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The Bunce-Deddens Algebras as Crossed Products

arXiv:funct-an/9302001v1 9 Feb 1993UNM–RE–005February 8, 1993Printed November 15, 2018The Bunce-Deddens Algebras as Crossed Productsby Partial AutomorphismsRuy Exel*Department of Mathematics and Statistics, University of New MexicoAlbuquerque, New Mexico 87131We describe both the Bunce-Deddens C *-algebras and their Toeplitz versions, as crossed prod-ucts of commutative C *-algebras by partial automorphisms.In the latter case, the commutativealgebra has, as its spectrum, the union of the Cantor set and a copy the set of natural numbers N,fitted together in such a way that N is an open dense subset. The partial automorphism is inducedby a map that acts like the odometer map on the Cantor set while being the translation by one on N.From this we deduce, by taking quotients, that the Bunce-Deddens C *-algebras are isomorphic to the(classical) crossed product of the algebra of continuous functions on the Cantor set by the odometermap.1.

IntroductionRecall from [4] that a weighted shift operator is a bounded operator on l2 = l2(N)given, on the canonical basis {en}∞n=0, by Sa(en) = an+1en+1, where the weight sequencea = {an}∞n=1, is a bounded sequence of complex numbers. A weighted shift is said to bep-periodic if its weights satisfy an = an+p for all n.Given a strictly increasing sequence {nk}∞k=0 of positive integers, such that nk dividesnk+1 for all k, the Bunce-Deddens-Toeplitz C∗-algebra A = A({nk}) is defined to be theC∗-algebra of operators on l2, generated by the set of all nk-periodic weighted shifts, forall k. These algebras were first studied by Bunce and Deddens in [5].

It was observedby them that the algebra K, of compact operators on l2, is contained in A and that thequotient A/K is a simple C∗-algebra. The latter became known as the Bunce-DeddensC∗-algebra and has been extensively studied (see, for example, [1], [2, 10.11.4], [3], [6], [9,p.

248] , [11], [12], [13]). We shall denote these algebras by B({nk}) or simply by B, ifthe weight sequence is understood.The goal of the present work is to describe both A({nk}) and B({nk}) as the crossedproduct of commutative C∗-algebras by partial automorphisms [7], in much the same wayas we have described general AF-algebras [8] as partial crossed products.In the case of A({nk}), we shall see that it is given by a curious (partial) dynamicalsystem consisting of a topological space X which can be thought of as a compactification ofthe (discrete) space N of natural numbers, the complement of N in X being homeomorphicto the Cantor set K. The transformation f of X, by which the partial crossed product is* On leave from the University of S˜ao Paulo.

2ruy exeltaken, leaves both N and K invariant. Its behavior on N is that of the translation by one,while the action on K is by means of the odometer map (see, for example, [12]) whichis defined as follows.

Given a sequence {qk}∞k=0 of positive integers (below we shall useqk = nk+1/nk), consider the Cantor set, as given by the modelK =∞Yj=0{0, 1, . .

., qj −1}.The odometer map is the mapO: K →K,given by formal addition of (1, 0, . .

.) withcarry over to the right.

Note thatO(q0 −1, q1 −1, . .

.) = (0, 0, .

. .

)since the carry overprocess, in this case, extends all the way to infinite. For further reference let us call by thename of “partial odometer” the restriction of O to a map from X −{(q0 −1, q1 −1, .

. .

)}to X −{(0, 0, . .

.)} so that, for this map, the carry over process always terminates in finitetime.As already mentioned, B({nk}) is the quotient of A({nk}) by K. But K can be seen tocorrespond to the restriction of the above dynamical system to N (see [7]).

So, we deducethat the Bunce-Deddens algebras B({nk}) are isomorphic to the crossed product of theCantor set by the odometer map. This result is already well known [2, 10.11.4] but it isinteresting to remark how little bookkeeping is necessary to deduce it from the machineryof partial automorphisms [7].

Moreover, this should be compared with [8], Theorem 3.2,according to which UHF-algebras correspond to the crossed product of the Cantor set bythe partial odometer map. One therefore obtains a crystal clear picture of the fact, alreadynoticed by Bunce and Deddens, that UHF-algebras sit as subalgebras of B({nk}) (see also[12]).2.

Circle ActionsFor each z in the unit circle S1 = {w ∈C: |w| = 1} let, Uz denote the diagonalunitary operator on l2, given by Uz(en) = znen. If S is any weighted shift, it is easy tosee that UzSU −1z= zS.

Thus, denoting by αz the inner automorphism of B(l2) given byconjugation by Uz, one finds that A is invariant under αz. Moreover, one can see thatthis gives a continuous action of S1 on A, in the sense of [10], 7.4.1 (even though thecorresponding action is not continuous on B(l2)).Let us denote the fixed point subalgebra by A0.

It is easy to see that A0 consistsprecisely of the operators in A which are diagonal with respect to the basis {en}. Now,given that the C∗-algebra of (bounded) diagonal operators is isomorphic to l∞= l∞(N), itis convenient to view A0 as a subalgebra of l∞.

For the purpose of describing A0, observethat, since K is contained in A, it follows that c0 (the subalgebra of l∞formed by sequencestending to zero) is contained in A0. Carrying this analysis a bit further one can prove thatA0 = c0 ⊕D where D is the subalgebra of l∞generated by all nk-periodic sequences, forall k.This decomposition is useful in determining the spectrum of A0.

Note, initially, thatspectrum of c0 is homeomorphic to N (with the discrete topology) while the spectrum of

bunce-deddens algebras and partial automorphisms3D is the Cantor set, here denoted K. This said, one has that the spectrum of A0 can bedescribed, at least in set theoretical terms, as the union X = N ∪K. Moreover, since c0 isan essential ideal in A0, one sees that N is an open dense subset of X.

To better grasp theentire topology of X we need a more precise notation. Assume, without loss of generality,that n0 = 1 and let qk = nk+1/nk for k ≥0.

Any integer n with 0 ≤n < nk has a uniquerepresentation asn =k−1Xj=0β(n)jnjwhere 0 ≤β(n)j< qj. Here the β(n)jplay the role of digits in a decimal-like representation,except that the base varies along with the position of each digit.

Accordingly, we let β(n) =(β(n)0, . .

., β(n)k−1) be the corresponding notation for n (which we shall use interchangeablywithout further warning). When convenient, we shall also view β(n) as an element of thesetKk =k−1Yj=0{0, 1, .

. ., qj −1}.For each k and each β in Kk, we denote by eβ the nk-periodic sequence (thus an elementof D) given byeβ(n) =1ifn ≡β(mod nk)0otherwiseNote that the length of β determines which nk should be used in the above definition.Clearly the set {eβ: β ∈S∞k=1 Kk} generates D. Making use of the notation introduced,we shall adopt for the Cantor set, the modelK =∞Yj=0{0, 1, .

. ., qj −1}so that, once we view D as the algebra of continuous functions on K via the Gelfandtransform, the support of the eβ form a basis for the topology of K. In fact, the supportof eβ is precisely the set of elements γ = (γi) in K such that γj = βj for all j = 0, .

. ., k −1(assuming that β is in Kk).

That is, γ is in the support of β if and only if its initialsegment coincides with β.Considering now, the whole of A0, note that it is generated by the set of all idempotentsp which, viewed as elements of l∞, have one of the two following forms: either it has afinite number of non-zero coordinates (in which case p is in c0), or it coincide with someeβ, except for finitely many coordinates.The set of such idempotents is closed undermultiplication, which therefore implies that their support, in the spectrum X of A0, forma basis for the topology of X. With this we have precisely described the topology of X:

4ruy exel2.1. Theorem.

The spectrum of A0 consists of the union of the Cantor set K and a copyof the set of natural numbers N. Each element of N is an isolated point and a fundamentalsystem of neighborhoods of a point γ = (γi) in K consists of the sets Vk defined to be theunion of the sets{ζ ∈K: ζi = γi, i < k}and{n ∈N: n ≥kandβ(n)i= γi, i < k}.Note the interesting interplay between the digital representation of the natural num-bers on one hand, and of elements of the Cantor set, on the other.3. The Main ResultRecall from [7], Theorem 4.21, that a regular semi-saturated action of S1 on a C∗-algebra, causes it to be isomorphic to the covariance algebra of a certain partial automor-phism of the fixed point subalgebra.

In the case of the action α of S1 on A, describedabove, it is very easy to see that it is regular and semi-saturated. The semi-saturationfollows immediately, since every weighted shift belongs to the first spectral subspace ofα, henceforth denoted A1.The fact that α is regular depends on the existence of anisomorphismθ: A∗1A1 →A1A∗1and a linear isometryλ: A∗1 →A1A∗1such that, forx, y ∈A1, a ∈A∗1A1 and b ∈A1A∗1(i) λ(x∗b) = λ(x∗)b(ii) λ(ax∗) = θ(a)λ(x∗)(iii) λ(x∗)∗λ(y∗) = xy∗(iv) λ(x∗)λ(y∗)∗= θ(x∗y).See [7], 4.3 and 4.4 for more information.

It is easy to see that θ and λ, given by θ(a) =SaS∗and λ(x∗) = Sx∗, satisfy the desired properties, where S denotes the unilateral(unweighted) shift.Note that, in the present case, A∗1A1 = A0 while A1A∗1 is the ideal of A0 formedby all sequences for which the first coordinate vanishes.That is, under the standardcorrespondence between ideals and open subsets of the spectrum, A1A∗1 corresponds toX−{0}. The isomorphismθ: C(X) →C(X−{0})therefore induces a homeomorphismf: X →X −{0}which we would now like to describe.If an integer n is thought of as an element of X, as seen above, then the elementδn of l∞, represented by the sequence having the nth coordinate equal to one and zeroseverywhere else, corresponds to the characteristic function of the singleton {n} and, giventhat θ(δn) = δn+1, we see that f(n) = n + 1.Now, if γ = (γi) is in K, let γ|k be the kth truncation of γ, that is γ|k = (γ0, .

. ., γk−1)so that we can consider eγ|k, as defined above.

Also let fk be the element of c0 given byfk = (1, . .

., 1, 0, . .

.) where the last “1” occurs in the position k −1, counting from zero.

bunce-deddens algebras and partial automorphisms5The support of the Gelfand transform of the idempotent element (1 −fk)eγ|k is preciselythe set Vk referred to in 2.1. So, as k varies, the intersection of these sets is precisely thesingleton {γ}.

Therefore, to find out what f(γ) should be, it is enough to look for theintersection of the supports of the Gelfand transforms of the elementsθ(1 −fk)eγ|k= S(1 −fk)eγ|kS∗.The reader is now invited to verify, using this method, that the effect that f has on γis precisely the effect of the odometer map. One should exercise special attention to checkthat the above method does indeed givef(q0 −1, q1 −1, .

. .) = (0, 0, .

. .

),in contrastwith the partial automorphisms that produce UHF-algebras [8], since (q0 −1, q1 −1, . .

. )is removed from the domain of the maps considered there.

Summarizing our findings sofar, we have:3.1. Theorem.

The Bunce-Deddens-Toeplitz C∗-algebra A({nk}) is isomorphic to thecrossed product ofC(X) by the partial automorphismθ: C(X) →C(X−{0})inducedby the (inverse of the) mapf: X →X −{0}acting like the odometer on K and liketranslation by one on N.Recall that the Bunce-Deddens C∗-algebras were defined to be the quotient B = A/K.If one identifies A with the crossed product above, it is easy to see that the ideal Kcorresponds to the crossed product of c0 by the corresponding restriction of θ. Therefore,the quotient can be described as the (classical) crossed product of the Cantor set by theodometer map. That is:3.2.

Theorem. The Bunce-Deddens C∗-algebra B({nk}) is isomorphic to the crossedproduct of C(K) by the automorphism induced by the odometer map.References[1] R. J. Archbold, “An averaging process for C∗-algebras related to weighted shifts”,Proc.

London Math. Soc.

35 (1977), 541–554. [2] B. Blackadar, “K-theory for operator algebras”, MSRI Publications, Springer–Verlag,1986.

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189 (1985), 55–63. [4] J. W. Bunce and J.

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[6] E. E. Effros and J. Rosenberg, “C∗-algebras with approximate inner flip”, Pacific J.Math. 77 (1978), 417–443.

[7] R. Exel, “Circle actions on C∗-algebras, partial automorphisms and a generalizedPimsner–Voiculescu exact sequence”, preprint, University of New Mexico, 1992.

6ruy exel[8], “Approximately finite C∗-algebras and partial automorphisms”, preprint,University of New Mexico, 1992. [9] P. Green, “The local structure of twisted covariance algebras”, Acta Math.

140 (1978),191–250. [10] G. K. Pedersen, “C∗-Algebras and their Automorphism Groups”, Academic Press,1979.

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421 (1991), 43–61. [12] I. F. Putnam, “The C∗-algebras associated with minimal homeomorphisms of theCantor set”, Pacific J.

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