On the entropy and index of the winding endomorphisms of \mathcal{Q}_p
aa r X i v : . [ m a t h . OA ] F e b ON THE ENTROPY AND INDEX OF THE WINDINGENDOMORPHISMS OF P-ADIC RING C ∗ -ALGEBRAS VALERIANO AIELLO AND STEFANO ROSSI
Abstract.
For p ≥ , the p -adic ring C ∗ -algebra Q p is the universal C ∗ -algebragenerated by a unitary U and an isometry S p such that S p U = U p S p and P p − l =0 U l S p S ∗ p U − l = 1 . For any k coprime with p we define an endomorphism χ k ∈ End( Q p ) by setting χ k ( U ) := U k and χ k ( S p ) := S p . We then compute theentropy of χ k , which turns out to be log | k | . Finally, for selected values of k wealso compute the Watatani index of χ k showing that the entropy is the naturallogarithm of the index. Introduction
First introduced by Adler, Konheim, and McAndrew [1], the topological entropyof a continuous map on a compact Hausdorff space soon proved to be a useful numer-ical invariant (under topological conjugacy) to tackle, for instance, dynamics thatmay be out of the reach of the celebrated Halmos-von Neumann theorem, whichonly settles those with topological discrete spectrum. Two results worth mentioningare that the entropy of any homeomorphism of the circle is null and the entropy ofa differentiable map on a Riemannian manifold is finite, the latter of which is alsoknown as Kushnirenko’s theorem. After a few years, Dinaburg and Bowen gave anovel yet equivalent definition for maps on metric spaces, which is particularly suitedto establishing connections with Kolmogorov’s measure theoretic entropy.It was not until the mid s, though, that Voiculescu [24] extended the origi-nal definition to endomorphisms, or more generally to completely positive maps, ofnuclear C ∗ -algebras, thought of as the natural non-commutative counterpart of com-pact Hausdorff spaces. However, the computations involved to find the exact valueof the entropy are often rather demanding, so much so that not as many examples ofendomorphisms as one would expect are known whose entropy has been computed.Of course, part of the difficulty also depends on the choice of the C ∗ -algebra. Nowthe Cuntz algebras O p , p ≥ , are a natural family of C ∗ -algebras to consider notleast because of their many connections with several research fields such as algebraicquantum field theory, index theory, and wavelets. The first example to be discussedwas the so-called canonical shift of O p . In [15] Choda showed that its entropy isgiven by log p , which is quite a remarkable fact as this value is nothing but theentropy of a Bernoulli shift on the alphabet { , , . . . , p } and the restriction of thecanonical shift to the diagonal subalgebra D p ⊂ O p is just such a Bernoulli shift. Soon after this result was obtained by Boca and Golstein [13] for shift-type endo-morphisms on arbitrary Cuntz-Krieger algebras by using a different technique, andmore recently by Skalski and Zacharias [23] for higher rank graph C ∗ -algebras. In[22] the last-mentioned authors provided an upper bound to the entropy of a generalclass of endomorphisms of O p that leave the UHF subalgebra F p invariant and satisfya "finite-range" condition. Furthermore, they found the exact value of the entropyfor all such endomorphisms of O associated with permutations of rank . In thispaper we aim to show that a suitable adaptation of the techniques employed in theaforementioned paper can be exploited to compute the entropy of a countable class ofendomorphisms acting on the so-called p -adic ring C ∗ -algebras Q p . These and theirgeneralizations have been of late the focus of much research [2, 3, 4, 5, 6, 8, 10, 7]and are here considered because they contain the Cuntz algebras in a natural way.Indeed, as we will see in the next section, each O p is contained in Q p . Moreover,the commutative C ∗ -algebra of continuous functions on the one-dimensional torus T appears as a maximal abelian subalgebra of each Q p . Now the endomorphisms dealtwith in our paper preserve this MASA, on which they simply act as T ∋ z z k ∈ T ,for some integer k . For this reason we will refer to them as the winding endomor-phisms . Quite interestingly, their entropy is completely determined by k . Moreprecisely, the main result of the present paper is that their non-commutative entropyis log | k | , which is exactly the classical entropy of the continuous map Φ k ( z ) = z k , z ∈ T , on the circle. This is much in the same spirit as Choda’s result on the entropyof the canonical shift we recalled above.Finally, in the last section we attack the problem of computing the Watatani index,too, of the winding endomorphisms so as to spot possible relations with the entropy,very much in line with what done in [16], where the quadratic permutation endo-morphisms of the Cuntz algebra O were studied. The technique we employ can beapplied only to values of k of the form ± ( p − i , i ∈ N , and the Watatani indexof the corresponding endomorphism turns out to be exactly | k | = ( p − i . Never-theless the index of the restriction of the winding endomorphisms to a remarkablesubalgebra of Q p , the so-called gauge invariant subalgebra Q T p , which is isomorphicwith the Bunce-Deddens algebra of type p ∞ , can be computed for all values of k and,again, is given by | k | . In particular, in all cases where the index can be computedthe entropy is the natural logarithm of the index.2. Preliminaries and notation
Let p be a natural number greater than or equal to . The p -adic ring C ∗ -algebra Q p is the universal C ∗ -algebra generated by a unitary U and an isometry S p suchthat U p S p = S p U and p − X l =0 U l S p S ∗ p U − l = 1 see also [19] for Q , and [8] for the general case. Note that U S ∗ p = S ∗ p U p , U ∗ S ∗ p = S ∗ p U − p , and P p k j =0 U j S kp ( S ∗ p ) k U − j = 1 for all k ∈ N . Furthermore, we also have N THE ENTROPY AND INDEX OF THE WINDING ENDOMORPHISMS OF Q p ( S ∗ p ) m U − i U j S mp = δ i,j , if ≤ i, j ≤ p m − . All of these equalities can also be checkedby means of the so-called canonical representation π : Q p → B ( ℓ ( Z )) defined by π ( S p ) e k := e pk and π ( U ) e k := e k +1 for all k ∈ Z , where { e k : k ∈ Z } is the canonicalbasis of ℓ ( Z ) , that is e k ( l ) := δ k,l for any k, l ∈ Z . The canonical representation ofa p -adic ring C ∗ -algebra is irreducible, see [8, Proposition 2.3], where the result isproved for a broad class of C ∗ -algebras, including all p -adic C ∗ -algebras, for whicha canonical representation is always defined. As is known, the Cuntz algebra O p isthe universal C ∗ -algebra generated by p isometries T j , j = 0 , , . . . , p − , such that P p − j =0 T j T ∗ j = 1 , [17]. We recall that O p injects into Q p through the ∗ -homomorphismthat sends T j to U j S p for j = 0 , . . . , p − . Henceforth, we will always think of O p as a subalgebra of Q p .The p -adic ring C ∗ -algebra is acted upon by T in a natural way through the so-called gauge automorphisms { α z : z ∈ T } . These are defined as α z ( S p ) := zS p and α z ( U ) := U . We denote by Q T p ⊂ Q p the subalgebra fixed by the gauge action of T , i.e. Q T p := { x ∈ Q p : α z ( x ) = x, for any z ∈ T } . By definition, it is easy to checkthat Q p is the norm closure of the linear span of monomials of the type U i S hp ( S ∗ p ) h U j , h ∈ N and i, j ∈ Z . Moreover, Q T p is known to be isomorphic with the Bunce-Deddensalgebra of type p ∞ , see [11, Remark 2.8]. Another notable subalgebra of Q p , whichwill play a key role in Section 4, is the so-called diagonal subalgebra, D p , whichis the abelian C ∗ -algebra generated by all projections of the form U i S mp ( S ∗ p ) m U − i .It turns out that D p is linearly generated by the above projections. Furthermore, D p is known to be maximal abelian [8]. The Gelfand spectrum of D p can be seento be homeomorphic with the Cantor set K , and the adjoint action of U restrictsto D p as the p -adic odometer, which throughout this paper we denote by T . Theendomorphisms of Q p we will be focused on are those that fix S p while mapping U to a power of it, say U k . Set ˜ U := U k and ˜ S p = S p . For such an endomorphismto exist, by universality it is necessary and sufficient that ˜ U and ˜ S p continue tosatisfy the defining relations. Now the relation ˜ U p ˜ S p = ˜ S p ˜ U does not cause anyrestriction on k since it is trivially satisfied. Because U p commutes with S p S ∗ p , therelation P p − l =0 ˜ U l ˜ S p ˜ S ∗ p ˜ U − l = 1 does entail a restriction on the possible values of k ,for we must have { [0] , [ k ] , [2 k ] , . . . , [( p − k ] } = Z p , where [ l ] denotes the congruenceclass of l modulo p . This condition is fulfilled if and only if k and p are coprime,namely when their greatest common divisor is , in which case we write ( k, p ) = 1 .This is a consequence of a simple result, which we single out below for the reader’sconvenience. Proposition 2.1.
Let p > be a fixed integer number. Then the group homomor-phism Ψ k defined on ( Z p , +) by Ψ k ([ n ]) := [ kn ] , for any [ n ] ∈ Z p , is surjective if andonly if ( k, p ) = 1 Thus, for any k coprime with p we can introduce the winding endomorphisms χ k : Q p → Q p given by χ k ( U ) := U k , χ k ( S p ) := S p . Except when k = ± , these are allproper endomorphisms, cf. [2, Proposition 6.1] . When p = 2 , these endomorphisms VALERIANO AIELLO AND STEFANO ROSSI were originally introduced in [2, Section 6] for Q . Note that χ k ◦ χ k = χ k k , forany pair of integers k , k coprime with p .We now recall Voiculescu’s definition of topological entropy, [24, Section 4]. Sincethe C ∗ -algebras dealt with in this paper are all unital and nuclear, we will limitourselves to recalling the definition for this class, although a more general definitioncan be given for arbitrary exact C ∗ -algebras, see [14].Given a nuclear C ∗ -algebra A and an endomorphism α : A → A , we denote byCPA ( A ) the set of triples ( φ, ψ, B ) , where B is a finite-dimensional C ∗ -algebra, φ : A → B , ψ : B → A are unital completely positive maps (u.c.p. for short). For any ǫ > and any finite subset ω ⊂ A (for brevity we write ω ∈ P f ( A ) ), we denote byCPA ( A , ω, ǫ ) the set of triples ( φ, ψ, B ) ∈ CPA( A ) such that k ( ψ ◦ φ )( a ) − a k < ǫ forall a ∈ ω . As is known, the nuclearity of A is equivalent to the existence of a triple ( φ, ψ, B ) ∈ CPA( A , ω, ǫ ) for any ω ∈ P f ( A ) and ǫ > . For a thorough accountof completely positive maps and nuclear (also known as amenable) C ∗ -algebras, werefer the reader to [20].The completely positive ǫ -rank of an endomorphism α is then defined by the followingformula rcp( ω, ǫ ) := inf { rank( B ) | ( φ, ψ, B ) ∈ CPA( A , ω, ǫ ) } where rank ( B ) denotes the dimension of a maximal abelian subalgebra of B . If weset ht( α, ω ; ǫ ) := lim sup n →∞ log rcp( ω ∪ α ( ω ) ∪ . . . ∪ α n − ( ω ); ǫ ) n ht( α, ǫ ) := sup ǫ> ht( α, ω ; ǫ ) the topological entropy of α is finally defined as ht( α ) := sup ω ∈P f ( A ) ht( α, ω ) One way to obtain a lower bound for the topological entropy is to consider a com-mutative C ∗ -algebra C of A that is invariant under α . Then, it holds ht( α ) ≥ ht( α ↾ C ) = h top ( T ) where T is the map induced by α ↾ C on the level of the spectrum of C , [24]. Sometimesa lower bound thus obtained is just the exact value of the entropy. However, in[21] examples are given of automorphisms on non-commutative C ∗ -algebras whoseentropy is in fact bigger than the supremum of the the lower bounds provided byconsidering the restriction to all classical subsystems. Another fundamental tool isthe so-called Kolmogorov-Sinai property which says that if ( ω i ) i ∈ I is a family of finitesubsets of A such that the linear span of ∪ i ∈ I,n ∈ N α n ( ω i ) is dense in A , then ht( α ) = sup ǫ> ,i ∈ I lim sup n →∞ (cid:18) n log rcp( α n ( ω i ) , ǫ ) (cid:19) N THE ENTROPY AND INDEX OF THE WINDING ENDOMORPHISMS OF Q p Main result
Theorem 3.1.
For any k coprime with p , the entropy of the winding endomorphimsis given by ht( χ k ) = log | k | . The proof requires some technical preliminary results, which are given below.First, introduce a countable family of finite sets whose linear span coincides with thewhole p -adic ring C ∗ -algebra.For any l, m, n ∈ N , the set A l,m,n is by definition the set of all monomials of theform U i S mp ( S ∗ p ) n U j , where | i | , | j | ≤ l , and one of the following three conditions holds(1) p m > p n > | j | .(2) | i | < p m < p n .(3) p m = p n > | j | .Finally, B l,m,n is the the vector space generated by A l,m,n . Remark 3.1.
The sets A l,m,n are mapped to A lk,m,n by the winding endomorphism χ k ∈ Aut( Q p ) . Indeed, we have χ k ( U i S mp ( S ∗ p ) n U j ) = U ik S mp ( S ∗ p ) n U kj . This meansthat the vector spaces B l,m,n , too, are mapped to B kl,m,n by χ k . Lemma 3.1.
The set ∪ ∞ l,m,n =0 A l,m,n linearly generates a dense subspace of Q p .Proof. The monomials { U i S mp ( S ∗ p ) n U j | i, j ∈ Z , m, n ∈ N } generate a dense sub-space of Q p (see [8, Section 2] and the references therein). The fact that we onlyneed to consider the three aforementioned cases is explained below.If p m ≥ p n ≤ | j | , then j = p n a + b (with | b | < p n ) and U i S mp ( S ∗ p ) n U j = U i S mp ( S ∗ p ) n U p n a + b = U i S mp U a ( S ∗ p ) n U b = U i + p m a S mp ( S ∗ p ) n U b If | i | ≥ p m < p n , then i = p m a + b (with | b | < p m ) and U i S mp ( S ∗ p ) n U j = U p m a + b S mp ( S ∗ p ) n U j = U b S mp U a ( S ∗ p ) n U j = U b S mp ( S ∗ p ) n U j + p n a where we used U S ∗ p = S ∗ p U p .If p m = p n ≤ | j | , then j = p m a + b (with | b | < p m ) and U i S mp ( S ∗ p ) m U j = U i S mp ( S ∗ p ) m U p m a + b = U i U p m a S mp ( S ∗ p ) m U b where we used U p m S mp ( S ∗ p ) m = S mp ( S ∗ p ) m U p m . (cid:3) In the sequel we will repeatedly make use of the natural identification between M n ( C ) ⊗ Q p and M n ( Q p ) .In the following lemma we single out an isomorphism between the p -adic ring C ∗ -algebra Q p and its tensor product with p h × p h matrices. This will be useful in someof the subsequent computations. Lemma 3.2.
For any p ≥ and for any h ≥ , the map Ψ h : Q p → M p h ( C ) ⊗ Q p given by Ψ h ( x ) := P p h − i,j =0 e i,j ⊗ ( S ∗ p ) h U − i xU j S hp , x ∈ Q p , is an isomorphism. VALERIANO AIELLO AND STEFANO ROSSI
Proof.
It is enough to check that the map is multiplicative Ψ h ( x )Ψ h ( y ) = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h U − i xU j S hp p h − X m,n =0 e m,n ⊗ ( S ∗ p ) h U − m yU n S hp = p h − X i,j,m,n =0 e i,j e m,n ⊗ ( S ∗ p ) h U − i xU j S hp ( S ∗ p ) h U − m yU n S hp = p h − X i,j,m,n =0 δ j,m e i,j e m,n ⊗ ( S ∗ p ) h U − i xU j S hp ( S ∗ p ) h U − m yU n S hp = p h − X i,j,n =0 e i,n ⊗ ( S ∗ p ) h U − i xU j S hp ( S ∗ p ) h U − j yU n S hp = p h − X i,n =0 e i,n ⊗ ( S ∗ p ) h U − i x p h − X j =0 U j S hp ( S ∗ p ) h U − j yU n S hp = p h − X i,n =0 e i,n ⊗ ( S ∗ p ) h U − i xyU n S hp = Ψ h ( xy ) Injectivity follows from the simplicity of Q p .As for the surjectivity, it suffices to show that, for all x ∈ Q p and i, j , the element e i,j ⊗ x is in the image of Ψ h . Indeed, we have Ψ h ( U i S hp x ( S hp ) ∗ U − j ) = p h − X i ′ ,j ′ =0 e i ′ ,j ′ ⊗ ( S ∗ p ) h U − i ′ ( U i S hp x ( S hp ) ∗ U − j ) U j ′ S hp = e i,j ⊗ x where in last step we used that { U i S hp : i = 0 , , . . . , p h − } is a family of mutuallyorthogonal isometries, as can be checked in the canonical representation. Indeed, forany k ∈ Z we have ( S ∗ p ) h U − j U i S hp e k = ( S ∗ p ) h e kp h +( i − j ) which means ( S ∗ p ) h U − j U i S hp e k = 0 if i = j because i − j is never a multiple of p h . (cid:3) In the following lemma we point out an inequality which will come in useful in thenext.
Lemma 3.3.
Let d be an integer number and { Q i } i =1 , { R j } j =1 ⊂ M d ( C ) . If wedefine A, B ∈ M d ( Q p ) as A := Q ⊗ S m − np + Q ⊗ S m − np U + Q ⊗ S m − np U ∗ + Q ⊗ U S m − np + Q ⊗ U ∗ S m − np B := R ⊗ R ⊗ U + R ⊗ U ∗ N THE ENTROPY AND INDEX OF THE WINDING ENDOMORPHISMS OF Q p for any pair of integer numbers with m > n , we have k Q i k ≤ k A k for any i =1 , , , , and k R j k ≤ k B k for any j = 1 , , .Proof. We will only treat A , for B can be dealt with even more easily. We think of M d ( Q p ) ∼ = M d ( C ) ⊗ Q p as a concrete C ∗ -algebra acting on the tensor Hilbert space C d ⊗ ℓ ( Z ) . For any x, x ′ ∈ C d and y, y ′ ∈ ℓ ( Z ) with k x k , k x ′ k , k y k , k y ′ k ≤ wehave: k A k ≥ | ( A ( x ⊗ y ) , x ′ ⊗ y ′ ) | = | ( Q x, x ′ )( S m − np y, y ′ ) + ( Q x, x ′ )( S m − np U y, y ′ ) + ( Q x, x ′ )( S m − np U ∗ y, y ′ )+( Q x, x ′ )( U S m − np y, y ′ ) + ( Q x, x ′ )( U ∗ S m − np y, y ′ ) | There are now five cases to consider. • Choosing y = e and y ′ = S m − np e = e p m − n the inequality simply becomes k A k ≥ | ( Q x, x ′ ) | as the remaining four terms are separately zero as theproduct of two factors, the second of which vanishes by construction. Takingthe sup on x, x ′ running on the unit ball of C d the inequality in the statementis obtained. • Choosing y = e and y ′ = S m − np U e = e p ( m − n ) we now find k A k ≥ | ( Q x, x ′ ) | and the conclusion follows. • Choosing y = e and y ′ = S m − np U ∗ e = e we now find k A k ≥ | ( Q x, x ′ ) | and the conclusion follows. • Choosing y = e and y ′ = U S m − np e = e p m − n +1 we now find k A k ≥ | ( Q x, x ′ ) | and the conclusion follows. • Choosing y = e and y ′ = U ∗ S m − np e = e p ( m − n ) − we now find k A k ≥| ( Q x, x ′ ) | and the conclusion follows. (cid:3) The following lemma is one of the main ingredients in the proof of an upper boundfor the entropy of the winding endomorphisms.
Lemma 3.4.
Let h, l, m, n ∈ N , with h > max { m, n } , l < p h , and x ∈ B l,m,n . Wehave: • If m > n , then Ψ h ( x ) = P p m − n − j =1 R j ⊗ U j S m − np + R ⊗ S m − np + ˜ R ⊗ S m − np U +ˆ R ⊗ U ∗ S m − np where R j , R , ˜ R , ˆ R ∈ M p h ( C ) , with k R j k ≤ k x k , k R k ≤ k x k , k ˜ R k ≤ k x k , k ˆ R k ≤ k x k . • If m = n , then Ψ h ( x ) = R ⊗ R ⊗ U + R ⊗ U ∗ where R , R , R ∈ M p h ( C ) ,and k R i k ≤ k x k for all i . • If m < n , then Ψ h ( x ) = P p n − m − j =1 R j ⊗ ( S n − mp ) ∗ U − j + R ⊗ S n − mp + ˜ R ⊗ S m − np U + ˆ R ⊗ U ∗ S m − np where R j , R , ˜ R , ˆ R ∈ M p h ( C ) , k R j k ≤ k x k , with k R k ≤ k x k , k ˜ R k ≤ k x k , k ˆ R k ≤ k x k .Proof. Without loss of generality, we may suppose that x = U a S mp ( S ∗ p ) n U d . Westart by dealing with the first case, that is m > n . In turn we will have to settle four VALERIANO AIELLO AND STEFANO ROSSI subcases depending on the signs of a and d . N THE ENTROPY AND INDEX OF THE WINDING ENDOMORPHISMS OF Q p Suppose that a ≥ , d ≤ , | d | < p n . We have a = p m b + r (with ≤ r < p m ).Note that ≤ b < p h − m . Then we have Ψ h ( x ) = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h U − i U a S mp ( S ∗ p ) n U d U j S hp = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − i U r U p m b S mp )( S ∗ p ) n U d U j S hp = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − i U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p m − X i =0 p h − m − X i =0 e i + i p m ,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − ( i + i p m ) U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p m − X i =0 p h − m − X i =0 e i + i p m ,j ⊗ ( S ∗ p ) h − m U − i (( S ∗ p ) m U − i U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p h − m − X i =0 e r + i p m ,j ⊗ ( S ∗ p ) h − m U − i + b ( S ∗ p ) n U d U j S hp = p h − X j =0 p h − m − X i =0 e r + i p m ,j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j S np ) S h − np = p n − X j =0 p h − n − X j =0 p h − m − X i =0 e r + i p m ,j + p n j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j + p n j S np ) S h − np = p n − X j =0 p h − n − X j =0 p h − m − X i =0 e r + i p m ,j + p n j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j S np ) U j S h − np = p h − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n j ⊗ ( S ∗ p ) h − m U − i + b U j S h − np = p h − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n j ⊗ ( S ∗ p ) h − m U − i + b + j S h − mp S m − np = p h − m − X j =0 p m − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n ( j + j p h − m ) ⊗ ( S ∗ p ) h − m U − i + b U j + j p h − m S h − mp S m − np = p h − m − X j =0 p m − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n ( j + j p h − m ) ⊗ ( S ∗ p ) h − m U − i + b U j S h − mp U j S m − np Now since − p h − m +1 ≤ − i + b + j ≤ p h − m − , there is only one nontrivial multipleof p h − m among the values taken by − i + b + j , namely p h − m itself. Therefore, thelast expression we had can be rewritten as p m − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n ( i − b + j p h − m ) ⊗ U j S m − np ++ p m − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n ( i − b + p h − m + j p h − m ) ⊗ U j +1 S m − np = p m − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n ( i − b + j p h − m ) ⊗ U j S m − np + p m − n X j =1 p h − m − X i =0 e r + i p m , − d + p n ( i − b + p h − m + j p h − m − p h − m ) ⊗ U j S m − np = p m − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n ( i − b + j p h − m ) ⊗ U j S m − np + p m − n − X j =1 p h − m − X i =0 e r + i p m , − d + p n ( i − b + p h − m + j p h − m − p h − m ) ⊗ U j S m − np + p h − m − X i =0 e r + i p m , − d + p n ( i − b + p h − m + p m − n p h − m − p h − m ) ⊗ U p m − n S m − np and in the last sum of the above expression we easily recognize a term of the form ˜ R ⊗ S m − np U .Suppose that a ≤ , d ≤ , | d | < p n . We have a = p m b + r (with ≤ r < p m ).Note that ≤ − b ≤ p h − m . Then we have Ψ h ( x ) = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h U − i U a S mp ( S ∗ p ) n U d U j S hp = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h U − i U r S mp U b ( S ∗ p ) n U d U j S hp N THE ENTROPY AND INDEX OF THE WINDING ENDOMORPHISMS OF Q p = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − i U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p m − X i =0 p h − m − X i =0 e i + i p m ,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − ( i + i p m ) U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p m − X i =0 p h − m − X i =0 e i + i p m ,j ⊗ ( S ∗ p ) h − m U − i (( S ∗ p ) m U − i U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p h − m − X i =0 e r + i p m ,j ⊗ ( S ∗ p ) h − m U − i + b ( S ∗ p ) n U d U j S hp = p h − X j =0 p h − m − X i =0 e r + i p m ,j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j S np ) S h − np = p n − X j =0 p h − n − X j =0 p h − m − X i =0 e r + i p m ,j + p n j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j + p n j S np ) S h − np = p n − X j =0 p h − n − X j =0 p h − m − X i =0 e r + i p m ,j + p n j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j S np ) U j S h − np = p h − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n j ⊗ ( S ∗ p ) h − m U − i + b + j S h − mp S m − np = p h − m − X j =0 p m − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n ( j + j p h − m ) ⊗ ( S ∗ p ) h − m U − i + b U j + j p h − m S h − mp S m − np = p h − m − X j =0 p m − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n ( j + j p h − m ) ⊗ ( S ∗ p ) h − m U − i + b U j S h − mp U j S m − np Now since − p h − m + 1 ≤ b + j − i ≤ p h − m − , there is only one nontrivial multipleof p h − m among the values taken by − i + b + j , namely - p h − m itself. Therefore, thelast expression we had can be rewritten as p m − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n ( i − b + j p h − m ) ⊗ U j S m − np + p m − n − X j =0 p h − m − X i =0 e r + i p m , − d + p n ( i − b − p h − m + j p h − m ) ⊗ U j − S m − np Now in the second sum of the above expression the summand corresponding to j = 0 accounts for the presence of a term of the type ˆ R ⊗ U ∗ S m − np , as in the statement.Now we assume a ≥ , d ≥ , | d | < p n . We have a = p m b + r (with ≤ r < p m ).Note that ≤ b < p h − m . Then we have Ψ h ( x ) = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h U − i U a S mp ( S ∗ p ) n U d U j S hp = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − i U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p m − X i =0 p h − m − X i =0 e i + i p m ,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − ( i + i p m ) U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p m − X i =0 p h − m − X i =0 e i + i p m ,j ⊗ ( S ∗ p ) h − m U − i (( S ∗ p ) m U − i U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p h − m − X i =0 e r + i p m ,j ⊗ ( S ∗ p ) h − m U − i + b ( S ∗ p ) n U d U j S hp = p h − X j =0 p h − m − X i =0 e r + i p m ,j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j S np ) S h − np = p n − X j =0 p h − n − X j =0 p h − m − X i =0 e r + i p m ,j + p n j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j + p n j S np ) S h − np = p n − X j =0 p h − n − X j =0 p h − m − X i =0 e r + i p m ,j + p n j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j S np ) U j S h − np = p h − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n j ⊗ ( S ∗ p ) h − m U − i + b U j +1 S h − np = p h − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n j ⊗ ( S ∗ p ) h − m U − i + b + j +1 S h − mp S m − np = p h − m − X j =0 p m − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n ( j + j p h − m ) ⊗ ( S ∗ p ) h − m U − i + b +1 U j + j p h − m S h − mp S m − np N THE ENTROPY AND INDEX OF THE WINDING ENDOMORPHISMS OF Q p = p h − m − X j =0 p m − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n ( j + j p h − m ) ⊗ ( S ∗ p ) h − m U − i + b +1 U j S h − mp U j S m − np Now since − p h − m + 2 ≤ − i + b + 1 + j ≤ p h − m − , there is only one nontriv-ial multiple of p h − m among the values taken by − i + b + j , namely p h − m itself.Therefore, the last expression we had can be rewritten as p m − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n ( i − b − j p h − m ) ⊗ U j S m − np + p m − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n ( i − b − p h − m + j p h − m ) ⊗ U j +1 S m − np Finally, we discuss the forth subcase: a ≤ , d ≥ , d < p n . We have a = p m b + r (with ≤ r < p m ). Note that ≤ − b ≤ p h − m . We have Ψ h ( x ) = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h U − i U a S mp ( S ∗ p ) n U d U j S hp = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − i U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p m − X i =0 p h − m − X i =0 e i + i p m ,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − ( i + i p m ) U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p m − X i =0 p h − m − X i =0 e i + i p m ,j ⊗ ( S ∗ p ) h − m U − i (( S ∗ p ) m U − i U r S mp ) U b ( S ∗ p ) n U d U j S hp = p h − X j =0 p h − m − X i =0 e r + i p m ,j ⊗ ( S ∗ p ) h − m U − i + b ( S ∗ p ) n U d U j S hp = p h − X j =0 p h − m − X i =0 e r + i p m ,j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j S np ) S h − np = p n − X j =0 p h − n − X j =0 p h − m − X i =0 e r + i p m ,j + p n j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j + p n j S np ) S h − np = p n − X j =0 p h − n − X j =0 p h − m − X i =0 e r + i p m ,j + p n j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) n U d U j S np ) U j S h − np = p h − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n j ⊗ ( S ∗ p ) h − m U − i + b +1 U j S h − np = p h − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n j ⊗ ( S ∗ p ) h − m U − i + b + j +1 S h − mp S m − np = p h − m − X j =0 p m − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n ( j + j p h − m ) ⊗ ( S ∗ p ) h − m U − i + b +1 U j + j p h − m S h − mp S m − np = p h − m − X j =0 p m − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n ( j + j p h − m ) ⊗ ( S ∗ p ) h − m U − i + b +1 U j S h − mp U j S m − np Now since − p h − m + 2 ≤ − i + b + 1 + j ≤ p h − m , there are two nontrivial multiplesof p h − m among the values taken by − i + b + j , namely ± p h − m itself. Therefore,the last expression we had can be rewritten as p m − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n ( i − b − j p h − m ) ⊗ U j S m − np + p m − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n ( i − b − − p h − m + j p h − m ) ⊗ U j − S m − np + p m − n − X j =0 p h − m − X i =0 e r + i p m ,p n − d + p n ( i − b − p h − m + j p h − m ) ⊗ U j +1 S m − np We now move on to treat the second case, namely when m = n . There is noloss of generality if we also suppose | a | , | d | < p h and | d | < p m . As in the first case,there are again four subcases to consider depending on the signs of a and d . As theyare very similar to one another, we treat in full one of these only: the case when a ≥ and d ≤ . We observe that a = p m b + r (with ≤ r < p m ). Note also that ≤ b < p h − m . Then we have Ψ h ( x ) = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h U − i U a S mp ( S ∗ p ) m U d U j S hp = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h U − i U r S mp U b ( S ∗ p ) m U d U j S hp = p h − X i,j =0 e i,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − i U r S mp ) U b ( S ∗ p ) m U d U j S hp N THE ENTROPY AND INDEX OF THE WINDING ENDOMORPHISMS OF Q p = p h − X j =0 p m − X i =0 p h − m − X i =0 e i + i p m ,j ⊗ ( S ∗ p ) h − m (( S ∗ p ) m U − ( i + i p m ) U r S mp ) U b ( S ∗ p ) m U d U j S hp = p h − X j =0 p m − X i =0 p h − m − X i =0 e i + i p m ,j ⊗ ( S ∗ p ) h − m U − i (( S ∗ p ) m U − i U r S mp ) U b ( S ∗ p ) m U d U j S hp = p h − X j =0 p h − m − X i =0 e r + i p m ,j ⊗ ( S ∗ p ) h − m U − i + b ( S ∗ p ) m U d U j S hp = p h − X j =0 p h − m − X i =0 e r + i p m ,j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) m U d U j S mp ) S h − mp = p m − X j =0 p h − m − X j =0 p h − m − X i =0 e r + i p m ,j + p m j ⊗ ( S ∗ p ) h − m U − i + b (( S ∗ p ) m U d U j + p m j S mp ) S h − mp = p h − m − X j =0 p h − m − X i =0 e r + i p m , − d + p m j ⊗ ( S ∗ p ) h − m U − i + b + j S h − mp Since − p h − m + 1 ≤ − i + b + j ≤ p h − m − , p h − m is the only nontrivial multipleamong the possible values of − i + b + j , which means the above expression rewritesas p h − m − X j =0 e r +( j + b ) p m , − d + p m j ⊗ p h − m − X j =0 e r +( j + b + p h − m ) p m , − d + p m j ⊗ U Finally, the case m < n requires no work, for it is easily reconducted to the firstthanks to the the fact that Ψ h is a ∗ -homomorphism.The inequalities involving the norms follow from the formulas arrived at above.Indeed, set A := R ⊗ S m − np + ˜ R ⊗ S m − np U + ˆ R ⊗ U ∗ S m − np . When m > n , we have k Ψ h ( x ) k = k p m − n − X j =1 R j ⊗ U j S m − np + A ∗ p m − n − X j =1 R j ⊗ U j S m − np + A k = k p m − n − X j =1 R ∗ j R j ⊗ A ∗ A k It follows that k x k = k Ψ h ( x ) k ≥ k R ∗ j R j k = k R j k ∀ j ∈ { , . . . , p m − n − }k x k ≥ k A ∗ A k = k A k From Lemma 3.3 we get k x k ≥ max {k R k , k ˜ R k , k ˆ R k} . The cases m < n and m = n are quite analogous. (cid:3) Going back to the computations of the entropy of our winding endomorphisms,the first thing we do is to provide a lower bound for it. As in [22], this can bedone by looking at the restriction of χ k to a suitable MASA of Q p . In our case, theconvenient MASA to consider is obviously C ∗ ( U ) (cf. [2, 8]). Lemma 3.5.
For any integer k coprime with p , one has ht( χ k ) ≥ log | k | Proof.
The claim follows by monotonicity ht( χ k ) ≥ ht( χ k ) ↾ C ∗ ( U ) = h top ( T k ) = log | k | . where T k ( z ) := z k , see e.g. [12]. (cid:3) We are now in a position to prove the main result of this paper.
Proof of Theorem 3.1.
Thanks to Lemma 3.5, all we have to do is show that theentropy of χ k is less than or equal to log | k | . For any l ∈ N , we set ω l := ∪ lq,r,s =0 A q,r,s .For n ∈ N we denote by ω ( n ) l the union ∪ nj =0 χ jk ( ω l ) . Fix δ > . Since Q p is nuclear,there exists ( φ , ψ , M C l ( C )) ∈ CPA( Q p , ω l , δ · p l ) . We want to find an m such thatthe exponents of U appearing in the elements in Ψ m ( ω ( n ) l ) ⊂ M p m ( Q p ) are smallerthan p m . This is certainly the case provided that | k | n l < p m , as follows from astraightforward application of Remark 3.1. The inequality can also be rewritten as n log p | k | + log p ( l ) < m , which is more suited to our purposes. For instance, we cansimply choose m = [ n log p | k | + log p ( l )] + 1 , where [ · ] denotes the integer part of areal number.Again, by nuclearity of Q p there exists a d ∈ N and u.c.p. map γ : Ψ m ( Q p ) = M p m ( Q p ) → M d ( C ) and η : M d ( C ) → Q p such that for all a ∈ ω ( n ) l the inequality k η ◦ γ (Ψ m ( a )) − a k < δ/ holds.Set ψ := (id ⊗ ψ ) ◦ Ψ m and φ := η ◦ γ ◦ (id ⊗ φ ) . Now for any x ∈ ω l and h ∈ N with h ≤ n , we have k φ ◦ ψ ( χ hk ( x )) − χ hk ( x ) k == k η ◦ γ ◦ (id ⊗ φ ◦ ψ ) ◦ Ψ m ( χ hk ( x )) − ( χ hk ( x ) k = k η ◦ γ ◦ (id ⊗ φ ◦ ψ ) ◦ Ψ m ( χ hk ( x )) − η ◦ γ ◦ Ψ m ( χ hk ( x )) k + k η ◦ γ ◦ Ψ m ( χ hk ( x )) − ( χ hk ( x ) k≤ k η ◦ γ ◦ (id ⊗ φ ◦ ψ ) ◦ Ψ m ( χ hk ( x )) − η ◦ γ ◦ Ψ m ( χ hk ( x )) k + δ ≤ k (id ⊗ ( φ ◦ ψ )) ◦ Ψ m ( χ hk ( x )) − Ψ m ( χ hk ( x ) k + δ (1)By Remark 3.1 χ hk ( x ) is in B q,r,s where q ≤ | k | h l (which is smaller than p m byconstruction), and r, s ≤ l . As of now we will also assume r > s since the case r ≤ s can be handled by means of similar computations. By Inequality (1) and Lemma 3.4 N THE ENTROPY AND INDEX OF THE WINDING ENDOMORPHISMS OF Q p we get k φ ◦ ψ ( χ hk ( x )) − χ hk ( x ) k < k p r − s − X j =1 R j ⊗ ( φ ◦ ψ )( U j S r − sp )++ R ⊗ ( φ ◦ ψ )( S r − sp ) + ˜ R ⊗ ( φ ◦ ψ )( S r − sp U ) + ˆ R ⊗ ( φ ◦ ψ )( U ∗ S r − sp ) − p r − s − X j =1 R j ⊗ U j S r − sp − R ⊗ S r − sp − ˜ R ⊗ S r − sp U − ˆ R ⊗ U ∗ S r − sp k + δ k p r − s − X j =1 ( R j ⊗ ( φ ◦ ψ )( U j S r − sp ) − U j S r − sp ) + R ⊗ (( φ ◦ ψ )( S r − sp ) − S r − sp )+ ˜ R ⊗ (( φ ◦ ψ )( S r − sp U ) − S r − sp U )++ ˆ R ⊗ (( φ ◦ ψ )( U ∗ S r − sp ) − U ∗ S r − sp ) k + δ ≤ p r − s − X j =1 k ( R j ⊗ ( φ ◦ ψ )( U j S r − sp ) − U j S r − sp ) k + k R ⊗ (( φ ◦ ψ )( S r − sp ) − S r − sp ) + ˜ R ⊗ (( φ ◦ ψ )( S r − sp U ) − S r − sp U )++ ˆ R ⊗ (( φ ◦ ψ )( S r − sp U ) − S r − sp U ) k + δ ≤ δ · p l p r − s − X j =1 + δ · p l δ ≤ δ · p l ( p r − s −
1) + 3 δ · p l + δ ≤ δ
16 + 3 δ
32 + δ δ < δ where we used that k R j i k ≤ for all i , k R k ≤ , k ˜ R k ≤ , k ˆ R k ≤ . The inequalityproved above shows that the triple ( φ, ψ, M p m ( C ) ⊗ M C l ( C )) is in CPA ( Q p , ω ( n ) l , δ ) .Therefore, rcp ( ω ( n ) l , δ ) must be less than or equal to C l p m . Accordingly, the naturallogarithm of the former quantity can be bounded in the following way: log rcp( ω ( n ) l , δ ) ≤ log( C l ) + m log( p ) = log( C l ) + log( p )( (cid:2) n log p | k | + log p ( l ) (cid:3) + 1)= log( C l ) + log( p ) (cid:18) log | k | log( p ) n + log p ( l ) (cid:19) + 2 log( p ) ≤ log( C l ) + n log | k | + log( l ) + 2 log( p ) , from which we find lim sup n →∞ (cid:16) n log rcp( ω ( n ) l , δ ) (cid:17) ≤ log | k | .The thesis is thus arrived at thanks to the Kolmogorov-Sinai property of non-commutative entropy. (cid:3) It is worth noting that all winding endomorphisms leave the Bunce-Deddens alge-bra Q T p invariant, which means one can also compute the entropy of the restrictionof the winding endomorphisms to this subalgebra. It turns out that the index of therestriction does not decrease. Indeed, we have the following Corollary 3.1.
For any integer k coprime with p , one has ht( χ k ↾ Q T p ) = log | k | Proof.
The claim follows directly from the monotonicity of the entropy: log | k | = ht( χ k ↾ C ∗ ( U ) ) ≤ ht( χ k ↾ Q T p ) ≤ ht( χ k ) = log | k | . (cid:3) On the Watatani index of the winding endomorphisms
Motivated by the work done in [16] on quadratic permutation endomorphisms ofthe Cuntz algebra O , in this section we undertake a study of the relation betweenthe entropy and index of the restriction to the Bunce-Deddens subalgebras of ourwinding endomorphisms.We are going to show that, as well as the entropy, the Watatani index (see [25] forthe definition and the main properties) of the restriction of χ k to the Bunce-Deddensalgebra Q T p can also be computed exactly. More precisely, for any integer k coprimewith p , the value of the index turns out to be | k | . Rather interestingly, the entropyof χ k ↾ Q T p is then given by the natural logarithm of the index of χ k ↾ Q T p .For the reader’s convenience we recall some basic definitions that we will makeuse of. We start with an inclusion of unital C ∗ -algebras A ⊂ B with a common unit I such that there exists a faithful conditional expectation E : B → A . Definition 4.1.
A finite family { u , u , . . . , u n } ⊂ B is said to be a quasi-basis for E if for any x ∈ B one has x = n X i =1 u i E ( u ∗ i x ) = n X i =1 E ( xu i ) u ∗ i Now our conditional expectation E : B → A has finite index if there exists aquasi-basis for E . The index is then defined as Ind( E ) := P ni =1 u i u ∗ i ∈ B . Despiteits definition, the index does not depend on the chosen quasi-basis. Moreover, theelement Ind( E ) sits in the center of B . In particular, if B has trivial center, then Ind( E ) is a positive real number greater than or equal to . The Watatani index ofthe inclusion A ⊂ B is then defined as the infimum of the set of the indices obtainedas above corresponding to any conditional expectation of finite index. It turns outthat this infimum is actually a minimum provided that both B and A have trivial N THE ENTROPY AND INDEX OF THE WINDING ENDOMORPHISMS OF Q p center, see [25].To accomplish our computation of the index, we will make use of the followingkey result, where the image of the winding endomorphism (restricted to Q T p ) is seento coincide with the fixed-point subalgebra of a finite-order automorphism, which wenext define. Fix z := e πi/k and let α : Q T p → Q T p be the automorphism mapping U to zU and fixing S hp ( S ∗ p ) h for all h ∈ N , see e.g. [18]. Note that α | k | = id Q T p , which allowsus to define a conditional expectation E from Q T p to ( Q T p ) α := { x ∈ Q T p : α ( x ) = x } as(2) E ( x ) := 1 | k | | k |− X l =0 α n ( x ) , x ∈ Q T p . Proposition 4.1.
The image χ k ( Q T p ) coincides with ( Q T p ) α . First of all, we need a preliminary lemma.
Lemma 4.1.
Let p, k ∈ Z be coprime integers. For any i, h ∈ N , there exist b, m ∈ Z such that i + bp h = mk .Proof. Since k and p are coprime, so are k and p h . This means that ck + dp h for some c, d ∈ Z . It follows that i = ick + idp h and so we may choose b as − id and m = ic . (cid:3) We now go back to the proof of Proposition 4.1.
Proof of Proposition 4.1.
Since χ k ( Q T p ) is generated by the monomials of the form U ki S hp ( S ∗ p ) h U kj , with i, j ∈ Z , h ∈ N , the inclusion χ k ( Q T p ) ⊂ ( Q T p ) α is clear.For the converse inclusion, we need to use the conditional expectation E : Q T p → ( Q T p ) α defined in (2). As already mentioned, the algebra Q T p is linearly generatedby the elements of the form U i S hp ( S ∗ p ) h U j , h ∈ N and i, j ∈ Z . So all we have todo is show that the images of these elements under E are in χ k ( Q T p ) . A straight-forward computation shows that E ( U i S hp ( S ∗ p ) h U j ) is zero, unless i + j is mod k ,that is j = − i + mk for some m ∈ Z . If E ( U i S hp ( S ∗ p ) h U j ) is not zero, then it isequal to U i S hp ( S ∗ p ) h U − i U km . Now U i S hp ( S ∗ p ) h U − i U km lies in χ k ( Q T p ) if and only if U i S hp ( S ∗ p ) h U − i does. Thanks to the chain of equalities U i S hp ( S ∗ p ) h U − i = U i S hp U b U − b ( S ∗ p ) h U − i = U i + bp h S hp ( S ∗ p ) h U − i − bp h which hold for any b ∈ N , our claim finally follows from Lemma 4.1. (cid:3) Remark 4.1.
The inclusion ( Q T p ) α ⊂ Q T p is an example of a non-commutative self-covering of the type considered in [9] . We are now in a position to compute the index of χ k relative to the conditionalexpectation E considered above. Theorem 4.1.
The index of χ k ( Q T p ) in Q T p relative to the conditional expectation E above is | k | .Proof. It is a direct consequence of the description of χ k ( Q T p ) as the fixed-pointalgebra under the action of the finite group Z | k | , which has order | k | . (cid:3) Our next goal is to show that the conditional expectation E we have used so faris actually unique. This will be a consequence of Corollary 1.4.3 in [25] once we haveascertained the set equality χ k ( Q T p ) ′ ∩ Q T p = C . Proposition 4.2.
For any k coprime with p , the relative commutant χ k ( Q T p ) ′ ∩ Q T p is trivial.Proof. Since D p is contained in χ k ( Q T p ) , we have χ k ( Q T p ) ′ ∩ Q T p ⊂ D p ∩ C ∗ ( U k ) ′ . Weare thus led to prove that any x ∈ D p such that xU k = U k x is actually a scalar.We will work in the canonical representation of Q p , which acts on the Hilbert space ℓ ( Z ) . Now any x ∈ D p is a diagonal operator w.r.t. the canonical basis { e i : i ∈ Z } of ℓ ( Z ) , that is xe i = x i e i , i ∈ Z , for suitable x i ∈ C . Since U k e i = e i + k , for any i ∈ Z , it is easy to see that any such x commutes with U if and only if x i = x i + hk for any i, h ∈ Z . In particular, the set { x i : i ∈ Z } is finite. In other terms, thespectrum of x is finite as well, which means its spectral projections belong to D p .Now write x = P k − i =0 x i P i , where P i is the orthogonal projection onto the subspace span { e i + hk : h, k ∈ Z } . As there can exist different values of i (in { , , , ..., k − } )giving the same x i , we rewrite the above sum as x := P λ ∈ σ ( x ) Q λ , where Q λ is thespectral projection associated with λ and Q λ := P i : x i = λ P i . But because σ ( x ) isfinite, each Q λ can be obtained via the continuous functional calculus of x , whichmeans Q λ belongs to D p for every λ ∈ σ ( x ) . As we next show, this implies that x must be a multiple of the identity. For, if it is not a scalar, then σ ( x ) containsat least two different values, say λ and µ . Now any projection in D p is a finitesum of projections of the type U m ( S p ) n ( S ∗ p ) n U − m , with m, n ∈ N , cf. [2, Lemma6.21]. In particular, Q λ ≥ U m ( S p ) n ( S ∗ p ) n U − m and Q µ ≥ U m ′ ( S p ) n ( S ∗ p ) n U − m ′ forsome n, m, m ′ ∈ N (there is no loss of generality is assuming that the power n ofthe isometry S p is the same in the two inequalities). Now take a := p n h + m and b := p n h + m ′ . Observe that by construction if e n lies in the range of Q λ , sodoes e n + k . We claim that a + lk ≡ b mod p n for some l ∈ Z . From this we find Q λ e a + lk = Q µ e a + lk = e a + lk , which is absurd since Q λ Q µ = 0 . The claim followsfrom the fact that k and p n are coprime: multiplying ks + p n t = 1 by ( b − a ) we get ( b − a ) sk + t ( b − a ) p n = b − a , that is a + ( b − a ) sk = b − t ( b − a ) p n and the claimis thus verified with l = ( b − a ) s . The proof is complete. (cid:3) Remark 4.2.
From the proof of the foregoing result we can single out the interestinginformation that all powers T k of the p -adic odometer T on the Cantor set K with k and p coprime enjoy the following property: any continuous T k -invariant functionis a constant. As an application of the previous result we also find the following.
N THE ENTROPY AND INDEX OF THE WINDING ENDOMORPHISMS OF Q p Theorem 4.2.
For any k coprime with p , the Watatani index of χ k ( Q T p ) in Q T p is | k | .Proof. Since the conditional expectation E : Q T p → χ k ( Q T p ) is unique, the index wecomputed in Theorem 4.1 is actually the Watatani index of the inclusion χ k ( Q T p ) ⊂Q T p . (cid:3) We would be inclined to believe that the index of χ k is still | k | at the level of thewhole p -adic ring C ∗ -algebras. Despite our efforts, though, we have not been able toascertain this equality in full generality. Even so, we do know the index is | k | when k = ± ( p − i , i ≥ (note that p and p − are always coprime).We start our analysis with k = p − . In this case by universality it is not difficultto see that for any z ∈ T such that z p − = 1 , β ( S p ) := S p and β ( U ) := zU defines anautomorphism of Q p . Again, β has finite order in that β p − = id Q p . In the followingwe take z as a primitive root of unity of order p − so that the corresponding β givesan automorphic action of Z p − . Let now F be the conditional expectation from Q p to Q βp given by F := p − P p − i =0 β i . We have the following result. Theorem 4.3.
Let p ≥ be a fixed integer. If k = p − , then χ k ( Q p ) = Q βp . Inparticular, the index of χ ( p − ( Q p ) relative to F is p − .Proof. The equality χ k ( Q p ) = Q βp is trivially satisfied. By the same argument as inthe proof of Theorem 4.1 it follows that the index of χ k is equal to p − k . (cid:3) Remark 4.3.
The case of a negative k , say k = − ( p − easily follows from the aboveresult as χ − k = χ − ◦ χ k and the index of χ − is because χ − is an automorphism. In order to conclude that the value of the index determined above is actually theWatatani index, we need to prove that F is the only conditional expectation of Q p onto Q βp . Proposition 4.3.
The conditional expectation F : Q p → Q βp is unique.Proof. Again, we need only prove that ( Q βp ) ′ ∩ Q p = C , which is a consequence ofthe equality C ∗ ( S p ) ′ ∩ Q p = C (see [8, Theorem, 4.6]) since β ( S p ) = S p . (cid:3) As a result, the following is now straightforward.
Theorem 4.4.
For any p ≥ , the Watatani index of χ k ( Q p ) in Q p is | k | if k = p − . Now iterating the procedure above, it is clear that there exists a conditional ex-pectation F j form χ jp − ( Q p ) onto χ j +1 p − ( Q p ) and the index of F j is still p − . Com-pounding these conditional expectations, for every i ≥ one obtains a conditionalexpectation from Q p onto χ i ( p − = χ ( p − i whose index is obviously ( p − i . Again,the relative commutant χ ( p − i ( Q p ) ′ ∩ Q p is trivial as χ ( p − i ( S p ) = S p , and so theconditional expectation is unique. Therefore, collecting everything together, thefollowing result is arrived at. Theorem 4.5.
For any k = ± ( p − i , i ∈ N , the Watatani index of χ k ( Q p ) in Q p is equal to k . Acknowledgements
V. A. acknowledges the support by the Swiss National Science foundation throughthe SNF project no. 178756 (Fibred links, L-space covers and algorithmic knottheory). The authors wish to thank Simone Del Vecchio for several useful discussionson the topic of this paper.
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Email address : [email protected] Stefano Rossi, Dipartimento di Matematica, Università degli studi Aldo Moro diBari, Via E. Orabona 4, 70125 Bari, Italy
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