Noncommutative topological entropy of endomorphisms of Cuntz algebras
aa r X i v : . [ m a t h . OA ] O c t NONCOMMUTATIVE TOPOLOGICAL ENTROPY OFENDOMORPHISMS OF CUNTZ ALGEBRAS
ADAM SKALSKI AND JOACHIM ZACHARIAS
Abstract.
Noncommutative topological entropy estimates are obtained forpolynomial gauge invariant endomorphisms of Cuntz algebras, generalisingknown results for the canonical shift endomorphisms. Exact values for theentropy are computed for a class of permutative endomorphisms related tobranching function systems introduced and studied by Bratteli, Jorgensen andKawamura.
In [Vo] D. Voiculescu defined noncommutative topological entropy for automor-phisms (or completely positive maps) of nuclear C ∗ -algebras based on completelypositive approximations which naturally extends the classical definition of topo-logical entropy for continuous maps of compact spaces. His definition has beensubsequently extended to the larger class of exact C ∗ -algebras (by Brown [Br]) andintensively studied in the past decade (we refer to the book [NS] for a comprehensivediscussion and many examples). For Cuntz algebras and its various generalisationsentropy estimates have only been obtained for canonical endomorphisms which maybe regarded as noncommutative extension of classical shift maps. In [Ch] M. Chodashowed that the noncommutative topological entropy of the canonical shift endo-morphism of the Cuntz algebra O N is equal to log N . Later in [BG] F. Boca andP. Goldstein used a different method which allowed them to compute entropy forthe shift-type endomorphisms on arbitrary Cuntz-Krieger algebras. Their tech-niques were extended in [SZ] to determine the values of noncommutative entropyand pressure for the multidimensional shifts on C ∗ -algebras associated with higher-rank graphs. In all of these results it turned out that the entropy of the canonicalshift endomorphism is the same as the entropy of the corresponding classical shift.In this paper we try to estimate entropies for more general endomorphisms ofCuntz algebras. Endomorphisms of Cuntz algebras have been studied intensivlyand have applications in subfactors, quantum field theory and other areas. Aparticularly interesting class is formed by those which leave F N , the canonicalUHF-subalgebra of O N , invariant. Besides the canonical shift this class contains therecently introduced and intensively studied permutative polynomial endomorphisms .We refer to [BJ] and [Ka] for connections with branching function systems andpermutative representations and [CS − ] and [CKS] for connections with subfactorsand Mathematical Physics. In particular [CS ] draws an interesting connectionbetween our entropy estimates and indices of endomorphisms on O .In the first part of this paper we give a general upper bound (in Theorem 2.2) forthe entropy of endomorphism which leave F N invariant and which verify a certain Permanent address of the first named author:
Department of Mathematics, University of L´od´z,ul. Banacha 22, 90-238 L´od´z, Poland.2000
Mathematics Subject Classification.
Primary 46L55, Secondary 37B40.
Key words and phrases.
Noncommutative topological entropy, Cuntz algebra, polynomialendomorphisms. finite-range’ condition (polynomial endomorphisms). Easy examples show that thisbound is sharp. We note that the same methods apply to similar endomorphismsof Cuntz-Krieger algebras, graph and even higher-rank graph C ∗ -algebras.In the second part we obtain exact values of the entropy for all polynomialendomorphisms of O coming from permutations of rank 2 (fully classified in [Ka]).The results of the entropy computations are given in the Table 1 in Section 3.Just as the canonical endomorphism they all leave the standard maximal abeliansubalgebra C of O invariant and thus correspond there to a transformations of itsspectrum, the Cantor set. But contrary to the case of the shift endomorphism itmay happen that the entropy of the endomorphism is greater than the entropy ofits restriction to C . However, for all permutative endomorphisms we consider wecan find other non-standard masas which are invariant and such that the restrictionto this masa has the same entropy as the whole endomorphism.Our results lead to the following questions about endomorphisms of Cuntz alge-bras which we have not been able to answer.(1) Are there polynomial (or more general) endomorphisms for which the non-commutative topological entropy is strictly greater than the supremum ofthe entropies over all classical (commutative) subsystems? In general thereare only few known examples of C ∗ -dynamical systems for which this mayhappen. (E.g. shifts on C ∗ -algebras of certain bitstreams have this prop-erty; c.f. Chapter 12 in [NS].)(2) Is the entropy of a polynomial (or more general gauge invariant) endomor-phism always equal to the entropy of its restriction to the UHF-algebra F N ? 1. Cuntz algebras and their endomorphisms
Definition and basics of Cuntz algebras.
Let N ∈ N . Recall ([Cu ]) thatthe Cuntz algebra O N is the unique C ∗ -algebra generated by isometries s , . . . , s N subject to the relations s ∗ i s j = δ i,j N X i =1 s i s ∗ i = 1 . O N is simple and nuclear. To describe elements in O N it is convenient to introducethe following multi-index notation. For k ∈ N the set of multi-indices of length k with values in { , . . . , N } will be denoted by J k , so that J k = { , . . . , N } k ; wewrite J = {∅} and J = ∪ k ∈ N J k (where N = N ∪ { } ). Multi-indices in J will be denoted either by capital Latin letters I, J, . . . (as in [BJ]) or little Greekletters µ, ν, . . . (as in [Cu ]), with the standard notation for concatenation: if I =( i , . . . , i k ) ∈ J k , J = ( j , . . . , j l ) ∈ J l , then IJ := ( i , . . . , i k , j , . . . , j l ) ∈ J k + l .The length of I ∈ J k is the number k denoted by | I | = k . We will often write p = | P | , l = | L | , etc.For J ∈ J k , J = ( j , . . . , j k ) we write s J = s j · · · s j k (and s ∅ := 1). It followsfrom the relations defining O N that every element in the ∗ -algebra generated by s , . . . , s N can be written as a linear combination of expressions of the form s I s ∗ J ,where I, J ∈ J . This dense ∗ -subalgebra of O N is the polynomial subalgebra P ( O N )of O N .Notice that the span of { s I s ∗ J : I, J ∈ J k } is isomorphic to M N k (also denoted F k ), where s I s ∗ J corresponds to the matrix unit e I,J . The subalgebra generated by k { s I s ∗ J : I, J ∈ J k } is a UHF-algebra of constant multiplicity N which will bedenoted by F N . There is a conditional expectation E : O N → F N which is given byintegration over the gauge action (c.f. [Cu ]). The standard masa (maximal abeliansubalgebra) is the algebra C N generated by { s I s ∗ I : I ∈ J } . It is maximal abelianin O N and F N . So we have a chain of natural inclusions C N ⊆ F N ⊆ O N Besides the standard masa C N we will consider some other masas in F N . Recall thata masa D in F N is said to be a product masa if it is of the form N ∞ i =1 C i , whereeach C i is a masa in M N . Any two product masas are approximately unitarilyequivalent but will be conjugate only in exceptional cases. Special product masas are of the type C = N ∞ i =1 C i , where C i = C j for all i, j ∈ N . It is not hard tosee that each such C is isomorphic to the algebra of continuous functions on theinfinite product Q ∞ i =1 { , . . . , N } of N letters, which is in turn is homeomorphic tothe Cantor set C . Endomorphisms of O N . The canonical shift endomorphism θ : O N → O N isgiven by the formula θ ( x ) = N X i =1 s i xs ∗ i , x ∈ O N . It leaves F N and C N invariant, and also every special product masa. It is easy to seethat if C is a special product masa and if we regard x ∈ C as a function on C then θ ( x ) = x ◦ T , where T : C → C is the usual one-sided shift on C = Q ∞ i =1 { , . . . , N } .It is well-known ([Cu ]) that there is a bijective correspondence between ∗ -endomorphisms of the Cuntz algebra O N and unitaries in O N . Given a unitary u ∈ O N , us , . . . , us N verify again the Cuntz algebra relations so that s i us i for i = 1 , . . . , N extends uniquely to a unital endomorphism ρ u . Conversely, givena unital endomorphism ρ , u ρ = P Ni =1 ρ ( s i ) s ∗ i is a unitary such that ρ u ρ = ρ and u ρ u = u . Easy examples are the gauge automorphism , where u = λ
1, with λ a complex number of modulus 1, or the canonical shift endomorphism, where u = P Ni,i =1 s i s j s ∗ i s ∗ j ∈ F N .A description of the action of ρ u on higher monomials can be given as follows:for k ∈ N if we define(1.1) u k = uθ ( u ) . . . θ k − ( u ) , where θ is the shift endomorphism defined above, then ρ u ( s I ) = u k s I for all multi-indices I ∈ J k and thus ρ u ( s I s ∗ J ) = u k s I s ∗ J u ∗ l for I ∈ J k and J ∈ J l .If x ∈ F N then ρ u ( x ) = lim k →∞ u k xu ∗ k ; it is also known that ρ u preserves F N iff u ∈ F N iff ρ u commutes with all gauge automorphisms ([Cu ]). Theseendomorphisms are thus called gauge invariant and for them we also have τ ◦ E ◦ ρ u = τ ◦ E where τ the unique trace on F N . Permutative endomorphisms of O N are defined as follows. Suppose that k ∈ N and σ is a permutation of the set J k . Put u σ = X J ∈J k s σ ( J ) s ∗ J . It is easy to check that u σ is a unitary in O N more precisely it lies in F k = M N k and can be regarded as the permutation matrix corresponding to σ . Thepermutative endomorphism ρ σ corresponding to the permutation σ is defined as u σ . For instance the canonical endomorphism θ is permutative. Several otherexamples will be discussed in detail in the last section of this note. Remark hereonly that the identity, canonical shift endomorphism and the flip automorphismsexchanging the generators all belong to this class.Finally the following is evident from the above correspondence between unitariesand endomorphisms. Remark 1.1. ρ u leaves the polynomial subalgebra P ( O N ) invariant iff u ∈ P ( O N ).Such endomorphisms are thus called polynomial endomorphisms and we will onlyconsider those in this paper. Three technical Lemmas.
Define for k, l ∈ N (1.2) A k,l = { s I s ∗ J : I ∈ J k , J ∈ J l } . and(1.3) F k,l = Lin A k,l . S k,l ∈ N A k,l is total in O N and the linear span Lin (cid:16)S k,l ∈ N A k,l (cid:17) = S k,l ∈ N F k,l isjust the polynomial subalgebra P ( O N ).There is a well-known ∗ -isomorphism between O N and M N k ⊗ O N defined byΨ k ( X ) = X K,M ∈J k e K,M ⊗ s ∗ K Xs M , X ∈ O N , where e K,M denote the matrix units in M N k as before.The following Lemma is elementary, but crucial for what follows. It shows thatneither in Lemma 2 of [BG] nor in Lemma 2.2 of [SZ] was it essential that one dealtwith the shift-type transformations. Lemma 1.2.
Let k, p, l ∈ N , k ≥ max { p, l } . Suppose that X ∈ F p,l . If p > l then (1.4) Ψ k ( X ) = X J ∈J p − l T J ⊗ s J , where T J ∈ M N k , k T J k ≤ k X k for each J ∈ J p − l ; if p < l then Ψ k ( X ) = X J ∈J l − p T J ⊗ s ∗ J , where T J ∈ M N k , k T J k ≤ k X k for each J ∈ J l − p . Finally if p = l then Ψ k ( X ) = T ⊗ , where T ∈ M N k , k T k ≤ k X k .Proof. Suppose that p > l and let X = P P ∈J p ,L ∈J l γ P,L s P s ∗ L , where γ P,L ∈ C .Then Ψ k ( X ) = X K,M ∈J k X P ∈J p ,L ∈J l γ P,L e K,M ⊗ s ∗ K s P s ∗ L s M = X P ∈J p ,L ∈J l X K ′ ∈J k − p ,M ′ ∈J k − l γ P,L e P K ′ ,LM ′ ⊗ s ∗ K ′ s M ′ = X P ∈J p ,L ∈J l X K ′ ∈J k − p ,J ∈J p − l γ P,L e P K ′ ,LK ′ L ′ ⊗ s J . his shows that (1.4) holds for some T J ∈ M N k .Observe now that for each n, k ∈ N and a family of complex n × n matrices { α J : J ∈ J m } we have k X J ∈J m α J ⊗ s J k = k ( X J ∈J m α J ⊗ s J ) ∗ ( X K ∈J m α K ⊗ s K ) k = k X J,K ∈J m α ∗ J α K ⊗ s ∗ J s K k = k X J ∈J m α ∗ J α J ⊗ k = k X J ∈J m α ∗ J α J k , so in particular for any fixed K ∈ J m we have k α K k ≤ k X J ∈J m α J ⊗ s J k . Now connecting the above with the fact that Ψ k is a ∗ -homomorphism, we get foreach J ∈ J m k T J k ≤ k Ψ k ( X ) k ≤ k X k and the proof is finished. The cases p < l and p = l follow in a similar way. (cid:3) Analogous statements remain true in the context of the Cuntz-Krieger, graphand higher-rank graph C ∗ -algebras. It can be used to estimate the entropy of acompletely positive contractive map on the C ∗ -algebra of one of the types listedabove if only we have control on ‘how far’ the map sends the canonical matrix units. Lemma 1.3.
Let k ∈ N and let u ∈ F k be a unitary. Then for all m, p, l ∈ N ρ mu ( F p,l ) ⊆ F p + m ( k − ,l + m ( k − . Proof.
It suffices to show this for m = 1. Using the formula ρ u ( s P s ∗ L ) = u p s P s ∗ L u ∗ l ,where p = | P | and l = | L | (and u p and u l are as in (1.1)), this follows since if u ∈ F k then u p ∈ F p + k − and u l ∈ F l + k − . (cid:3) If u is a permutation matrix we have a stronger statement. Lemma 1.4.
Let σ be a permutation of J k , p, l, m ∈ N . Then (1.5) ρ mσ ( A p,l ) ⊆ A p + m ( k − ,l + m ( k − . Further ρ σ leaves both F N and C N invariant.Proof. Similar to Lemma 1.3. (cid:3)
If a unitary u ∈ F k is not a permutation matrix, the endomorphism ρ u need notleave C N invariant.2. An Entropy Bound for Gauge Invariant PolynomialEndomorphisms
Topological Entropy.
Let A be a unital C ∗ -algebra. We say that ( φ, ψ, C ) isan approximating triple for A if C is a finite-dimensional C ∗ -algebra and both φ : C → A , ψ : A → C are unital and completely positive (ucp). This willbe indicated by writing ( φ, ψ, C ) ∈ CP A ( A ). Whenever Ω is a finite subset of A (Ω ∈ F S ( A )) and ε > φ, ψ, C ) ∈ CP A ( A , Ω , ε ) means that( φ, ψ, C ) ∈ CP A ( A ) and for all a ∈ Ω k φ ◦ ψ ( a ) − a k < ε. uclearity of A is equivalent to the fact that for each Ω ∈ F S ( A ) and ε > φ, ψ, C ) ∈ CP A ( A , Ω , ε ). For such algebras one can definercp(Ω , ε ) = min { rank C : ( φ, ψ, C ) ∈ CP A ( A , Ω , ε ) } , where rank C denotes the dimension of a maximal abelian subalgebra of C . Let usrecall the definition of noncommutative topological entropy in nuclear unital C ∗ -algebras, due to D.Voiculescu ([Vo]). Assume that A is nuclear and γ : A → A is aucp map. For any Ω ∈ F S ( A ) and n ∈ N let(2.1) orb n (Ω) = Ω ( n ) = n [ j =0 γ j (Ω) . Then the (Voiculescu) noncommutative topological entropy is given by the formula:ht( γ ) = sup ε> , Ω ∈ F S ( A ) lim sup n →∞ (cid:18) n log rcp(Ω ( n ) , ε ) (cid:19) . As shown in [Vo] Proposition 4.8 the approximation entropy coincides with classicaltopological entropy in the commutative case (see [Wa], Chapter 7 for the definitionof the latter). Another important property to be used in the sequel is the fact thatthe entropy decreases when passing to invariant subalgebras. More precisely,
Remark 2.1.
Let A be a nuclear C ∗ -algebra, γ : A → A ucp and B ⊂ A a nuclearsubalgebra such that γ ( B ) ⊆ B . Then ht( γ | B ) ≤ ht( γ ), where ht denotes theVoiculescu entropy. Proof.
The assertion holds true if ht denotes Brown’s entropy and A , hence B , areexact ([Br]). But for unital completely positive maps on nuclear C ∗ -algebras thedefinition of the entropy given above coincides with that due to N. Brown. (cid:3) We will also use the following version of the
Kolmogorov-Sinai property : if (Ω i ) i ∈ I is a family of finite subsets of A such that S i ∈ I,n ∈ N orb n (Ω i ) is total in A thenht( γ ) = sup ε> ,i ∈ I lim sup n →∞ (cid:0) n log rcp(orb n (Ω i ) , ε ) (cid:1) . The proof is an easy mod-ification of the one of Theorem 6.2.4 of [NS]. For further details and proofs andvarious related topics we refer to [NS]. Note that O N as a unital nuclear C ∗ -algebrafalls into the class considered above. The Entropy Bound.
The main general result of this note is the following the-orem. The proof is a generalisation of that of Theorem 2.4 of [SZ] (see also [BG]),now using Lemmas 1.2 and 1.3 instead of Lemma 2.2 of that paper. For the con-venience of the reader we give a terse reproduction of the proof with the necessarychanges.
Theorem 2.2.
Let k ∈ N and u ∈ F k be a unitary. Thenht ( ρ u ) ≤ ( k −
1) log N. In particular if σ is a permutation of J k , then this estimate holds true for thecorresponding permutation endomorphism ρ σ .Proof. Put for each n ∈ N ω l = l [ p,q =1 A p,q . ix l ∈ N , δ >
0. As O N is nuclear, there exists a triple ( φ , ψ , M C l ) ∈ CP A ( O N , ω l , N l δ ).Fix further n ∈ N and let ω ( n ) l = n [ p =0 ρ pu ( ω l ) . Put m = n ( k −
1) + l . Nuclearity of O N implies that there exists d ∈ N and ucpmaps γ : Ψ m ( O N ) → M d and η : M d → O N such that for all a ∈ ω ( n ) l , k η ◦ γ ( a ) − Ψ − m ( a ) k < δ . Let µ : M N m ⊗ O N → M d be a ucp extension of γ . Consider the following diagram: O N Ψ m ( O N ) M N m ⊗ O N O N Ψ m ( O N ) M N m ⊗ O N M d M N m ⊗ M C l Ψ m ✲ ❅❅❘ id ⊗ ψ ❅❅❅❘ id ⊗ φ (cid:0)(cid:0)(cid:0)✒ ∩ ❍❍❥ γ ✏✏✏✶ η Ψ − m ✲✟✟✯ µ ❍❍❍❍❍❍❍❍❍❍❍❍❍❥ ψ ✟✟✟✟✟✟✟✟✟✟✟✟✯ φ Consider now any X ∈ ω l and let p ∈ N , p ≤ n . Then k φ ◦ ψ ( ρ pu ( X )) − ρ pu ( X ) k = k η ◦ µ ◦ (id ⊗ φ ◦ ψ ) ◦ Ψ m ( ρ pu ( X )) − (Ψ − m ◦ Ψ m )( ρ pu ( X )) k≤ k (id ⊗ φ ◦ ψ ) ◦ Ψ m ( ρ pu ( X )) − Ψ m ( ρ pu ( X )) k + δ . Lemma 1.3 implies that ρ pu ( X ) ∈ F q,r , where q, r ≤ l + ( k − p ≤ m . We canassume that for example q > r . Then Lemma 1.2 implies that k φ ◦ ψ ( ρ pu ( X )) − ρ pu ( X ) k < (cid:13)(cid:13)(cid:13) X J ∈J q − r T J ⊗ (cid:0) ( φ ◦ ψ )( s J ) − s J (cid:1)(cid:13)(cid:13)(cid:13) + δ < N m δ N m + δ δ, and we proved that(2.2) ( φ, ψ, M N m ⊗ M C l ) ∈ CP A ( O N , ω ( n ) l , δ ) . This shows that rcp( ω ( n ) l , δ ) ≤ C l N m ,log rcp( ω ( n ) l , δ ) ≤ C l + m log N = C l + (( k − n + l ) log N and finally lim sup n →∞ (cid:18) n log rcp( ω ( n ) l , δ ) (cid:19) ≤ ( k −
1) log N. The Kolmogorov-Sinai property for noncommutative entropy ends the proof. (cid:3)
The proof above remains valid for any unital completely positive map on O N which satisfies the conclusions of Lemma 1.3, and again can be suitably adaptedto the context of (higher-rank) graph algebras O Λ . As we are not aware of any nteresting and natural examples of such ucp maps for O Λ (apart from the canonicalshifts in various directions analysed in [SZ]), we decided to present the result in thecontext of specific endomorphisms of O N .If the endomorphism ρ u leaves the canonical masa C N invariant, it is also possibleto obtain, exactly as in [SZ], estimates for the noncommutative pressure ([KP]) ofany selfadjoint element of C N . Note that in this case however the estimate willnot necessarily be optimal nor will it have to coincide with the pressure of thecorresponding element computed in C N viewed as the commutative subalgebra.This can be deduced from results in the next section.It is easy to see that it is both possible to have ht( ρ u ) = 0 (for the identityendomorphism) and ht( ρ u ) = ( k −
1) log N (for the endomorphism given by ( k − ρ u need not leave the canonical masain O N invariant, we cannot always use the classical topological entropy to obtainthe estimates from below (as was done in [BG] and [SZ]).Let us stress that the examples below will show that even if ρ u leaves C N invariantit is not necessarily true that ht( ρ u ) = ht( ρ u | C N ) although we will exhibit in eachof the examples a special product masa C = ⊗ ∞ i =1 C i , where C i = C j for all i, j ∈ N such that ht( ρ u ) = ht( ρ u | C ). Notice that for the canonical shift endomorphismht( θ ) = ht( θ | C ) for each special product masa C .3. Entropy Values for certain permutative Endomorphisms of O This section is devoted to computing the entropy of all permutative endomor-phisms ρ σ : O → O , where σ is a permutation of J . These 24 endomorphismswere listed and classified up to unitary equivalence in [Ka]. Note first that our en-tropy estimate in Theorem 2.2 implies that ht( ρ σ ) ≤ log 2. Below we will computethe actual value of ht( ρ σ ) (and of ht( ρ σ | F ), ht( ρ σ | C )). It turned out that in allcases either ρ σ restricted to the standard masa or a suitable product masa givesentropy log 2 hence ht( ρ σ ) = log 2 or we could show directly that the entropy of ρ σ is 0, so that our methods are essentially commutative and do not use Voiculescu’sdefinition.The notation will coincide with that of [Ka]; we identify J = { (1 , , (1 , , (2 , , (2 , } with { , , , } . The subscripts in the symbols denoting permutations (e.g. σ (12)(34) )will then correspond to the cycle decomposition of the permutation. Thus for in-stance u (1 , is given by: u (1 , = s s ( s s ) ∗ + s s ( s s ) ∗ + s s ( s s ) ∗ + s s ( s s ) ∗ = s s s ∗ s ∗ + s s s ∗ s ∗ + s s s ∗ s ∗ + s s s ∗ s ∗ . Therefore ρ (1 , acts on the generators s , s as follows ρ (1 , ( s ) = u (1 , s = s s s ∗ + s s s ∗ and ρ (1 , ( s ) = u (1 , s = s s s ∗ + s s s ∗ = s . The following table, modelled on that of [Ka], summarises the results of the entropycomputations. (In this table s ij,k denotes s i s j s ∗ k .)Notice that all automorphisms in this table have entropy 0 (these are ρ id , ρ (12)(34) , ρ (13)(24) , ρ (14)(23) ). It is possible that permutative endomorphisms always have en-tropy 0. onti and Szyma´nski ([CS ]) have recently extended the table below by deter-mining the indices of the endomorphisms which shows that entropy and index seemto be related, although several combinations may occur. Table 1. Entropy of the ‘rank 2’ permutative endomorphisms of O . ρ σ ρ σ ( s ) ρ σ ( s ) ht( ρ σ ) ht( ρ σ | C ) ρ id s s ρ s , + s , s log 2 0 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s s , + s , log 2 0 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s , + s , log 2 log 2 ρ s s , + s , log 2 0 ρ s , + s , s , + s , log 2 log 2 ρ s , + s , s log 2 0 ρ s , + s , s , + s , log 2 log 2 ρ (12)(34) s , + s , s , + s , ρ (13)(24) s s ρ (14)(23) s , + s , s , + s , By Lemma 1.4 each ρ σ leaves the canonical masa invariant and we always have a‘commutative model’ for our endomorphism. To be more precise, the algebra C is ∗− isomorphic to the algebra C ( C ) of continuous functions on the infinite productspace C = { ( w k ) ∞ k =1 : w k ∈ { , }} (equipped with the usual metric making it a0-dimensional compact space). The standard isomorphism is given by the linearextension of the map s I s ∗ I χ Z I , where Z I denotes the set of those sequences in C which begin with the finite sequence I . Each of the endomorphisms ρ σ restricted to C corresponds therefore to a continuous map T σ : C → C via ρ σ ( x ) = x ◦ T σ , wherewe regard x ∈ C as a continuous function on C . Whilst it is well-known and easyto see that for the shift endomorphism T is simply the left shift on the sequence( w k ) ∞ k =1 we have in general only somewhat indirect information on how exactly T σ looks like for a given σ . We note the following useful observations. Lemma 3.1. (i) Suppose that ρ σ ( s I s ∗ I ) = s I s ∗ I + · · · + s I m s ∗ I m for a given I ∈ J . Then s I s ∗ I , · · · , s I m s ∗ I m are pairwise orthogonal and T σ restrictsto a bijection T σ : m [ j =1 Z I j → Z I . ii) Suppose that for a given k ∈ N and i ∈ { , } ρ σ ( X J ∈J k ,j k = i s J s ∗ J ) = n X l =1 s I l s ∗ I l . Then ( T σ ( w )) k = i if and only if w ∈ n [ l =1 Z I l . We will often also make use of the following.
Lemma 3.2.
Suppose that T : C → C is a continuous map for which the followingimplication holds true: if k ∈ N , v, w ∈ C and w | k +1] = v | k +1] then either ( T w ) | k ] =( T v ) | k ] or w | k ] = v | k ] . Then h top ( T ) ≥ log 2 .Proof. Follow the usual argument using ( n, ǫ )-separated subsets of C (see [Wa]). (cid:3) The rest of the paper is devoted to showing how to obtain the entropy values listedin Table 1.
Identity map.
Entropy 0.
Shift endomorphism.
Arises from σ . The entropy is equal to log 2 (see [BG]).The shift θ leaves the canonical masa invariant and ht( θ ) = ht( θ | F ) = ht( θ | C ). Flip transformation.
Arises from σ (13)(24) , acts as ρ σ ( s ) = s , ρ σ ( s ) = s . Theentropy is 0; this follows immediately from general results of [DS], but can be alsoeasily deduced directly. The transformation induced by σ . Denote it simply by ψ . It is defined by ψ ( s ) = s s s ∗ + s s s ∗ , ψ ( s ) = s . We have ( n, m ∈ N ) ψ ( s n ) = s s n s ∗ + s s n − s s ∗ , ψ ( s m ) = s m (the first formula can be easily shown inductively, and the second is obvious). Thisleads to ψ ( s i s j · · · s j k − s i k ) = s s i − s s j − · · · s s j k − − s ( s i k − s s ∗ + s i k s ∗ ) ,ψ ( s i s j · · · s j k − s i k s j k ) = s s i − s s j − · · · s s j k − − s s i k − s s j k − ( k ∈ N , i , . . . , i k , j , . . . , j k ∈ N ). Further if s ν = s i s j · · · s j k − s i k then ψ ( s ν s ∗ ν ) = s ν s ∗ ν + s ν s ∗ ν , where s ν = s s i − s s j − · · · s s j k − − s s i k − s ,s ν = s s i − s s j − · · · s s j k − − s s i k . This shows immediately that ψ ( s ν s ∗ ν ) = s e ν s ∗ e ν , where s e ν = s s i − s s j − · · · s s j k − − s s i k − . If we have an index ending with 2, so that s µ = s i s j · · · s j k − s i k s j k , then ψ ( s µ s ∗ µ ) = s e µ s ∗ e µ , here s e µ = s s i − s s j − · · · s s j k − − s s i k − s s j k − . Note that each occurrence of s in ψ ( s µ s ∗ µ ) is caused by a ‘change’ in the sequencerepresented by µ (if we assume that all sequences µ have s as the 0-th element).This observation implies that any sequence µ ending in 2 gives an output sequencewith an even number of 1’s (even number of changes), so that ψ | C is induced bythe transformation of C given by( T ψ ( w )) k = ( ♯ { j ≤ k : w j = 1 } is odd,2 if ♯ { j ≤ k : w j = 1 } is even.It is easy to see that h top ( T ψ ) = 0, so also ht( ψ | C ) = 0.We will see that there is another masa in O N which is left invariant by ψ andsuch that the corresponding restriction has entropy log 2. Let X = s s ∗ + s s ∗ .Then ψ ( X ) = s s s ∗ s ∗ + s s s ∗ s ∗ + s s s ∗ s ∗ + s s s ∗ s ∗ = s Xs ∗ + s Xs ∗ . Moreover if θ : O → O denotes the canonical shift endomorphism then ψ ( θ ( X )) = ψ ( s Xs ∗ + s Xs ∗ ) = ψ ( s )( s Xs ∗ + s Xs ∗ ) ψ ( s ∗ )+ s ψ ( X ) s ∗ = θ ( ψ ( X )) . We will now show that for each k ∈ N (3.1) θ k ( ψ ( X )) = ψ ( θ k ( X ))Suppose we have shown for some n ∈ N that θ n ( ψ ( X )) = ψ ( θ n ( X )). Then ψ ( θ n +1 ( X )) = X µ ∈J n +1 ψ ( s µ Xs ∗ µ )= X ν ∈J n ( ψ ( s ) ψ ( s ν Xs ∗ ν ) ψ ( s ∗ ) + s ψ ( s ν Xs ∗ ν ) s ∗ )= ψ ( s ) X ν ∈J n s ν ψ ( X ) s ∗ ν ψ ( s ) ∗ + s X ν ∈J n s ν ψ ( X ) s ∗ ν s ∗ = X µ ∈J n +1 s µ ψ ( X ) s ∗ µ = θ n +1 ( ψ ( X )) . The second last equality follows if we notice that ψ ( s ) s ν = s s e ν , where e ν equals ν but with the first letter ‘switched’.The formula (3.1) will become useful when we view the UHF algebra F as thetensor product N ∞ i =1 M ( i )2 . Define first E = 12 ( I + X ) , F = 12 ( I − X ) . It is clear that E and F are minimal projections in the first matrix factor of theUHF algebra. Thus the algebra generated by { θ n ( E ) , θ n ( F ) : n ∈ N } is a masa,further denoted by C E,F . Because of (3.1) we immediately see that also θ k ( ψ ( E )) = ψ ( θ k ( E )) , θ k ( ψ ( F )) = ψ ( θ k ( F ))for all k ∈ N . As in the tensor picture ψ ( X ) = X ⊗ X , it is easy to see that ψ ( E ) = E ⊗ E + F ⊗ F, ψ ( F ) = F ⊗ F + E ⊗ E. In conjunction with the previous statement this implies that ψ leaves C E,F invariant.The algebra C E,F is isomorphic to C ( C ). The isomorphism may be given for example y identifying E with χ Z and F with χ Z so that for example E ⊗ F ⊗ E ⊗ E ismapped to χ Z . It is easy to show that T E,F , the induced continuous map on C ,is given by the formula: ( T E,F ( w )) k = ( w k = w k +1 , w k = w k +1 . By lemma 3.2 ht( ψ C E,F ) = h top ( T E,F ) = log 2. Together with Theorem 2.2 thisimplies that ht( ψ ) = ht( ψ | F ) = log 2 . The transformation induced by σ . Let ψ denote again the endomorphisminduced by σ and let ψ ′ denote the one induced by σ . Then ψ ′ ( s ) = ψ ( s ) , ψ ′ ( s ) = ψ ′ ( s ) . Note that this implies in particular that on the masa C E,F introduced earlier theendomorphisms ψ ′ and ψ coincide. Indeed, ψ ( E ) = ψ ′ ( E ) and also ψ ′ ( θ n ( E )) = ψ ′ ( X µ ∈J n s µ Es ∗ µ ) = X µ ∈J n ψ ′ ( s µ ) ψ ′ ( E ) ψ ′ ( s ∗ µ )= X µ ∈J n ψ ( s µ ) ψ ( E ) ψ ( s ∗ µ ) = ψ ( θ n ( E )) . Thus ht( ψ ′ ) ≥ ht( ψ ′ | C E,F ) = ht( ψ | C E,F ) = log 2 and we obtainht( ψ ) = ht( ψ | F ) = log 2 . Note that as ψ ′ ( P J ∈J k ,j k =1 s J s ∗ J ) = ψ ( P J ∈J k ,j k =2 s J s ∗ J ), the restriction of ψ ′ to C is isomorphic to the map given by( T ψ ′ ( w )) k = ( ♯ { j ≤ k : w j = 1 } is odd1 if ♯ { j ≤ k : w j = 1 } is evenand thus ht( ψ ′ | C ) = 0. The transformation induced by σ (14)(23) . It is shown in [Ka] that this endo-morphism is given by the conjugation with the unitary s s ∗ + s s ∗ . Thus it leaveseach of the subspaces F p,l ( p, l ∈ N ) invariant and one can easily deduce using theKolmogorov-Sinai property that its entropy is 0. The transformation induced by σ (12)(34) . This one is the composition of theflip automorphism and ρ (14)(23) . As they both leave F p,l invariant, the entropy is0. The transformations induced by σ , σ , σ , σ , σ , σ , σ . Let σ be one of the permutations from the above list. It is easy to show inductively thatfor any k ∈ N and J ∈ J k we have(3.2) ρ σ ( s J ) = s J s ∗ + s J s ∗ , where J , J are certain multi-indices in J k . This implies, as we will show below,that in some special cases the formula for the map T ρ σ : C → C induced by therestriction of ψ to C is determined already by the value of ψ ( s s ∗ ). Let us formulateit as a lemma: emma 3.3. Suppose that the endomorphism ρ σ : O → O satisfies the condition (3.2) . If ρ σ ( s s ∗ ) = s s s ∗ s ∗ + s s s ∗ s ∗ then ( T ρ σ ( w )) k = ( if w k +1 = 2 , if w k +1 = 1 . If ρ σ ( s s ∗ ) = s s s ∗ s ∗ + s s s ∗ s ∗ then ( T ρ σ ( w )) k = ( if w k +1 = 1 , if w k +1 = 2 . Proof.
It is enough to consider the first case, the second follows in an analogousway. Suppose that J ∈ J and J , J are as in the formula (3.2). Then ρ σ ( s J s s ∗ s ∗ J ) = ( s J s ∗ + s J s ∗ )( s s s ∗ s ∗ + s s s ∗ s ∗ )( s s ∗ J + s s ∗ J )= s J s s ∗ s ∗ J + s J s s ∗ s ∗ J . This implies that ρ σ ( X J ∈J k ,j k =1 s J s ∗ J ) = X I ∈J k +1 ,i k +1 =2 s I s ∗ I and we can finish the proof using 3.1.(ii). (cid:3) The analysis of the values at s s ∗ together with Theorem 2.2, Lemma 3.3 and3.2 show that for any σ from the list σ , σ , σ , σ , σ , σ , σ we haveht( ρ σ ) = ht( ρ σ | C ) = log 2 . The transformation induced by σ . Let σ = σ . It is easy to see that actuallyin this case the formula (3.2) can be made more precise so that we obtain for any k ∈ N and J ∈ J k ρ σ ( s J ) = s J s s ∗ + s J s s ∗ , where now J , J ∈ J k − . Moreover we have ρ σ ( s s ∗ ) = s s s ∗ s ∗ + s s s ∗ s ∗ , so that ρ σ ( s J s s ∗ s ∗ J ) = ( s J s s ∗ + s J s s ∗ )( s s s ∗ s ∗ + s s s ∗ s ∗ )( s J s s ∗ + s J s s ∗ ) ∗ = s J s s s ∗ s ∗ s ∗ J + s J s s s ∗ s ∗ s ∗ J . This implies easily that ρ σ restricted to C is induced by the map( T ρ σ ( w )) k = ( w k = w k +1 , w k = w k +1 . As in the last subsection we obtainht( ρ σ ) = ht( ρ σ | C ) = log 2 . he transformation induced by σ . The endomorphism is the compositionof the inner automorphism ρ (1243) with ρ (13) . This implies that ρ σ restricted to C is induced by the map( T ρ σ ( w )) k = ( k ≥ w k = w k +1 or k = 1 and w = w , k ≥ w k = w k +1 or k = 1 and w = w , and we obtain ht( ρ σ ) = ht( ρ σ | C ) = log 2 . The transformation induced by σ . The endomorphism is given by the for-mulas ρ σ ( s ) = s s s ∗ + s s s ∗ , ρ σ ( s ) = s s s ∗ + s s s ∗ , so that ρ σ ( s s ∗ ) = s s s ∗ s ∗ + s s s ∗ s ∗ ,ρ σ ( s s ∗ ) = s s s ∗ s ∗ + s s s ∗ s ∗ . It is also easy to check that it has the property described in (3.2). Moreover thefollowing lemma holds true.
Lemma 3.4.
Let k ∈ N , J ∈ J k . Then ρ σ ( s J s s ∗ s ∗ J ) = s J s ∗ J + s J s ∗ J , where J , J are certain indices in J k +1 such that the number of constant segmentsof 1’s and 2’s in J and in J is even.Proof. The statement will be proved by induction on k . The case k = 0 followsfrom the explicit formulae before the lemma. Let now J ∈ J k and compute ρ σ ( s s J s ∗ J s ∗ ) = ( s s s ∗ + s s s ∗ )( s J s ∗ J + s J s ∗ J )( s s s ∗ + s s s ∗ ) ∗ . Suppose first that J = 1 K for some K ∈ J k . Then( s s s ∗ + s s s ∗ )( s J s ∗ J )( s s s ∗ + s s s ∗ ) ∗ = s s s K s ∗ K s ∗ s ∗ . We want to count the constant segments in the multi-index 12 K . A moment ofthought shows that it has either equally many constant segments as 1 K (if K beganwith 2) or two more (if K began with 1). Similarly if J = 2 K for some K ∈ J k − we have ( s s s ∗ + s s s ∗ )( s J s ∗ J )( s s s ∗ + s s s ∗ ) ∗ = s s s K s ∗ K s ∗ s ∗ , and again the multi-index 21 K has either equally many constant segments as 2 K (if K originally began with 1) or two more (if K originally began with 2).It remains to consider what happens when on the left J is extended by 2 insteadof 1: ρ σ ( s s J s ∗ J s ∗ ) = ( s s s ∗ + s s s ∗ )( s J s ∗ J + s J s ∗ J )( s s s ∗ + s s s ∗ ) ∗ . Suppose first that J = 1 K for some K ∈ J k . Then( s s s ∗ + s s s ∗ )( s J s ∗ J )( s s s ∗ + s s s ∗ ) ∗ = s s s K s ∗ K s ∗ s ∗ . The multindex 11 K has obviously equally many constant segments as 1 K . Ananalogous argument suffices if J = 2 K for some K ∈ J k − and the inductive proofis finished - the parity of the number of constant segments in the multi-indicesappearing in the ( k + 1)-th stage is the same as in those which appeared in the k -thstage. (cid:3) he lemma above implies that the map on C induced by ρ σ | C is given by theformula:( T ρ σ ( w )) k = ( w | k +1] is even,2 if the number of constant segments in w | k +1] is odd . This implies again that ht( ρ σ ) = ht( ρ σ | C ) = log 2 . The transformation induced by σ . We will apply a method analogous tothat used for σ . Here the endomorphism is given by the formulas ρ σ ( s ) = s s s ∗ + s s s ∗ , ρ σ ( s ) = s s s ∗ + s s s ∗ , so that ρ σ ( s s ∗ ) = s s s ∗ s ∗ + s s s ∗ s ∗ ,ρ σ ( s s ∗ ) = s s s ∗ s ∗ + s s s ∗ s ∗ . It is also easy to check that it has the property described in (3.2).The next lemma is analogous to Lemma 3.4 and its proof is omitted.
Lemma 3.5.
Let k ∈ N , J ∈ J k . Then ρ σ ( s J s s ∗ s ∗ J ) = s J s ∗ J + s J s ∗ J , where J , J are certain indices in J k +1 such that k + the number of constant seg-ments in J is even and k + the number of constant segments in J is even. It follows that( T ρ σ ( w )) k = ( k + the number of constant segments in w | k +1] is odd,2 if k + the number of constant segments in w | k +1] is even . Thus we once more obtain ht( ρ σ ) = ht( ρ σ | C ) = log 2 . The transformations induced by σ , σ , σ , σ , σ and σ . Arisefrom σ , σ , σ , σ , σ and σ respectively, just by 1 and 2 switchingplaces. The entropy values can thus be read off from the earlier computations. Remark 3.6.
As in all the cases above the maximal value of the topological entropyis attained on a commutative subalgebra, the variational principle has to hold forall ρ σ . Recall that this means that ht( ρ σ ) = sup φ h φ ( ρ σ ), where the supremum istaken over all states on O left invariant by ρ σ and h φ ( ρ σ ) denotes the dynamicalstate entropy of Connes, Narnhofer and Thirring (see [NS]). It can be easily seenthat in each case the supremum is realised by the state τ ◦ E . References [BG] F. Boca and P. Goldstein, Topological entropy for the canonical endomorphisms of Cuntz-Krieger algebras,
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