New York Journal of Mathematics

New York J. Math. 15 (2009) 431–434.

Regular AF subalgebras of some crossed products

Catalin Rada

Abstract. Let C(X) ⋊_{T} Z be the crossed product associated to a dynamical system (X, T). We characterize the regular AF subalgebras of C(X) ⋊_{T} Z that can arise as the algebra A_{Y} = ⟨C(X), uC_{0}(X\Y)⟩ for some closed subset Y of X. We also characterize the minimal homeomorphisms in A_{Y} terms.

Contents

1. Regular AF subalgebras 431 2. Minimality 433 References 434

1. Regular AF subalgebras

Let X be the Cantor set and let T : X → X be a minimal homeomorphism. Denote by u the canonical unitary implementing the action Z ∋ n → α_{T}^{n} ∈ Aut(C(X)), where α_{T} : C(X) → C(X) is given by α_{T} (f) = f ∘ T^{−1} for every f ∈ C(X) and by C(X) ⋊_{T} Z the crossed product C(X) ⋊_{α_{T}} Z. Set C_{0}(X\Y) = {f ∈ C(X)|f|_{Y} = 0}. Recall the following theorem of Putnam ([2]):

Theorem 1.1. For any nonempty closed subset Y of X, the C*-subalgebra of C(X) ⋊_{T} Z generated by C(X) and uC_{0}(X\Y) is an AF algebra.

Set A_{Y} = ⟨C(X), uC_{0}(X\Y)⟩. For any C*-subalgebra A of C(X) ⋊_{T} Z containing C(X), the normalizer of C(X) in U(A) is given by N(C(X), A) = {v ∈ U(A)|vC(X)v^{∗} = C(X)}. (Here U(A) is the set of unitary elements in A.) The algebra A is called regular if N(C(X), A) generates A. It is

431^{ISSN 1076-9803/09}

Received August 21, 2009.

Mathematics Subject Classification. Primary 46L05; Secondary 37B05.

Key words and phrases. Crossed products, C*-algebras.

well-known that A_{Y} is regular. Another property of A_{Y} is contained in the following result of Poon (see [1]):

Theorem 1.2. Let Y be a nonempty subset of X. For every n ⩾ 1 and any clopen subset W of X one has that u^{n}χ_{W} (= uχ_{T^{n−1}(W)}uχ_{T^{n−2}(W)} … uχ_{W}) ∈ A_{Y} if and only if uχ_{T^{n−1}(W)}, uχ_{T^{n−2}(W)}, …, uχ_{W} ∈ A_{Y}. (Here χ_{W} denotes the characteristic function of W.)

We say that a C*-subalgebra A of the crossed product C(X) ⋊_{T} Z has property ⋆ if for any n ⩾ 1 and any clopen subset W of X,

uχ_{T^{n−1}(W)}, uχ_{T^{n−2}(W)}, …, uχ_{W} ∈ A

whenever u^{n}χ_{W} ∈ A.

Theorem 1.3. For any regular AF subalgebra A of C(X)⋊_{T} Z that contains C(X) and has property ⋆, there is a closed nonempty set Y of X such that A_{Y} = A.

Let us find the natural candidate for Y. Consider the set

I = {f ∈ C(X) | uf ∈ A} ⊆ C(X).

Lemma 1.4. The subset I is a closed ideal of C(X).

Proof. If f, g ∈ I then uf, ug ∈ A, and since A is an algebra u(f + g) = uf + ug ∈ A, so f + g ∈ I. If f ∈ I and g ∈ C(X) then uf ∈ A and since g ∈ C(X) ⊆ A we get ufg ∈ A, hence fg ∈ I. Remark that since g ∘ T^{−1} ∈ C(X) ⊆ A and uf ∈ A we get that ugf = (g ∘ T^{−1})uf ∈ A, hence gf ∈ I. Suppose that f_{n} → f and f_{n} ∈ I. Then uf_{n} → uf, and since uf_{n} ∈ A it follows that uf ∈ A, hence f ∈ I. □

Since I is an ideal in C(X) we know that I = C_{0}(X\Y) for some closed subset Y of X. Since A is AF we see that u is not in A, so Y is not empty.

Lemma 1.5. The inclusion A_{Y} ⊆ A is always valid.

Proof. It is sufficient to show that the generators of A_{Y} are in A. By the definition of A we know that C(X) ⊆ A. If f ∈ C_{0}(X\Y) = I then uf ∈ A, hence uC_{0}(X\Y) ⊆ A. Therefore A_{Y} ⊆ A. □

Lemma 1.6. The inclusion A ⊆ A_{Y} holds.

Proof. Since A is a regular algebra it is enough to show that each unitary in N(C(X), A) is also in A_{Y}. Letting v be a unitary in N(C(X), A), it is known from Lemma 5.1 of [2] that v = f ∑_{n∈Z} p_{n}u^{n}, where f is a unitary in

C(X), every p_{n} is a projection in C(X), only finitely many p_{n} are different than 0, p_{n}p_{k} = 0 for n ≠ k and ∑_{n∈Z} p_{n} = 1. Since the projections p_{n} are

orthogonal, we obtain that p_{n}u^{n} ∈ A for every n ∈ Z.

Regular AF subalgebras of some crossed products 433

We show that p_{n}u^{n} ∈ A_{Y}. Suppose that n ⩾ 1. If p_{n} = χ_{W} for a clopen subset W of X, from the chain of equalities p_{n}u^{n} = u^{n}(p_{n} ∘ T^{n}) = u^{n}χ_{T^{−n}(W)} = uχ_{T^{−1}(W)}uχ_{T^{−2}(W)} … uχ_{T^{−n}(W)} and the ⋆ condition,

uχ_{T^{−1}(W)}, uχ_{T^{−2}(W)}, …, uχ_{T^{−n}(W)} ∈ A.

Hence we see that χ_{T^{−1}(W)}, χ_{T^{−2}(W)}, …, χ_{T^{−n}(W)} ∈ I = C_{0}(X\Y). From the decomposition p_{n}u^{n} = uχ_{T^{−1}(W)}uχ_{T^{−2}(W)} … uχ_{T^{−n}(W)} conclude that p_{n}u^{n} ∈ A_{Y}. If p_{−n}u^{−n} ∈ A for some n ⩾ 1, we note that u^{n}p_{−n} = (p_{−n}u^{−n})^{∗} ∈ A, and as above one gets that u^{n}p_{−n} ∈ A_{Y}, and therefore p_{−n}u^{−n} ∈ A_{Y}. □

Remark 1.7. (1) Condition ⋆ can not be removed. To this end, let T be a minimal homeomorphism of X such that T^{2} is also minimal. We embed C(X) ⋊_{T^{2}} Z into C(X) ⋊_{T} Z via C(X) ∋ f → f ∈ C(X) and v → u^{2}, where v is the canonical unitary in C(X) ⋊_{T^{2}} Z implementing the action generated by T^{2}. Let Y be a closed, nonvoid subset of X, with Y ≠ X, and consider the regular AF subalgebra ⟨C(X), vC_{0}(X\Y)⟩ of C(X) ⋊_{T^{2}} Z. We can see this subalgebra as ⟨C(X), u^{2}C_{0}(X\Y)⟩ in the crossed product C(X) ⋊_{T} Z. The last algebra is then again regular and AF, not satisfying the condition ⋆ and not of the form ⟨C(X), uC_{0}(X\Z)⟩ for any nonempty closed subset Z of X.

(2) If ⋆ is removed and Y is a singleton it was noted by Poon (see Corollary 4.4 of [1]) that Lemma 1.6 still holds.

2. Minimality

In this section we give a characterization of the minimal homeomorphisms in terms of approximately finitely-dimensionality of A_{Y} s. If X is a zerodimensional compact space and T a self-homeomorphism of X, not necessarily minimal, then we can still define the C*-subalgebra ⟨C(X), uC_{0}(X\Y)⟩. It was proved in [1] that A_{Y} is an AF algebra iff X = ⋃_{n∈Z} T^{n}(W) for every

clopen subset W containing Y.

Proposition 2.1. Let X be a zero-dimensional compact space and T : X → X a self-homeomorphism. Then T is minimal iff A_{Y} is AF for any closed nonempty subset Y of X.

Proof. Suppose that A_{Y} is AF for any nonempty closed subset Y of X. If there is a proper closed subset Y of X such that T (Y) = Y (i.e., Y ≠ ∅, Y ≠ X) then Y is not clopen. Pick an element x_{0} ∈ X\Y and note again that Y ⋃{x_{0}} ≠ X (otherwise Y were clopen). Let W be a clopen subset of X such that Y ∪ {x_{0}} ⊆ W ⊆ X and Y ∪ {x_{0}} ≠ W, W ≠ X.

Since x_{0} is not in Y and x_{0} ∈ W there is a clopen subset U such that x_{0} ∈ U ⊆ W and U∩Y = ∅. But W^{c} ⋃{x_{0}} is a closed subset of X that is included in the clopen subset W^{c} ⋃ U. By Poon’s result (cited above) one gets that X = ⋃_{n∈Z} T^{n}(W^{c} ⋃ U). Since Y is not empty let y_{0} ∈ Y, we get that y_{0} ∈

T^{n_{0}}(W^{c} ⋃ U) for some n_{0} ∈ Z. We distinguish two cases. If y_{0} ∈ T^{n_{0}}(W^{c}), taking into account that also y_{0} ∈ Y implies that y_{0} ∈ T^{n_{0}}(Y), one gets that y_{0} ∈ T^{n_{0}}(W^{c}) ⋂ T^{n_{0}}(Y) = T^{n_{0}}(W^{c} ⋂ Y) = T^{n_{0}}(∅) = ∅, a contradiction. If y_{0} ∈ T^{n_{0}}(U) then again y_{0} ∈ T^{n_{0}}(U) ⋂ T^{n_{0}}(Y) = T^{n_{0}}(U ⋂ Y) = T^{n_{0}}(∅) = ∅, contradiction.

The other implication is always true by [2]. □

Acknowledgements. We are deeply indebted to Professor David Handelman for many useful discussions and continuous support.

References

[1] Poon, Yiu Tung. AF subalgebras of certain crossed products. Rocky Mountain J. Math. 20 (1990) 527–537. MR1065849 (91k:46078), Zbl 0727.46044.

[2] Putnam, Ian F. The C^{∗}-algebras associated with minimal homeomorphisms of the Cantor set. Pacific J. Math. 136 (1989) 329–353. MR0978619 (90a:46184), Zbl 0631.46068.

University of Ottawa, Department of Mathematics and Statistics, Ottawa, ON, Canada catalin.rada@alumni.uottawa.ca

This paper is available via http://nyjm.albany.edu/j/2009/15-24.html.