New York Journal of Mathematics

New York J. Math. 8 (2002) 99–109.

On C^{∗}-Algebras Associated to Certain Endomorphisms of Discrete Groups

Ilan Hirshberg

Abstract. Let α : G → G be an endomorphism of a discrete amenable group such that [G : α(G)] < ∞. We study the structure of the C^{∗} algebra generated by the left convolution operators acting on the left regular representation space, along with the isometry of the space induced by the endomorphism.

Contents

1. Introduction 99 2. The γ-invariant subalgebra 101

2.1. An abelian subalgebra 104 3. A crossed product representation 105 References 109

1. Introduction

Some interesting examples of non-invertible topological dynamical systems, such as the n-fold covering of the circle, and the one-sided full symbolic shifts on n letters, arise as surjective endomorphisms of compact abelian groups.

If H is such an abelian group, and T : H ↠ H is the map, then T induces an isometry on L^{2}(H, Haar) via the pull-back. The algebra C(H) acts on L^{2} by multiplication, and we want to study the structure of the C^{∗}-algebra generated by those multiplication operators and the isometry, which is analogous to (a representation of) the crossed product algebra in the case of an action of an automorphism.

Such surjective endomorphisms are in duality with injective endomorphisms of the (discrete) dual groups. Thus this problem can be conveniently reformulated as studying the structure of the C^{∗}-algebra generated by the convolution operators on the l^{2} space of a discrete group, along with the isometry induced on the space by the given injective endomorphism of the discrete group. Aside from the fact that this is technically more convenient, it allows an immediate generalization of the question by dropping the requirement that the group be commutative.

99^{ISSN 1076-9803/02}

Received January 31, 2002.

Mathematics Subject Classification. 46L55.

Key words and phrases. endomorphisms, crossed products.

The situation is analogous to that arising in the construction of the crossed product by an endomorphism, however the algebra is not a crossed product by an endomorphism in the sense alluded to in [C]. The main difference is that the endomorphism here not implemented by the associated isometry (i.e., if α is the endomorphism, S is the associated isometry and a is an element of the group C^{∗}- algebra, then SaS^{∗} ≠ α(a)), but rather is only intertwined by it (Sa = α(a)S). Additional conditions relevant to the situation seemed required here, so as to limit the size of the algebra.

The construction considered in this paper is related to a situation considered by Deaconu ([De]) concerning self covering maps of compact Hausdorff spaces (the two situations overlap when our group is abelian, in which case we can dualize to get a map from the dual group onto itself — see above). Deaconu used a groupoid approach to that problem. Exel’s approach to the crossed product by an endomorphism ([E]) places the examples considered in this paper in a more general framework (see Remark 1.4 below).

Definition 1.1. Let G be a discrete group, and let α : G ↪ G be an injective homomorphism. We call α pure if ⋂_{n=0}^{∞} α^{n}(G) = {1}.

We will assume throughout that [G : α(G)] < ∞.

We consider the universal C^{∗}-algebra E_{α} generated by ℂG and and an isometry S, satisfying the relations:

1. Sδ_{x} = δ_{α(x)}S for x ∈ G.

2. S^{∗}δ_{x}S = 0 if x ∉ α(G).

3. For any complete list of right coset representatives x_{1}, …, x_{n} ∈ G of α(G)

∑_{k=1}^{n} δ_{x_{k}^{−1}}SS^{∗}δ_{x_{k}} = 1.

Here, δ_{x} denotes the image of x in ℂG.

Remark 1.2. Note that we could fix in relation (3) an arbitrary list of coset representatives; relation (1) shows that

∑ δ_{x_{k}^{−1}}SS^{∗}δ_{x_{k}} = ∑ δ_{y_{k}^{−1}}SS^{∗}δ_{y_{k}}

if {x_{1}, …, x_{n}}, {y_{1}, …, y_{n}} are two sets of representatives.

There is an action γ of 𝕋 on E_{α} given by

γ_{t}(S) = tS, γ_{t}(a) = a ∀a ∈ ℂG , t ∈ 𝕋 ⊆ ℂ.

Remark 1.3. Note that E_{α} has a represenation λ extending the left regular representation of the group G, given by the usual left regular representation on l^{2}(G), λ(g)ξ_{h} = ξ_{gh}, and with λ(S)ξ_{h} = ξ_{α(h)}, where {ξ_{h} | h ∈ G} is the standard basis for l^{2}(G). It is easily verified that this representation satisfies relations (1)–(3). In particular, this implies that if G is amenable then E_{α} always contains a copy of C^{∗}(G). We shall assume from now on that G is amenable.

Remark 1.4. E_{α} can be described using the language of [E] as the crossed product of C^{∗}(G) by α with transfer operator corresponding to the expectation of C^{∗}(G) onto α(C^{∗}(G)) (see [E], Proposition 2.6). We will not make use of results from [E] in this paper, though.

C^{∗}-Algebras Associated to Certain Endomorphisms 101

2. The γ-invariant subalgebra

Let F denote the γ-invariant subalgebra. We denote P_{n} = S^{n}S^{∗}^{n}.

Claim 2.1. F = \overline{span\{δ_{x}P_{n}δ_{y} | x, y ∈ G, n ∈ ℕ\}}

Proof. We have an expectation map E_{F} : E_{α} → F defined by

E_{F}(a) = ∫_{𝕋} γ_{t}(a)dt.

E_{α} is densely spanned by the words with letters in {S, S^{∗}, δ_{x}, x ∈ G}, and it is clear that if w is such a word, then E_{F}(w) = 0 unless w has the same number of S’s and S^{∗}’s. If w is a word with n S’s and n S^{∗}’s, then it is easy to see that it can be rewritten as a word of the form δ_{x}P_{n}δ_{y}, by using the commutation relations defining the algebra. □

Now fix n and let F_{n} = \overline{span\{δ_{x}P_{n}δ_{y} | x, y ∈ G\}}. It is clear that F_{n} is a subalgebra of F. The spanning elements of F_{n} satisfy the following multiplication rule:

δ_{x}P_{n}δ_{y}δ_{z}P_{n}δ_{w} = \begin{cases}δ_{xyz}P_{n}δ_{w} & yz ∈ α^{n}(G) \\ 0 & otherwise. \end{cases}

Notice that the words δ_{x}P_{n}δ_{y} are not all distinct - clearly, δ_{x}P_{n}δ_{y} = δ_{xz}P_{n}δ_{z^{−1}y} for any z ∈ α^{n}(G). We can force uniqueness by fixing representatives for the conjugacy classes of α^{n}(G)\G and always choosing y to be one of them, for example. A more ‘symmetric’ way will be to introduce the following notation. Let R_{n} be fixed sets of representatives of right cosets, chosen so that α(R_{n}) ⊆ R_{n+1}. This is done as follows: we fix R_{1}, and then define recursively R_{n+1} = {α(s)r | s ∈ R_{n} , r ∈ R_{1}}. We may assume that 1 ∈ R_{1}.

Notation. (x, k, y) = δ_{xα^{n}(k)}P_{n}δ_{y}, where y ∈ R_{n}, x ∈ R_{n}^{−1}, k ∈ G.

In this notation, the multiplication rule assumes the simpler form:

(x, k, y)(z, l, w) = \begin{cases}(x, kl, w) & y = z^{−1} \\ 0 & otherwise. \end{cases}

Note that this multiplication rule is simply that of matrices with entries in ℂG. More explicitly, we can define a map M_{|R_{n}|}(ℂG) → F_{n} by ge_{xy} ↦ (x^{−1}, g, y) where we index the matrix entries by elements of R_{n} and e_{xy} denotes the usual matrix unit. This map will extend to an isomorphism from

M_{|R_{n}|}(C^{∗}(G)) ≅ K(l^{2}(α(G)\G))^{⊗n} ⊗ C^{∗}(G)

onto F_{n} (by Remark 1.3).

Relation (3) implies that F_{n+1} ⊇ F_{n}, and F = \overline{⋃_{n=0}^{∞} F_{n}}.

Denote by ι the inclusion map F_{n} → F_{n+1}. We now describe ι in terms of the ‘triples’ notation, so that we can effectively identify F_{n} with the familiar algebra K(l^{2}(α(G)\G))^{⊗n}⊗C^{∗}(G). Since we have to consider words with P_{n} and with P_{n+1} simultaneously, we’ll put a subscript to keep track of that and denote (x, k, y)_{j} = δ_{x}P_{j}δ_{α^{j}(k)y}, x^{−1}, y ∈ R_{j}.

Now, (x, k, y)_{n}(z, l, w)_{n+1} = 0 unless yz ∈ α^{n}(G), in which case the product is δ_{xα^{n}(k)yzα^{n+1}(l)}P_{n+1}δ_{w}. We need to convert this to ‘triples’ notation. Let z = y^{−1}α^{n}(j). So, now, the product is

δ_{xα^{n}(kj)α^{n+1}(l)}P_{n+1}δ_{w}.

Let kj = qα(g) for q ∈ R_{1}^{−1}. Recall that x ∈ R_{n}^{−1}, so x = r_{0}α(r_{1}) … α^{n−1}(r_{n−1}) for some r_{0}, …, r_{n−1} ∈ R_{1}^{−1}. Therefore xα^{n}(q) ∈ R_{n+1}^{−1}. So, the above expression in ‘triples’ notation is:

(xα^{n}(q), gl, w)_{n+1}.

So, we obtain

ι((x, k, y)_{n}) = ∑_{q∈R_{1}^{−1}}(xα^{n}(q), p, α^{n}(s)y)_{n+1}

where p ∈ G,s ∈ R_{1} are given by the equation α(p)s = q^{−1}k.

Remark 2.2. We know that F_{n+1} ≅ F_{n} ⊗ K(l^{2}(α(G)\G)). Writing F_{n} ≅ F_{0} ⊗ K(l^{2}(α(G)\G))^{⊗n}, F_{n+1} ≅ F_{1} ⊗ K(l^{2}(α(G)\G))^{⊗n}, by looking at the formula for ι, we can see that ι = ι_{0} ⊗ id, where ι_{0} is the inclusion map from F_{0} to F_{1}.

Theorem 2.3. If G is amenable and α is pure then F is simple.

To prove the theorem, we need a lemma. We first fix some notation.

Denote by tr the von Neumann trace on C^{∗}(G) (which is faithful since G was taken to be amenable). α extends to a unital endomorphism of C^{∗}(G), which we denote by α as well. For a ∈ C^{∗}(G) and g ∈ G we denote by â(g) the g-th Fourier coefficient of a, i.e., â(g) = tr(aδ_{g^{−1}}). Denote by α^{∗}(a) the element of C^{∗}(G) satisfyting { \atop α^{∗}(a)}(g) = â(α(g)). Observe that (1, 1, 1)_{1}ι(a)(1, 1, 1)_{1} = α^{∗}(a)(1, 1, 1)_{1} (when here C^{∗}(G) is identified with F_{0}).

Lemma 2.4. If a ∈ C^{∗}(G) is an element such that â(1) = 0, then ∥α^{∗n}(a)∥ → 0.

Proof. We know that the image of ℂG is dense in C^{∗}(G). Fix ϵ > 0, and let a′ ∈ ℂG satisfy ∥a′ − a∥ < ϵ. We can assume without loss of generality that â′(1) = 0. Observe that α^{∗} is a contraction. Since α is pure, there is some m′ such that for all m ≥ m′, α^{∗m}(a′) = 0 (since all its Fourier coefficients vanish). Therefore, for all m ≥ m′, ∥α^{∗m}(a)∥ < ϵ, which is what we wanted to show. □

Corollary 2.5. For all a ∈ C^{∗}(G), α^{∗n}(a) → â(1)1 in norm.

Proof of Theorem 2.3. Suppose J ◁ F. We know that

J = \overline{⋃_{n=0}^{∞} J ∩ F_{n}}

where we think of F_{n} as being inside F (the connecting maps are injective). Denote J_{n} = J ∩ F_{n}. Notice that ι(J_{n}) ⊆ J_{n+1}, and that J_{n} ≅ J_{0} ⊗ K(l^{2}(α(G)\G))^{⊗n}. By the observation before the lemma, this implies that if a ∈ J_{0} then α^{∗}(a) ∈ J_{0} as well.

Suppose J_{0} ≠ 0. Let a ∈ J_{0} be a nonzero positive element. Since tr is faithful, â(1) = tr(a) > 0. By Corollary 2.5, since J_{0} is closed, we see that 1 ∈ J_{0}, so J_{0} is trivial, and hence J_{n} are trivial for all n, hence J is trivial, hence F is simple. □

Examples.

1. Take G = ℤ, and α to be multiplication by a positive integer N > 1. We identify δ_{n} with the function z^{n} (where z denotes both the complex variable and the inclusion map 𝕋 → ℂ). We pick R_{1} = {0, 1, 2, …, N − 1}, and we identify K(l^{2}(ℤ/Nℤ)) with M_{N}, where we label the columns 0, 1, 2, …, N −1.

By the above remark, in order to see what the map ι looks like, it is enough to look at the map from F_{0} ≅ C(𝕋) to F_{1} ≅ M_{N}(C(𝕋)).

The only triples in F_{0} are of the form (0, n, 0)_{0} so we simply write them as z^{n}, and we write (−a, n, b)_{1}, a, b ∈ {0, 1, …, N − 1} as z^{n}e_{a,b}, where e_{a,b} is the standard matrix unit (the matrix whose entry in the a-th row and b-th column is 1, and is zero everywhere else).

Since C(𝕋) is generated by z, it suffices to see to where z maps. Translating the above formula to additive notation and to our special case, and replacing q by −q for convenience, we see that z maps to ∑_{q=0}^{N−1} z^{p}e_{q,s} where p,s are given by the equation Np + s = q + 1. It is easy to solve this equation: we have p = 0, s = q + 1 except for the last term, in which case we have p = 1, s = 0.

Writing this in matrix form, we get

z ↦ ( \begin{matrix}0 & 1 & 0 & 0 & … & 0 & 0 \\ 0 & 0 & 1 & 0 & … & 0 & 0 \\ 0 & 0 & 0 & 1 & … & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & … & 1 & 0 \\ 0 & 0 & 0 & 0 & … & 0 & 1 \\ z & 0 & 0 & 0 & … & 0 & 0 \end{matrix} ) .

This is the same map which appears in the construction of the Bunce-Deddens algebras, so F is isomporphic to the Bunce-Deddens algebra corresponding to the supernatural number N^{∞}.

2. Fix a finite group H, let G = H × H × H × … (the algebraic sum, so each element has only finitely many nontrivial terms), and let α be the left shift α(a_{1}, a_{2}, a_{3}, …) = (1, a_{1}, a_{2}, a_{3}, …). We pick R_{1} to be H × 1 × 1 × …, which we identify with H in the obvious way. As in the previous example, it is enough to understand the map F_{0} → F_{1}. We denote the images of G in F_{0} = C^{∗}(G) by δ_{a,b⃗}, with a ∈ H and b⃗ denoting the tail. The formula yields in this case: δ_{a,b⃗} ↦ δb⃗ ∑_{q∈H} e_{q,qa}, i.e., the image is δb⃗ tensored by the permutation matrix coming from the image of a under the right regular representation of H (notice that we write δ_{z}e_{x,y} = (x^{−1}, z, y), x, y ∈ H).

Since a, b⃗ has only finitely many nonzero entries, after sufficiently many applications of ι to this, we will have a zero-one matrix. ι applied to a scalar matrix simply embeds it in the standard multiplicity |H| embedding into the higher matrix algebra. Therefore, the image of each element of the dense subalgebra ℂG ⊗ K(l^{2}(α(G)\G))^{⊗n} ⊆ F_{n} in F is the same as an image of a scalar matrix in some higher F_{m}. Therefore, the images of the scalar matrices are dense in F, so F = \overline{⋃_{k=0}^{∞} M_{|H|^{k}}}, with the appropriate inclusion, and therefore F is isomorphic to the UHF algebra with supernatural number |H|^{∞}.

3. Let G = ℤ_{2} ∗ ℤ_{2} = ⟨a, b | a^{2} = b^{2} = 1⟩, and let α : G → G be given by α(a) = aba, α(b) = bab. It is easily verified that [G : α(G)] = 3, and R_{1} = R_{1}^{−1} = {1, a, b} is a list of both left and right coset representatives. Notice that α(G) is not normal in G in this case. It is known ([B] 6.10.4) that K_{0}(C^{∗}(G)) ≅ ℤ^{3}, and K_{1}(C^{∗}(G)) = 0, and that if we denote P_{a} = (1 + δ_{a})/2, P_{b} = (1 + δ_{b})/2, then [1],[P_{a}],[P_{b}] generate K_{0}(C^{∗}(G)) as a free abelian group. Identifying F_{1} with M_{3}(C^{∗}(G)), and making the first row in the matrix correspond to 1 ∈ R_{1}, the second to a ∈ R_{1} and the third to b ∈ R_{1}, we see that

ι(1) = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ι(δ_{a}) = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & δ_{b} \end{matrix} ) ι(δ_{b}) = ( \begin{matrix}0 & 0 & 1 \\ 0 & δ_{a} & 0 \\ 1 & 0 & 0 \end{matrix} ) .

so

ι(P_{a}) = ( \begin{matrix}\frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & P_{b} \end{matrix} ) ι(P_{b}) = ( \begin{matrix}\frac{1}{2} & 0 & \frac{1}{2} \\ 0 & P_{a} & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} \end{matrix} )

and therefore the induced map on K_{0} is given by

ι_{∗}([1]) = 3 · [1]  ι_{∗}([P_{a}]) = [1] + [P_{b}]  ι_{∗}([P_{b}]) = [1] + [P_{a}]

which in matrix form is given by

( \begin{matrix}3 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) .

K_{0}(F) is the inductive limit of the inductive system given by ℤ^{3} → ℤ^{3} → ℤ^{3} → ⋯, where the maps are given by this matrix. This group is isomorphic to ℤ [\frac{1}{3}] ⊕ ℤ ⊕ ℤ.

2.1. An abelian subalgebra. Let A_{n} = span{(x^{−1}, 1, x)_{n}|x ∈ R_{n}} ⊆ F_{n}. A_{n} is an abelian finite dimensional algebra (of dimension |R_{n}|). Notice that if we identify F_{n} with C^{∗}(G) ⊗ M_{m} (where m = |R_{n}| = [G : α^{n}(G)]), then A_{n} is identified with ℂ1 ⊗ Diag, Diag being the diagonal subalgebra of M_{m}. Let Δ denote the standard faithful expectation map M_{m} → Diag, and let E_{A_{n}} : F_{n} → A_{n} be the map given by E_{A_{n}} = tr(·)1 ⊗ Δ, then E_{A_{n}} is a faithful expectation. Notice that A_{n} ⊆ A_{n+1} and furthermore, the E_{A_{n}}’s are consistent with the inclusion maps, i.e., E_{A_{n+1}}ι = ι⌈_{A_{n}}E_{A_{n}}. Therefore, this consistent system of faithful expectation maps passes on to the direct limit, and so we obtain a faithful expectation E_{A} : F → A, where A is the direct limit of the A_{n}’s.

Â, the Gelfand spectrum of A, is homeomorphic to the Cantor set. It can be thought of as a product of a countable number of copies of R_{1} with the Tychonof topology, where there Gelfand transform of (x^{−1}, 1, x)_{n}, where

x = α^{n−1}(r_{n−1})α^{n−2}(r_{n−2}) ⋯ α(r_{1})r_{0},

r_{0}, …, r_{n−1} ∈ R_{1}, is the characteristic function of the basic cylindrical clopen set {r_{0}} × {r_{1}} × ⋯ × {r_{n−1}} × R_{1} × R_{1} × · · ·.

For any a ∈ A, and any x ∈ G, we have δ_{x}aδ_{x^{−1}} ∈ A. Denote this left action by β: β_{x}(a) = δ_{x}aδ_{x^{−1}}. That gives rise to an action of G on Â. Let us describe this action. Notice that G acts on R_{1}, where r · x = r′ for the unique r′ ∈ R_{1} such that rx ∈ Gr′. Similarly, we have an action on R_{2}, and the action on R_{1} is the quotient action (because of the way we constructed R_{2}), and so on. Â ≅ lim_{←} R_{n}, so the consistency gives us an action on Â, which is the dual of the action on A. If ω = (r_{0}, r_{1}, …) ∈ Â (when we identify it with R_{1} × R_{1} × ⋯), then the stabilizer of ω is the decreasing intersection of the stabilizers of ω_{n} = (r_{0}, …, r_{n}) ∈ R_{n+1}. Write x_{n} = α^{n}(r_{n}) ⋯ α(r_{1})r_{0}, then the stabilizer of ω_{n} is x_{n}^{−1}α^{n+1}(G)x_{n}. Therefore the stabilizer of ω is ⋂_{n=0}^{∞} x_{n}^{−1}α^{n+1}(G)x_{n}.

Lemma 2.6. F ≅ A ×_{β} G.

Proof. Any element of the form δ_{x}P_{n}δ_{y} can be written in the form δ_{t^{−1}}P_{n}δ_{t}δ_{z}, for some t ∈ R_{n}. If we take two such elements, δ_{t^{−1}}P_{n}δ_{t}δ_{z}, δ_{s^{−1}}P_{n}δ_{s}δ_{w}, then their product satisfies δ_{t^{−1}}P_{n}δ_{t}δ_{z}δ_{s^{−1}}P_{n}δ_{s}δ_{w} = δ_{t^{−1}}P_{n}δ_{t}(δ_{z}δ_{s^{−1}}P_{n}δ_{s}δ_{z^{−1}})δ_{z}δ_{w} = δ_{t^{−1}}P_{n}δ_{t}β_{z}(δ_{s^{−1}}P_{n}δ_{s})δ_{zw} where β denotes the action of G on A.

So, span{aδ_{x} | a ∈ A , x ∈ G} is a dense subalgebra of F, which is isomorphic to the twisted convolution algebra used in the definition of the crossed product. Therefore, F is a quotient of the crossed product A ×_{β} G. Let φ : A ×_{β} G → F denote this quotient map. It is clear that E_{A} ∘ φ coincides with the canonical expectation A ×_{β} G → A, and therefore kerφ = 0, which is what we wanted. □

Remark 2.7. When the endomorphism is the multiplication by n map on the integers, the action of ℤ on the Cantor set we obtain is the odometer action (with n digits). This gives us another way to see that the Bunce-Deddens algebras can be obtained as crossed products by the odometer actions (see [Da]).

3. A crossed product representation

Define Φ : F → F by Φ(a) = SaS^{∗}. Φ is a non-unital injective endomorphism of F. Following the procedure from [C], we repeat Φ and form a direct system F^{(0)} ⟶ F^{(1)} ⟶ F^{(2)} ⟶ ⋯ where the F^{(n)} are just copies of F. We denote the direct limit by F⃗, and we denote by Φ⃗ the automorphism of F⃗ induced by Φ. We denote by P the image of 1_{F^{(0)}} in F⃗.

Let Ẽ_{α} = F⃗ ×Φ⃗ ℤ, then E_{α} ≅ P Ẽ_{α}P, so E_{α} and Ẽ_{α} are strongly Morita equivalent. Notice that in ‘triples’ notation, Φ((x, k, y)_{n}) = (α(x), k, α(y))_{n+1}, and therefore

Φ is unitarily equivalent to the multiplicity one embedding of F into the upper corner of M_{[G:α(G)]}(F) ≅ F, and in particular, it induces an isomorphism on Ktheory, so K(F) ≅ K(F⃗).

The map Φ, thought of as a map on the inductive system ⋯ → F_{n} → F_{n+1} → ⋯ has the effect of shifting the sequence by 1 (and stabilizing by M_{|R_{n}|}), and so the action on K_{∗}(F) will be ι_{∗}^{−1} (the action of the shift). The Pimsner-Voiculescu 6-term exact sequence for crossed products by the ℤ now gives us a procedure to compute K_{∗}(E_{α}).

Examples. We return to the examples from the previous section.

1. Let us compute K_{∗}(E_{α}). It can be seen from the inductive limit description above that in this case, K_{0}(F) ≅ ℤ [\frac{1}{N} ], K_{1}(F) ≅ ℤ. Φ⃗ induces the identity map on K_{1}, and the multiplication by \frac{1}{N} map on K_{0}. An application of the Pimsner-Voiculescu sequence yields then that

K_{0}(E_{α}) ≅ ℤ ⊕ Z_{N−1}, K_{1}(E_{α}) ≅ Z.

2. Here the algebra E_{α} we obtain is isomorphic to the Cuntz algebra O_{n}, with n = |H|.

3. The induced action on K_{0} is given by

( \begin{matrix}3 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} )^{−1}

Applying the Pimsner-Voiculescu sequence yields

K_{0}(E_{α}) ≅ ℤ ⊕ Z_{2}, K_{1}(E_{α}) ≅ ℤ.

Unlike the first two cases, it is not clear at this time whether E_{α} is simple.

The expectation E_{F} is the restriction of the faithful canonical expectation map Ẽ_{α} ⟶ F⃗ from the general theory of crossed products, and therefore it is faithful as well.

Denote E = E_{A} ∘ E_{F} : E_{α} ⟶ A. This is also a faithful expectation.

Let π be a nondegenerate representation of E_{α}. π⌈_{A} is a representation of A, and as such is given by a spectral measure Λ on Â. We henceforth view vectors of the representation space H_{π} as sections over Â. Since F is simple, assuming π ≠ 0, π must be faithful when restricted to F, and therefore it must be faithful when restricted to A as well, so we must have \overline{supp}(Λ) = Â, and therefore

∥π(a)∥ = sup{|â(ω)| | ω ∈ Â}.

We know that S, δ_{x} normalize A (i.e., SAS^{∗} ⊆ A, S^{∗}AS ⊆ A, δ_{x}Aδ_{x}^{∗} ⊆ A for all x ∈ G), and therefore the semigroup generated by those elements acts (partially) on Â. Denote by p · U the action of this semigroup on Â, where p ∈ Â and U is in the semigroup.

Observation 3.1. Let U be as above, and let ξ ∈ H_{π}, then \overline{supp}(Uξ) ⊆ \overline{supp} ξ · U.

Proof. It suffices to verify this for ξ such that \overline{supp} ξ is a cylindrical subset of Â of the form B = {r_{1}} × {r_{2}} × ⋯ × {r_{n}} × R_{1} × R_{1} × · · ·. For the purpose of this proof, we freely identify elements of A with functions in C(Â). So, ξ = χ_{B}ξ. Notice that U^{∗}U ∈ A. Write f = U^{∗}U ∈ C(Â). So, Uξ = UU^{∗}Uξ = Ufξ = Ufχ_{B}ξ = Uχ_{B}fξ = (Uχ_{B}U^{∗})Uξ = χ_{B·U}Uξ, and \overline{supp} χ_{B·U}Uξ ⊆ \overline{supp} χ_{B·U} = B · U = \overline{supp} ξ · U which is what we wanted. □

For x ∈ G we can associate an element (x_{0}, x_{1}, …) of R_{1}^{ℕ} (≅ Â) by requiring that x ∈ α^{n+1}(G)α^{n}(x_{n}) ⋯ α(x_{1})x_{0}. Clearly this is well defined.

Definition 3.2. We call the above sequence the R-sequence of x, and denote it by R(x).

This gives us a map R : G ⟶ Â.

Lemma 3.3. The map R is injective.

Proof. We must show that if R(x) = R(y), x, y ∈ G, then x = y. R(x) = R(y) is equivalent to saying that α^{n}(G)x = α^{n}(G)y for all n ∈ ℕ, i.e., xy^{−1} ∈ α^{n}(G) for all n, and since α was assumed to be pure, we obtain xy^{−1} = 1. □

Definition 3.4. We say that the endomorphism α is totally normal if α^{n}(G) is normal in G for all n.

From now on, we assume that α is totally normal. Notice that this implies that the action of G on Â is free (from the remarks in the previous section).

We define maps Ψ_{k}^{n} : R_{1}^{n+k} → R_{1}^{n} (for n > k) by taking Ψ_{k}^{n}(x_{1}, …, x_{n+k}) to be the element of R_{1}^{n} corresponding to the coset representative of

x_{1}^{−1}α(x_{2}^{−1}) ⋯ α^{n−1}(x_{n}^{−1})α^{n−1}(x_{n+k}) ⋯ α(x_{k+2})x_{k+1}.

From the normality codition, it follows that Ψ_{k}^{n} is |R_{1}|^{k}–to–1. Furthermore, the following diagram commutes.

R_{1}^{n+k+1} ⟶^{Ψ_{k}^{n+1}} R_{1}^{n+1}
π↓  ↓^{π}
R_{1}^{n+k} ⟶^{Ψ_{k}^{n}} R_{1}^{n}

where the π’s denote the projections onto the first n + k and n coordinates, respectively.

Therefore those maps form a consistent system, and so we get maps on the projective limit Ψ_{k} : Â → Â.

Let μ be the probability product measure on Â (thought of as R_{1}^{ℕ}), where the measure on each factor is the one giving all points the same measure (\frac{1}{|R_{1}|}). The

canonical maps Â ⟶ R_{1}^{m} for m ∈ ℕ induce product push-forward probability measures μ_{m} on R_{1}^{m} (assigning the measure 1/|R_{1}|^{m} to each point). The fact that the maps Ψ_{k}^{n} are |R_{1}|^{k}–to–1 implies that they are measure preserving with those measures, and therefore the maps Ψ_{k} are all measure perserving.

Definition 3.5. We say that a point ω ∈ R_{1}^{ℕ} ≅ Â is tail intersecting if there is some nontrivial x ∈ G such that ωx is a tail of ω.

Lemma 3.6. If ω is tail intersecting then there is some k such that Ψ_{k}(ω) ∈ R(G) (i.e., is an R-sequence for some group element).

Proof. Suppose ω = (x_{1}, x_{2}, …), and there is some y ∈ G such that ωy = (x_{k}, x_{k+1}, …). Then for each n ∈ ℕ, we must have

y ∈ α^{n}(G)x_{1}^{−1}α(x_{2}^{−1}) ⋯ α^{n−1}(x_{n}^{−1})α^{n−1}(x_{n+k}) ⋯ α(x_{k+2})x_{k+1},

and therefore, we must have Ψ_{k}(ω) = R(y). □

Corollary 3.7. {ω | ω is not tail intersecting} is dense in R_{1}^{ℕ}.

Proof. Since R(G) is countable, and the measure μ is nonatomic, it has μ-measure 0. Since the Ψ_{k} are measure preserving, we see that μ(⋃_{k} Ψ_{k}^{−1}(R(G))) = 0, and therefore its complement, which contains the set in which we’re interested, has full measure, and in particular is dense. □

Lemma 3.8. For all a ∈ E_{α}, ∥π(a)∥ ≥ ∥π(E(a))∥.

Before proving the lemma, we state and prove its main consequence.

Theorem 3.9. If G is amenable and α is pure and totally normal then E_{α} is simple.

Proof of Theorem 3.9 (assuming Lemma 3.8). Let π be a representation of E_{α}, and let a ∈ ker(π) be a positive element, then E(a) ≠ 0, since E is faithful. By the lemma, E(a) ∈ ker(π), so ker(π) ∩ F ≠ 0, and therefore ker(π) ⊇ F, so 1 ∈ ker(π), so π = 0. So any nonzero representation of E_{α} must be faithful, i.e., E_{α} is simple. □

Proof of Lemma 3.8. By the remarks preceeding the lemma, it suffices to show that ∥π(a)∥ ≥ |{ \atop E(a)}(ω)| for a dense collection of ω ∈ Â. It also suffices to verify it for a dense collection of a ∈ E_{α}. So we assume that a is taken from the *- algebra generated by S and the δ_{x}, x ∈ G. Each such element is in the span of the semigroup from above. Notice that if ω is not tail intersecting then the only semigroup elements that fix it are the idempotents which have it in their domain. We pick ω to be in this dense set.

We can decompose the representation space H_{π} into a direct sum H_{π} = L^{2}(Â, ν)⊕ H′, where \overline{supp}(ν) = Â, and A acts on L^{2}(Â, ν) by multiplication (by the Gelfand transform). We also assume, without loss of generality, that ν(Â) = 1. Suppose ω corresponds to the sequence r_{1}, r_{2}, r_{3}, . . .. Denote B_{n} = {r_{1}} × {r_{2}} × ⋯ × {r_{n}} × R_{1} × R_{1} × ⋯, let ξ′_{n} be the vector χ_{B_{n}} ⊕ 0, and let ξ_{n} = \frac{ξ′_{n}}{∥ξ′_{n}∥}. Denote ξ_{0} = χÂ ⊕ 0 (and notice that ∥ξ_{0}∥ = 1), so ξ′_{n} = π(χ_{B_{n}})ξ_{0}.

For any a ∈ A, we have

â(ω) = lim_{n→∞} ⟨π(a)ξ_{n}, ξ_{n}⟩ = lim_{n→∞} \frac{1}{∥ξ_{n}∥^{2}} ⟨π(χ_{B_{n}}aχ_{B_{n}})ξ_{0}, ξ_{0}⟩

If U in the semigroup is not a projection, then because of our choice of ω, for sufficiently large n, B_{n} ∩ (B_{n} · U) = ∅, so χ_{B_{n}}Uχ_{B_{n}} = 0. Also, notice that the intersection of the semigroup with A is exactly the projections. Therefore, for an element a in this dense subalgebra and for such an ω, for all sufficiently large n we have ⟨π(χ_{B_{n}}aχ_{B_{n}})ξ_{0}, ξ_{0}⟩ = ⟨π(χ_{B_{n}}E(a)χ_{B_{n}})ξ_{0}, ξ_{0}⟩. So,

∥π(a)∥ ≥ | ⟨π(a)ξ_{n}, ξ_{n}⟩ | = | ⟨π(E(a))ξ_{n}, ξ_{n}⟩ | → |{ \atop E(a)}(ω)|

which is what we wanted. □

Remark 3.10. A groupoid approach could have been used in this case. We represented the γ-invariant subalgebra as a crossed product with an action of a discrete group on a compact Hausdorff set, and therefore it can be viewed as the groupoid algebra of the associated transformation groupoid. We could then present E_{α} as the groupoid algebra of a groupoid which is the restriction of the semi-direct product groupoid of the above transformation groupoid by an action of the integers (cf. [R]). Equivalently, we could view this groupoid as a certain restriction of the universal groupoid associated to the inverse semigroup generated by G and S (cf. [Pa]). In this language, we showed that under our conditions, the groupoid thus obtained is essentially principal ([R]) with no invariant sets in the unit space. The proof of the last lemma is essentially that of [R], with the simplifications that arise from the fact that our situation is less general.

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Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720,USA

ilan@math.berkeley.edu

This paper is available via http://nyjm.albany.edu:8000/j/2002/8-6.html.