be of finite index with characteristic number K(Ẽ_{(r)}) > 1. Then

(E′_{0} ∘ (K(E) · Ẽ_{(r)} − id_{A ×_{(r)} G}))(a) = ((K(E) · E − id_{A}) ∘ E_{0})(a) ≥ 0

for every a ∈ (A ×_{(r)} G)_{h}^{+}. Since E_{0} is a faithful positive mapping we derive the positivity of the element (K(Ẽ_{(r)}) · E − id_{A ×_{(r)} G})(a) for every a ∈ (A ×_{(r)} G)_{h}^{+}, and K(Ẽ_{(r)}) ≥ K(E). This proves the equality K(Ẽ_{r}) = K(Ẽ) = K(E)

The following assertion extends another result of M. Khoshkam about conditional expectations which are algebraically of finite index to the case of those of finite index, cf. [10: Prop. 3.5].

Proposition 1.3. Let A ⊇ B ⊇ C be C*-algebras with the same unit. Suppose the maps E : A → B ⊆ A and F : B → C ⊆ B are faithful conditional expectations relating these three C*-algebras. Then F ∘ E : A → C ⊆ A has finite index if and only if E and F both have finite indices. Moreover, the inequality K(F ∘ E) ≤ K(F) · K(E) is valid. Finally, the composition F ∘ E is algebraically of finite index if and only both F and E are so.

Proof. Suppose E and F to be of finite index in the sense of [6], i.e. there exists real numbers

K(E) = inf{K : (K · E − id_{A}) is positive on A} ≥ 1 ,

K(F) = inf{K : (K · F − id_{B}) is positive on B} ≥ 1 .

Since K(E) = 1 implies E = id_{A} we may assume K(E) > 1 without loss of generality. Because the inequality K(F) · F ≥ id_{B} is valid by assumption we conclude that

(K(F) · K(E) · F ∘ E − id_{A}) ≥ K(E) · E − id_{A} ≥ 0 ,  (2)

which implies F ∘ E to be of finite index. Conversely, let F ∘ E be of finite index. Then there exists a finite real number K(F ∘ E) such that the map (K(F ∘ E) · F ∘ E − id_{A}) is positive. Applying the positive map E from the left we obtain

0 ≤ E ∘ (K(F ∘ E) · F ∘ E − id_{A}) = (K(F ∘ E) · F − id_{B}) ∘ E ,

which implies the positivity of the map (K(F ∘ E) · F − id_{B}) for the finite real number K(F ∘ E), and hence proves F to be of finite index. Furthermore, assume that E is not of finite index. Equivalently, there exists a sequence {a_{i} : i ∈ N} ∈ A_{h}^{+} such that the sequence {∥E(a_{i})∥ : i ∈ N} is bounded, but the sequence {∥a_{i}∥ : i ∈ N} is not. Then the sequence {∥F (E(a_{i}))∥ : i ∈ N} is bounded since F was shown to be of finite index. However, F (E(a_{i})) = (F ∘ E)(a_{i}), and F ∘ E was supposed to be of finite index, too. Consequently, the sequence {∥a_{i}∥ : i ∈ N} has to be bounded in contradiction to its choice. The only way out is the assumption of the existence of a finite real number K(E) such that the inequality K(E) · ∥E(a)∥ ≥ ∥a∥ holds for every a ∈ A_{h}^{+}. This condition is equivalent to the property of E to be of finite index. For the special case of E and F or F ∘ E to be algebraically of finite index we refer to M. Khoshkam’s result. The inequality K(F ∘ E) ≤ K(F) · K(E) can be derived from (2) above. An example below shows that equality does in general not hold

Example 1.4. Consider the faithful conditional expectations of finite index

\begin{matrix}E : M_{2}(C) → C^{2} & , E : ( \begin{matrix}a & b \\ c & d \end{matrix} ) → ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) , \\ F : C^{2} → C & , F : ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) → \frac{a + d}{2} · ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) . \end{matrix}

The corresponding characteristic numbers are K(E) = K(F) = K(F ∘ E) = 2, and, therefore, K(F ∘ E) < K(F) · K(E).

For a discrete group G and a subgroup H ⊆ G we can define conditional expectations E^{H} (resp., E_{r}^{H}) from A × G (resp., from A ×_{r} G) onto A × H (resp., A ×_{r} H) induced by the canonical projection of l^{1}(G, A) onto l^{1}(H, A), see [10] for explanations. By [10: Th. 3.4] the conditional expectations E^{H} and E_{r}^{H} have finite index if and only if [G : H] < ∞, and Ind(E_{(r)}^{H}) = [G : H] in this case. Applying Proposition 1.3 to the conditional expectation E_{(r)}^{H} ∘ Ẽ_{(r)} : A ×_{(r)} G → B ×_{(r)} H we can formulate the following result, (cf. [10: Th. 3.6] for the algebraic case):

Corollary 1.5. In the described situation the conditional expectation E_{(r)}^{H} ∘ Ẽ_{(r)} is of finite index if and only if E is of finite index and [G : H] < ∞. It is algebraically of finite index if and only if E is algebraically of finite index and [G : H] < ∞. In the latter case the indices can be counted and Ind(Ẽ_{(r)})) = [G : H] · Ind(E).

2. Conditional expectations arising from topological group actions

For a locally compact Hausdorff space X and a continuous action of a topological group G on X we denote the C*-subalgebra of G-invariant continuous functions on X by C^{G}(X) and the stabilizer subgroup of x ∈ X by G_{x} = {g ∈ G : gx = x}. We want to consider situations for which the group action of G on X gives rise to a well-defined conditional expectation E_{G} on the corresponding C*-algebra C_{0}(X). For this aim we need the following definitions:

Definition 2.1. We say that an action of a topological group G on a locally compact Hausdorff space X is uniformly continuous if for every point x ∈ X and every neighbourhood U_{x} of x there exists a neighbourhood V_{x} of x such that g(V_{x}) ⊆ U_{x} for every g ∈ G_{x}.

Example 2.2. Let the space X ⊂ R^{2} be defined in polar coordinates (r, ϕ) by

X = \{(\frac{1}{n}, \frac{π}{m}) : n, m ∈ N\} ∪ \{(\frac{1}{n}, 0) : n ∈ N\} ∪ \{(0, 0)\} .

As X is a closed bounded subset in R^{2} it is a compact normal space. For convinience we denote its components by

\begin{matrix}X_{m} = \{(\frac{1}{n}, \frac{π}{m}) : n ∈ N\} ∪ \{(0, 0)\} & , m ∈ N , \\ X_{0} = \{(\frac{1}{n}, 0) : n ∈ N\} ∪ \{(0, 0)\} & , X = ∪_{0}^{∞}X_{m} . \end{matrix}

Let G = ⊕_{1}^{∞}Z_{2} be the group of sequences g = (g_{1}, …, g_{k}, …)with entries equal to zero or one, where only a finite number of elements differs from zero. By g_{(k)} we denote the sequence with the unique non-zero element on the k-th place. Define an action of G on X by following rules:

(i) if x ∈ X_{m} and m ≠ k, then g_{(k)}x = x,

(ii) if x ∈ X_{k}, x = (\frac{1}{n}, \frac{π}{k} ) and n ≠ 1, n ≠ k, then again g_{(k)}x = x,
(iii) g_{(k)} transposes the points (1, \frac{π}{k} ) and (\frac{1}{k} , \frac{π}{k} ).

Under the action of G the orbit of every point of X\X_{0} consists of exactly two points, and the points of X_{0} are fixed. The described action is continuous (when G is considered as a discrete group), however this action does not satisfy the condition of uniform continuity as every neighbourhood of the origin contains points which are moved by some element of G to points located at a distance from the origin equal to one.

Note that a continuous action of a compact group is always uniformly continuous.

Definition 2.3. An action of an amenable topological group G on a locally compact Hausdorff space X generates a conditional expectation E_{G,am} on C_{0}(X) if the formula

E_{G,am}(f)(x) = m_{G}(f(g^{−1}x))  (3)

defines a continuous function for any f ∈ C_{0}(X) and for some fixed (left) invariant mean m_{G} on G.

Examples 2.4. Let the discrete group Z act on the manifold S^{1} by an irrational rotation. Then every orbit of a point x ∈ S^{1} consists of infinitely many points, and the conditional expectation E_{Z} is well-defined by the integral formula. The positive linear functional E_{Z} turns out to be the standard integral of continuous functions over S^{1}. Consider the discrete group Z acting on itself in the natural way. The related conditional expectation E_{G,am} coincides with the invariant mean m_{Z} selected. The mapping E_{G,am} derived from the group action in Example 2.2 by (3) is discontinuous at (0, 0) what shows the need of the additional assumption of uniform continuity to the action of G on X.

Beside the proof of well-definedness of this type of conditional expectations we want to obtain some sufficient conditions for them to be of finite index in the sense of [3,6].

Lemma 2.5. Let G be an (amenable) topological group acting on a locally compact Hausdorff space X. Suppose that the orbit Gx consists of a finite number of points for every x ∈ X. Then the connected component H of the identity of G acts trivially on x, i.e. it belongs to the stabilizer G_{x} of x ∈ X.

Proof: A finite subset of a Hausdorff space X consisting of more than one point is not connected. Consider the following mapping

H → H/H_{x} → Hx , h ↦ hH_{x} ↦ hx ,

where H_{x} is the stabilizer of x under the action of H. The first mapping in this combination is an epimorphism, the second one is a homeomorphism. So Hx is connected and hence coincides with {x}

Lemma 2.6. Let G be an amenable topological group which acts on a locally compact normal space X in such a way that it generates a conditional expectation E_{G,am} on C(X), cf. (3). Suppose, for some x ∈ X that there exists an orbit Gx consisting of an infinite number of points, and the corresponding mean on Gx is normal, namely for every ε > 0 there exists a positive function f with f(x) = 1 and m_{Gx}(f) < ε. Then the characteristic number K(E_{G,am}) of the conditional expectation E_{G,am} is infinite.

Proof: The mentioned functions exist by [18: V.17.4], and they satisfy the inequality

1 = f(x) ≤ K(E_{G,am}) · E_{G,am}(f)(x) < K(E_{G,am})ε .

So the claim is obvious since ε > 0 was arbitrarily chosen

Note that the conditions of Lemma 2.6 are satisfied for the Examples 2.4.

Theorem 2.7. Let G be an amenable topological group which acts on the locally compact normal space X in such a way that a conditional expectation E_{G,am} is generated on C(X) for a certain (left) invariant mean m_{G} on G. Suppose that the corresponding mean on Gx is normal there for every x ∈ X. Then there are exactly two cases:

(i) For the connected component of the identity H of G the action of G factorizes into the composition

G × X → G/H × X
↘  ↙
X  .

That is, the action of G on X can be described by the action of the completely disconnected group G/H, alternatively. In this case the characteristic value K(E_{G,am}) can in general be finite.

(ii) In all other cases only infinite values K(E_{G,am}) appear.

Proof. First of all, H is a normal subgroup of G by [17: p. 128]. If there is an infinite orbit, then by Lemma 2.6 we obtain the second situation. Otherwise we have the first case by Lemma 2.5

The consequence of Theorem 2.7 is: the consideration of conditional expectations of finite index arising from the actions of amenable topological groups on locally compact normal spaces X leads to the investigation of actions of amenable completely disconnected groups on these spaces. We may restrict to the latter kind of actions to look for results. Note that the factor group G/H of a locally compact group by its connected component of the identity H is not necessarily discrete (however, this is true for Lie groups). For example, any infinite product of finite groups is a non-discrete group with trivial connected component. We will, nevertheless, concentrate on the case of finite and discrete group actions during our further considerations. The most promising situation is characterized by the existence of an upper bound for the number of points in any orbit.

Theorem 2.8. Let X be a locally compact Hausdorff space equipped with an action of a finite group G. Then for the conditional expectation E_{G,am} we have the estimate

K(E_{G,am}) ≤ k := max_{x∈X} #(Gx) .

If X is a normal space, then we have the equality K(E_{G,am}) = k.

Proof: Let x ∈ X be an arbitrary element and k_{x} = #Gx. Then

\begin{aligned}kE_{G,am}(f)x & = k \frac{1}{|G|} ∑_{g∈G} f(g^{−1}x) = k \frac{1}{|G|} ∑_{[g]∈G/G_{x}} |G_{x}| · f(gx) \\ & ≥ \frac{k}{k_{x}} ∑_{[g]∈G/G_{x}} f(g^{−1}x) ≥ f(x) . \end{aligned}

Suppose that K(E_{G,am}) < k and X is normal. Take an element x ∈ X realizing k_{x} > K(E_{G,am}). Then there exists a neighbourhood U_{x} of x such that (gU_{x}) ∩ U_{x} = ∅ for some element g ∈ G with gx ≠ x. Since X is supposed to be normal there exists a continuous non-negative function f with support inside U_{x}, cf. [18: V.17.4]. Then

K(E_{G,am})E_{G,am}(f)x = K(E_{G,am})\frac{1}{|G|} ∑_{g∈G} f(gx)

= K(E_{G,am})\frac{1}{|G|} ∑_{[g]∈G/G_{x}} |G_{x}| · f(gx)

< k_{x} \frac{1}{k_{x}} ∑_{[g]∈G/G_{x}} f(gx) = f(x) .

This contradicts the definition of the characteristic number K(E_{G,am}), and completes the proof

Lemma 2.9. Suppose a discrete amenable group G acts uniformly continuously on a locally compact Hausdorff space X in such a way that each orbit has length #Gx with #Gx ≤ k for some fixed k ∈ N. Then E_{G,am} is well-defined.

Proof: Let H ⊆ G be a subgroup of finite index and let T be a transversal of H, i.e. a choice of representatives in cosets. We have the equality

\begin{aligned}m_{G}(f(gx)) & = m_{G} ( ∑_{g_{a}∈T} f(hg_{a}x)χ_{g_{a}H} ) = \\ & = ∑_{g_{a}∈T} f(g_{a}x) · m_{G}(χ_{g_{a}H}) = c · ∑_{g_{a}∈T} f(g_{a}x) . \end{aligned}

Set f = 1. Then for every x ∈ X the equality 1 = m_{G}(1) = c · #(G/H) holds, which implies c = #(G/H)^{−1} and

m_{G}(f(gx)) = \frac{1}{#(G/H)} · ∑_{g_{a}∈T} f(g_{a}x) .

Now, fix ε > 0 and x ∈ X. Choose y so close to x that the estimate |f(gx) − f(gy)| < ε holds for every g ∈ G. Because of the uniform continuity of the action of G on X this choice is possible, since we can take y from such a small neighbourhood U of x that the variation of f is less than ε on g_{1}V = V, g_{2}V, ..., g_{m}V for {g_{1} = e, g_{2}, …, g_{m}} = T and G_{x}V ⊆ U. In this situation every g ∈ G can be decomposed as g = g_{i}h for some h ∈ H, and gy ∈ g_{i}V. Consequently, the subgroup D = G_{x} ∩ G_{y} has finite index and the set of cosets G/D coincides with the set of pairwise intersections gG_{x} ∩ hG_{y}. Moreover, the index of D in G is lower-equal the product of indices of G_{x} and G_{y}, cf. [12: p. 62]. The functions ϕ(g) = f(gx) and ψ(g) = f(gy) are constant on D, and so we obtain the equalities

E_{G,am}(f)(x) = \frac{1}{#(G/D)} · ∑_{g_{a}∈T} f(g_{a}x) ,

and analogously

E_{G,am}(f)(y) = \frac{1}{#(G/D)} · ∑_{g_{a}∈T} f(g_{a}y)

for the selected elements x, y ∈ X. Hence

|E_{G,am}(f)(x) − E_{G,am}(f)(y)| < \frac{1}{#(G/D)} · #(G/D) · ε = ε .

Definition 2.10. Suppose, an arbitrary topological group G acts uniformly continuously on a locally compact Hausdorff space X in such a way that each orbit has length #Gx with #Gx ≤ k for some fixed k ∈ N. Similar as in Lemma 2.9 we define a conditional expectation E_{G} : C_{0}(X) → E_{G}(C_{0}(X)) ⊆ C_{0}(X) by

E_{G}(f)(x) = \frac{1}{#(G/G_{x})} · ∑_{g_{a}∈T_{x}} f(g_{a}x) , (x ∈ X) .

Just the same argument as in Lemma 2.9 applies to prove the following statement.

Lemma 2.11. Suppose, a discrete group G acts uniformly continuously on a locally compact Hausdorff space X in such a way that each orbit has length #Gx ≤ k for some fixed k ∈ N. Then E_{G} is well-defined.

Theorem 2.12. Suppose a discrete group G acts on a locally compact Hausdorff space X in such a way that k := max{#(Gx) : x ∈ X} < +∞. If X is a normal space, then K(E_{G}) = k.

Proof. Let x ∈ X be an arbitrary element and k_{x} = #Gx. Then

k · E_{G}(f)(x) = k \frac{1}{k_{x}} · ∑_{g_{a}∈T} f(g_{a}x) ≥ f(x) .

Assume, K(E_{G}) < k for a normal space X. Choose a point x ∈ X with k_{x} > K(E_{G}). Then we can obtain a neighbourhood U_{x} of x such that g_{i}U_{x} ∩ g_{j}U_{x} = ∅ for i ≠ j, {g_{1} = 1, g_{2}, …, g_{m}} = T_{x}. For a continuous non-negative function f with support inside U_{x} (cf. [18: V.17.4] for the existence) we have the inequality

K(E_{G}) · E_{G}(f)(x) = K(E_{G})\frac{1}{k_{x}} · ∑_{g_{a}∈T} f(g_{a}x) < f(x) .

This contradicts the definition of K(E_{G}), and completes the proof

3. Conditional expectations arising from discrete group actions

In this place we investigate conditional expectations of finite index arising from several discrete group actions on compact Hausdorff spaces. The investigations are inspired by results of Y. Watatani [20: Prop. 2.8.1, 2.8.2] on actions of finite groups G on compact Hausdorff spaces X giving rise to conditional expectations of finite index E on C(X),

E_{G}(f)(x) = \frac{1}{|G|} ∑_{g∈G} f(gx) , x ∈ X ,

which are sometimes non-algebraically of finite index. We would like to give a more detailed analysis of actions of amenable discrete groups G on compact Hausdorff spaces X for which the orbit of every x ∈ X is finite. For the definition of a corresponding conditional expectation E_{G} on the C*-algebra C(X) we refer to Lemma 2.10 and 2.12. The goal is to give a more detailed analysis of the appearing phenomena from the point of view of Hilbert C*-module theory adapting techniques from [20].

First, we want to give two examples of conditional expectations of finite index which are not algebraically of finite index in the sense of Y. Watatani’s definition. They are derived from [20: Prop. 2.8.1, 2.8.2] as special cases giving new arguments for proving their non-triviality.

Example 3.1. Let {[0, 1]_{i} : i = 1, ..., n} be n copies of the unit interval equipped with the standard topology. As the basic compact Hausdorff space X we want to consider these n unit intervals with towed together into one point zero ends. Let A be the C*-algebra C(X) of all continuous functions on X. Consider the normal conditional expectation E on A defined by taking the mean with respect to the cyclic group action

E(f)(x_{1}) = ... = E(f)(x_{n}) = \frac{f(x_{1}) + ... + f(x_{n})}{n} for x_{i} ∈ [0, 1]_{i} , i = 1, ..., n .

The image C*-algebra B can be identified with the subset of functions of A {f ∈ A : f(x_{1}) = ... = f(x_{n}) for x_{i} ∈ [0, 1]_{i} , i = 1, ..., n} inheriting the C*-structure from A. Note that {A, E(⟨., .⟩_{A})} is not self-dual as a Hilbert B-module since the bounded Blinear mapping (with f ∈ A)

E(f_{0}^{∗}f)(x_{1}) = ... = E(f_{0}^{∗}f)(x_{n}) = \frac{f_{0}^{∗}(x_{1})f(x_{1}) + ... + f_{0}^{∗}(x_{n})f(x_{n})}{n} , x_{i} ∈ [0, 1]_{i}, i ∈ N,

f_{0}(x) = \{ \begin{matrix}1 & : x_{1} ∈ (0, 1]_{1} \\ −1 & : x_{2} ∈ (0, 1]_{2} \\ 0 & : elsewhere \end{matrix}

maps A into B, but f_{0} does not belong to A. Consequently, the Hilbert B-module {A, E(⟨., .⟩_{A})} cannot be finitely generated and projective, and E is not of finite index in the sense of Y. Watatani’s algebraic definition. Nevertheless, E has a very good property: The mapping (n·E −id_{M}) is obviously completely positive. Beside this a careful analysis done below yields the B-reflexivity of the Hilbert B-module {A, E(⟨., .⟩_{A})}. The characteristic number of E is K(E) = n.

Now let us have a look on the index value: for this we extend E to its canonical normal conditional expectation E^{∗∗} : A^{∗∗} → B^{∗∗} ⊆ A^{∗∗} (see [18,4]) and count the index of the extended conditional expectation E^{∗∗} as an element of the center of A^{∗∗}. The corresponding constants

K(E^{∗∗}) = K(E) = inf{K : (K · E − id_{A}) is positive on A} ,
L(E^{∗∗}) = L(E) = inf{L : (L · E − id_{A}) is completely positive on A}

are preserved by monotonicity, [14].

The discrete part of the bidual linear space and W*-algebra A^{∗∗} of A is isomorphic to l_{∞}(X) by the Gel’fand theorem. The formula defining E^{∗∗} is the same as for E. The index can be algebraically counted. The resulting value is

Ind(E) = Ind(E^{∗∗})_{discr.} = \{ \begin{matrix}2 & : x_{i} ≠ 0 , i = 1, ..., n \\ 1 & : x = 0 \end{matrix} ∈ l_{∞}(X) ,

and Ind(E) ∉ A ↪ A_{discr.}^{∗∗}. Note that the index value Ind(E) ∈ A_{discr.}^{∗∗} gives rise to a positive mapping (Ind(E) · E^{∗∗} − id_{A_{discr.}^{∗∗}}) on A_{discr.}^{∗∗}.

Since A^{∗∗} is commutative (hence, type I) let us look at the non-discrete part of it. By the decomposition theory of commutative W*-algebras it is a direct sum of W*- algebras L^{∞}([0, 1], λ), where λ denotes the Lebesgue measure, cf. [18]. Consider the canonical embedding A = C(X) ⊂ L^{∞}(X, λ) ≅ ∑ ⊕L^{∞}([0, 1]_{i}, λ). Again, the index of E^{∗∗} reduced to L^{∞}(X, λ) can be found, it equals the function f(x) ≡ 2, what is different from the value obtained for the discrete part of Ind(E^{∗∗}).

Example 3.2. Consider two copies of the usually topologized interval [−1, 1] and denote their disjoint union by X. Let the cyclic group G = Z_{4} act on X in such a way that the elements of X = [−1, 1]_{1} ∪ [−1, 1]_{2} are mapped by the rule

\begin{aligned}x ∈ [−1, 1]_{1} & → −x ∈ [−1, 1]_{2} , \\ x ∈ [−1, 1]_{2} & → x ∈ [−1, 1]_{1} . \end{aligned}

The orbits of all points x ∈ X \ {0 ∈ [−1, 1]_{1}, 0 ∈ [−1, 1]_{2}} have length 4, the two exceptional points have an orbit of length 2. Now, switch again to the derived action of G on the corresponding unital C*-algebra A = C(X) and consider the canonically defined conditional expectation

E(f)(x) = \frac{1}{4}(f(ex) + f(gx) + f(g^{2}x) + f(g^{3}x))

(where g ∈ Z_{4} is a generator) which possesses the characteristic constant K(E) = 4. If we consider a bounded B-linear functional on the Hilbert B-module {A, E(⟨., .⟩)} defined by the rule E(f_{1}^{∗}f) for f ∈ A and

f_{1}(x) = \{ \begin{matrix}1 & : x ∈ [−1, 0)_{1} \\ −1 & : x ∈ (0, 1]_{1} \\ 0 & : elsewhere \end{matrix}

we obtain the non-self-duality of {A, E(⟨., .⟩)} as Hilbert B-module, again. Therefore, E cannot be algebraically of finite index, despite of the non-existence of fixed points under the action of G on X.

Two other examples show that actions of infinite amenable discrete groups can yield non-trivial conditional expectations of finite index, as well as mappings E_{G} with images outside the C*-algebra C(X). That is, we have always to keep track of the elements of the construction. Very often we can adapt examples with locally compact spaces X to the compact situation switching to its one-point-compactification αX. However, additional fixed points of the action of G can appear during this process.

Example 3.3. Let G = ⊕_{i=1}^{∞}(Z_{2})_{i}, an infinite discrete group. Let g_{i} be the generator of the i-th copy of Z_{2}. Let Y := \{\frac{1}{n} : n =∈ N\}∪\{0\}, X := Y ×I, where I = [0, 1]. If the canonical pointwise identification maps a_{j}^{i} of {1/i} × I and {1/j} × I are used we can define the action of G on X by

g_{i}(x) = \{ \begin{matrix}a_{2i}^{2i−1}(x) & : x ∈ \frac{1}{2i−1} × I \\ a_{2i−1}^{2i}(x) & : x ∈ \frac{1}{2i} × I \\ x & : elsewhere \end{matrix} .

Then the cardinality of any orbit is less or equal to 2, however the action of G does not factorize through any action of any compact or finite group. Now, define a natural conditional expectation E on A = C_{0}(X) by the formula

E(f)(x) = lim_{n→∞} \frac{1}{2^{n}} ∑_{g∈⊕_{i=1}^{n}(Z_{2})_{i}} f(gx)

= lim_{n→∞} \frac{1}{2^{n}} · 2^{(n−1)} ( ∑_{g∈Z_{2}} f(gx))

= \frac{f(e ∘ x) + f(g ∘ x)}{2} , x ∈ X .

We obtain a natural conditional expectation of finite index with structural constant K(E) = 2, which is algebraically of finite index, too. The intrinsic point is that E can be induced alternatively by an action of the group Z_{2} on X which is different from the group action we started with, despite the fact that the original group action cannot be replaced by any finite group action.

Example 3.4. Let G = Z act on X = R as shifts. Let f : R → C be the following positive continuous even function:

f(x) := \{ \begin{matrix}1 & : x ∈ [n + \frac{1}{n+1}, n + 1 − \frac{1}{n+1}] , n ≥ 0 \\ 0 & : x = n , n ≥ 0 \\ linear & : on the remaining intervals. \end{matrix}

Then E(f)(0 + i) = 0 for any i ∈ Z. For x ∉ Z there exists a number j_{0} ∈ N such that f(x + j) = 1 for j with |j| > j_{0}. Hence

E(f)(x) = lim_{n→∞} \frac{1}{2n + 1} ∑_{j=−n}^{j=n} f(x + i)

= lim_{n→∞} \frac{1}{2n + 1} ∑_{j=−k_{0}}^{k_{0}} f(x + i) + lim_{n→∞} \frac{1}{2n + 1} ∑_{k_{0}<|j|≤n} f(x + i)

= 0 + lim_{n→∞} \frac{2(n − k_{0})}{2n + 1} = 1 .

Summing up we obtain

E(f)(x) = \{ \begin{matrix}1 & : x ∉ Z \\ 0 & : x ∈ Z , \end{matrix}

which is a discontinuous function.

The following example describes an action of an amenable group G on a (locally) compact Hausdorff space X, which induces a non-trivial (i.e. non-state) conditional expectation E on the C*-algebra C_{0}(X), that cannot be induced by an action of a finite group on X .

Example 3.5. Let G = ⊕_{k=1}^{∞}Z_{k}, a separable discrete group. Denote by g_{k} be the generator of the k-th summand Z_{k} ⊂ G. Let X be the infinite disjoint union (summing over k ∈ N) of disjoint unions of k copies of the interval I_{k} = [0, 1], respectively. If a_{j}^{i}(k) denotes the pointwise identification function of the intervals I_{k}^{i} and I_{k}^{j} at the kth position, then we describe the action of G on X by the formula

g_{k}(x) = \{ \begin{matrix}a_{i+1 mod k}^{i}(k)(x) & : x ∈ I_{k}^{i} \\ x & : elsewhere . \end{matrix}

Of course, in this situation the characteristic number K(E) = +∞.

Theorem 3.6. Let X be a compact Hausdorff space and G a discrete group acting uniformly continuously on X so that all orbits consist of the same finite number of points. Then the Hilbert C^{G}(X)-module {C(X), E(⟨·, ·⟩)} is finitely generated and projective.

Proof. We want to imitate the original proof of Y. Watatani in the case of a free action of a finite group [20: Prop. 2.8.1]. By [18: VI.22.8] every compact Hausdorff space is normal. So Urysohn’s lemma can be used, cf. [18: V.17.4]. However, first we need the following two lemmata.

Lemma 3.7. Let X be a compact Hausdorff space and G be a discrete group acting uniformly continuously on X so that all orbits consist of the same finite number of points. Then for x ∈ X and g ∈ G_{x} there exists an open neighbourhood U_{x} of x on which g acts as the identity map.

Proof: Consider the stabilizer subgroup G_{x} ⊂ G. It is of finite index, and the index is G : G_{x} = #(Gx) = n. Let x_{1}, …, x_{n} ∈ X be the orbit of x and h_{i} ∈ G be elements such that h_{i}x = x_{i}. Fix g_{0} ∈ G_{x} and assume that every neighbourhood U_{x} of x contains some point y ∈ U_{x} such that g_{0}y ≠ y. Fix some neighbourhoods U_{x_{i}} of the points x_{i} satisfying U_{x_{i}} ∩ U_{x_{j}} = ∅ for i ≠ j. There exists a neighbourhood V_{x} ⊆ U_{x} of x such that h_{i}(V_{x}) ⊆ U_{x_{i}}. By assumption G acts on X uniformly continuously, so let W_{x} ⊆ V_{x} be such a neighbourhood of x that g(W_{x}) ⊆ V_{x} for every g ∈ G_{x}. For y ∈ W_{x} with g_{0}y ≠ y the orbit Gy contains at least n + 1 distinct points {g_{i}y ∈ U_{x_{i}} : i = 1, …, n} and {y, g_{0}y ∈ V_{x}}. This contradicts the boundedness of orbits by n, and proves the lemma

Lemma 3.8. Let X be a compact Hausdorff space and G be a discrete group acting uniformly continuously on X so that all orbits consist of equal finite number of points. For any x ∈ X there exists a neighbourhood V_{x} such that the action of G_{x} on V_{x} is the identical map.

Proof. For every g ∈ G_{x} set U_{x}(g) = {y ∈ X : gy = y}. Suppose at the contrary that the set ∩_{g∈G_{x}}U_{x}(g) does not contain any neighbourhood of the point x, i.e. every neighbourhood U_{x} of x contains some point z such that for some g_{z} ∈ G_{x} we have g_{z}z ≠ z. Consider a neighbourhood V_{x} of x and neighbourhoods {U_{h_{i}x}} for fixed representatives {h_{i} ≠ e} ∈ G of the cosets of G/G_{x} such that their pairwise intersection is empty and h_{i}(V_{x}) ⊆ U_{h_{i}x}, cf. the proof of Lemma 3.7. By assumption G acts uniformly continuously on X, i.e. there exists a neighbourhood W_{x} ⊆ V_{x} of x such that g(W_{x}) ⊆ V_{x} for every g ∈ G_{x}. Define U_{x} = ∪_{g∈G_{x}}g(W_{x}) ⊆ V_{x}. It is an G_{x}-invariant open neighbourhood of x ∈ X. By construction the assumption g_{z}z ≠ z for some z ∈ U_{x}, g_{z} ∈ G_{x} leads to an orbit of z consisting of at least n + 1 points, what contradicts the supposition

Let U_{x} ⊂ V_{x} be neighbourhoods of x such that the action of G_{x} on V_{x} is the identical map. Then there exists a function f_{x} ∈ C(X) such that supp f_{x} ⊂ V_{x} and f_{x}|_{U_{x}} = 1. For any g ∈ G we have either (gV_{x}) ∩ V_{x} = ∅ or gV_{x} ≡ V_{x} which implies that

α_{g}(f_{x}) · f_{x} = \{ \begin{matrix}α_{g}(f_{x})^{2} & : if gx = x , \\ 0 & : otherwise , \end{matrix}

where α denotes the action of the group G on functions, α_{g}(f)(x) = f(gx). Fix a finite covering {U_{x_{1}}, …, U_{x_{k}}} of X by such type of sets and set

v = ∑_{i=1}^{k} f_{x_{i}} ≥ 1 , u_{i} = v^{−1/2}(f_{x_{i}})^{1/2} ∈ C(X) .

Note that for representatives g_{i} of the cosets of G/G_{x} we have a well-defined mapping

E_{G}(f)(x) = \frac{1}{#(Gx)} ∑_{i=1}^{n} f(g_{i}x) ,

for every x ∈ X, f ∈ C(X) due to the fixed length of orbits on X and Lemma 3.7. It is a conditional expectation on C(X). Furthermore, for every f ∈ C(X) we obtain the equalities

∑_{i=1}^{n} u_{i} · E_{G}(u_{i}^{∗}f) = ∑_{i=1}^{n} (\frac{1}{n} ∑_{j=1}^{n} u_{i}(x)u_{i}^{∗}(g_{j}x)f(g_{j}x)) = f , E_{G}(u_{i}^{∗}u_{j}) = δ_{i,j} .

Hence, {u_{1}, ..., u_{n}} is a quasi-basis of the Hilbert C^{G}(X)-module {C(X), E_{G}(⟨., .⟩)}, which is therefore shown to be finitely generated and projective

Let us remark that finite generatedness and projectivity implies self-duality for the Hilbert C^{G}(X)-module {C(X), E_{G}(⟨., .⟩)}. In the case of a more general group actions we cannot expect that the resulting action on the C*-algebra A = C(X) leads to a conditional expectation E_{G} on A turning it into a self-dual Hilbert C^{G}(X)-module. However, if K(E_{G}) < ∞ exists (i.e. when A is complete with respect to the norm given by the corresponding C*-valued inner product) we can expect that it might be C*-reflexive, i.e. the C*-bidual Hilbert C^{G}(X)-module A′′ of A coincides with A, where A′ denotes the dual Banach C^{G}(X)-module of bounded C^{G}(X)-homomorphisms from A to C^{G}(X). We can show the following fact:

Theorem 3.9. Let X be a compact Hausdorff space and G a discrete group acting uniformly continuously on X. If G acts in such a way that the lengths of all orbits are uniformly bounded by some number n and the number of points whose orbit is shorter than n is finite, then the Hilbert C^{G}(X)-module {C(X), E_{G}(⟨., .⟩)} is C^{G}(X)-reflexive.

Proof. Suppose that X has isolated points {x_{1}, ..., x_{k}} with orbit shorter than n. By assumption this set is invariant under the action of G on X as well as its complement X′ ⊂ X. The Hilbert C^{G}(X)-module {C(X), E_{G}(⟨., .⟩)} splits into the direct sum C(X′) ⊕ C({x_{1}, ..., x_{k}}), and the theorem is obvious for the second direct summand. So we may assume that X does not contain any isolated point with orbit shorter than n switching from the C*-algebra of coefficients C^{G}(X) to the smaller C*-algebra of coefficients C^{G}(X′). We start with a description of the C*-dual Banach C^{G}(X)-module A′. Let x_{1}, …, x_{n} be the points whose orbits have length less than n. We can take some open neighbourhoods U_{1}, …, U_{n} of these points such that, first, these neighbourhoods are invariant under the action of elements of G_{x_{i}} (i.e. under the action of those which do not move the point x_{i}, i = 1, …, n) and, secondly, such that if for some h ∈ G we have hx_{i} = x_{j}, then hU_{i} = U_{j}. Denote the compact space X \ (U_{1} ∪ … ∪ U_{n}) by Y. Notice that Y is also G-invariant. Let F ∈ A′ be an C^{G}(X)-valued functional on C(X). Due to commutativity we can consider its restriction to the Hilbert C^{G}(Y)-module {C(Y), E_{G}(⟨., .⟩)}. More precisely, for g ∈ C(Y) we can take its extension g̃ ∈ C(X) and define F |_{Y} by F |_{Y} (g) = F (g̃)|_{Y}. The restriction defined in this way does not depend on the choice of extension. If Y′ ⊃ Y is also a compact G-invariant subset of X not containing the points x_{1}, …, x_{n}, then obviously (F |_{Y′})|_{Y} = F |_{Y}. However, since every point in Y has an orbit of the same length n we can use the self-duality of the Hilbert C^{G}(Y)-module {C(Y), E_{G}(⟨., .⟩)} to conclude that F |_{Y} ∈ C(Y) by Theorem 3.6. Denote by C(X \ {x_{1}, …, x_{n}}) the set of continuous functions on the non-compact space X \ {x_{1}, …, x_{n}}. Then taking restrictions we get the map

A′ ⟶ C(X \ {x_{1}, …, x_{n}}) .  (4)

If F_{1}, F_{2} ∈ A′ coincide on X \ (x_{1}, …, x_{n}), then (as they are mappings of continuous functions to continuous functions) they coincide on X, hence the map (4) is a monomorphism.

Now we study local properties of functionals from A′ near the singular points x_{1}, …, x_{n}. Let x_{0} be one of these points. There exists a neighbourhood U_{x_{0}} of the point x_{0} such that for gx_{0} = x_{0} the set identity gU_{x_{0}} = U_{x_{0}} follows. The group G_{x_{0}} contains a normal subgroup G_{0} of elements which leave the points of U_{x_{0}} invariant. Fix one element g_{i} in every coset G/G_{x_{0}}. Then outside the point x_{0} the action of a functional F ∈ A′ is given by the equality

F (f)(x) = \frac{1}{n} ∑_{i=1}^{n} F^{∗}(g_{i}x) · f(g_{i}x) ,  (5)

and this action is extended to the point x_{0} by continuity. Let x_{0} = x^{0}, x^{1}, …, x^{k−1} be the orbit of x_{0}. Then the sum (5) can be written as

F (f)(x) = \frac{1}{n} ∑_{j=0}^{k−1} ( ∑_{i:g_{i}x_{0}=x^{j}} F^{∗}(g_{i}x) · f(g_{i}x)) .

Taking the norm limit which exists by assumption we obtain

F (f)(x_{0}) = lim_{x→x_{0}} \frac{1}{n} ∑_{j=0}^{k−1} ( ∑_{i:g_{i}x_{0}=x^{j}} F^{∗}(g_{i}x) · f(x^{j})) ,

which implies the existence of the limit

\frac{1}{n} lim_{x→x_{0}} ∑_{i:g_{i}x_{0}=x^{j}} F (g_{i}x)

for the special function f ≡ 1 of C(X) and for every x ∈ X \ {x_{1}, ..., x_{n}}. Recall that the function F (x) was defined only outside the point x_{0}. If we want to write the action of F on the Hilbert C^{G}(Y)-module {C(Y), E_{G}(⟨., .⟩)} in the form (5) everywhere on X we have to define the function F (x) at the point x_{0} by

F (x_{0}) = \frac{1}{n} lim_{x→x_{0}} ∑_{i:g_{i}x_{0}=x^{j}} F (g_{i}x) .  (6)

Lemma 3.10. The module A′ is isomorphic to the module of all bounded functions F (x) on X which are continuous on X\{x_{1}, …, x_{n}} and which satisfy the limit condition (6).

Proof. The remaining fact is to prove that the image of the monomorphism (4) consists of bounded functions. Suppose the contrary. Then we can find some point x̄ such that |F (x̄)| > n · ∥F ∥, where ∥F ∥ defines the norm of F in the C*-dual Banach C^{G}(X)-module A′. Moreover, we can choose a neighbourhood Ux̄ of x̄ such that the intersection Ux̄ ∩ g_{i}Ux̄ is empty whenever g_{i}x̄ ≠ x̄. Consider a function f ∈ C(X) such that f(x̄) = 1 and supp f ⊂ Ux̄. For f we derive

F (f)(x̄) = \frac{1}{n}F^{∗}(x̄) · f(x̄),

from the equality (5), and the resulting inequality

|F (f)(x̄)| = \frac{1}{n}|F^{∗}(x̄)| · |f(x̄)| = \frac{1}{n}|F (x̄)| > ∥F ∥,

gives rise to a contradiction

On conditional expectations 19

Now, equipped with a description of the elements in the C*-dual module A′ of the Hilbert C^{G}(Y)-module {C(Y), E_{G}(⟨., .⟩)} we can finally describe the corresponding C*- bidual module A′′. Since A′′ is a canonical subset of A′ and since the C*-valued inner product of every Hilbert C*-module can be continued to a C*-valued inner product on its C*-bidual Banach C*-module turning the latter into a Hilbert C*-module ([13]), we have only to find out to what kind of functions of A′ the C^{G}(X)-valued inner product can be extended to. Consider a function F from A′ ∩ A′′. Without loss of generality (adding some continuous function from A) we can assume that F (x_{0}) = 0. Then the value of the C^{G}(X)-valued inner product of F with itself has to be contained in C^{G}(X) and is given by the formula

⟨F, F ⟩(x) = E(|F (x)|^{2}) = \frac{1}{n} ∑_{i:g_{i}x_{0}=x^{j}} |F (g_{i}x)|^{2}  (7)

for every x ∈ X. But, since ⟨F, F ⟩(x_{0}) = 0 the assumption F ∈ A′′ on F yields

∥.∥ − lim_{x→x_{0}} F (g_{i}x) = 0

for every summand in the equality (7). Consequently, by the equality (6) we obtain the continuity of F at x_{0}, and F has to belong to A

Remark. Let X be a compact Hausdorff space. Suppose a discrete group G acts on X in such a way that the length of any orbit is not more than n. Denote by Z ⊂ X the set of points whose orbits have less than maximal length. If the set X \ Z is dense in X, then in a similar way we can prove that the dual Banach C^{G}(X)-module A′ is isomorphic to some C*-subalgebra of the algebra C_{b}(X \ Z) of all bounded continuous functions on X \ Z. The C*-algebra C_{0}(X \ Z) of continuous functions on X vanishing on Z is an ideal in C_{b}(X \ Z). It would be interesting to describe the structure of the factor-algebra A′/C_{0}(X \ Z) in terms of the group action.

Acknowledgement. The research was supported by the Volkswagen-Stiftung (Research-in-Pairs program at Oberwolfach). The authors are very grateful to the staff of the Mathematisches Forschungsinstitut Oberwolfach for the good conditions supporting the fruitful work.

The first author is grateful to Eberhard Kirchberg for joint work and fruitful discussions leading to the progress made in [6]. The authors are beholden to G. K. Pedersen, M. A. Rieffel and Y. Watatani for useful comments.

Preliminary joint research was partially supported by the local DAAD project ”Noncommutative geometry” at the University of Leipzig joint with the Moscow State University. The research of the second and the third authors was partially supported by the RBRF (Grant 96-01-00182-a). The research of the third author was partially supported by the INTAS (Project 94-3420 ”Geometry of discrete groups”).

The authors are most grateful to the referee for his valuable comments on the first version of the present paper. We are indebted to C. Abbey for final proof-reading of the manuscript.

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Received 09.01.1997, revised 30.09.97.