PACIFIC JOURNAL OF MATHEMATICS Vol. 190, No. 2, 1999 ENTROPY OF CUNTZ'S CANONICAL ENDOMORPHISM Marie Choda Let fs i g n i=1 be generators of the Cuntz

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1 PACIFIC JOURNAL OF MATHEMATICS Vol. 190, No. 2, 1999 ENTROPY OF CUNTZ'S CANONICAL ENDOMORPHISM Marie Choda Let fs i g n i=1 be generators of the Cuntz algebra OP n and let n be the *-endomorphism of O n dened by (x) = i=1 S ixs : i Then both of Connes-Narnhofer-Thirring's entropy h () and Voiculescu's topological entropy ht() are log n; where is the unique log n-kms state of O n : Also Longo's canonical endomorphism for N M have the same entropy log n; where the inclusion N M comes from O n. 1. Introduction. Connes-Strmer entropy H() extended the entropy invariant of Kolmogorov-Sinai to trace preserving automorphisms of nite von Neumann algebras ([CS]). Replacing a nite trace to an invariant state, Connes-Narnhofer- Thirring entropy h () is dened for automorphisms of C -algebras as a generalization of H() ([CNT]). These entropies depend on an invariant state under a given automorphism. The rst typical interesting example to compute the entropy is the Bernoulli shift n on the innite product space of n-point sets. In the context of operator algebras (von Neumann algebras or C -algebras), the non-commutative Bernoulli shift n takes the place of the the Bernoulli shift n : It is the shift automorphism on the innite tensor product A = N 1 i=?1 A i (where A i is the nn-matrix algebra) and H( n ) = log n = h ( n ) ([CS], [CNT]), where is the unique tracial state of A: Ler be an aperiodic automorphism of an algebra B: Then there exists an implimenting unitary operator u for in the crossed product M = B o Z : The inner automorphism Ad u ; (Ad u (x) = uxu ) of M is an extension of to M: In general, the entropy of is less than the entropy of Ad u : Strmer [S] asked if the equality between the entropies of and Ad u holds. Voiculescu [V] dened topological entropy ht() for automorphisms of nuclear C -algebras (cf. [Hu], [T]), which does not depend on any state but is based on approximations. As an application, he showed that his topological entropy satises the equality for the Bernoulli shift n ; so that Connes-Narnhofer-Thirring entropy does too. In this paper, we show the equality for both of the automorphism n and the unital *-endomorphism of the type of the non-commutative Bernoulli shift. 235

2 236 MARIE CHODA In x3, we denote only the fact that H(Ad u ) = h (Ad u ) = ht(ad u ) = log n; where is the unique tracial state of the reduced crossed product A o n Z : These are proved by similar method as in x4 and x5. The denition of Connes-Strmer entropy is available to trace preserving *-endomorphisms on nite von Neumann algebras. Similarly, we can apply the denition of Connes-Narnhofer-Thirring entropy to unital and state preserving *-endomorphisms of C -algebras, and also Voiculescu's topological entropy to unital *-endomorphisms of nuclear C -algebras. We apply here, in particular, to the unital *-endomorphism which is an extension of the *-endomorphism coming from the non-commutative Bernoulli shift n as follows. If we restrict our algebra A to the half N side innite C -tensor product (or 1 von Neumann tensor product) B = i=0 A i of matrix algebras, then the restriction of n to B denes a unit preserving *-endomorphism n of B; which is canonical in the sense of [Ch2, Ch3]. Then we have the extension algebra hb; n i of B by n ([Ch2, Ch3]). In the case of C algebras, hb; n i is the crossed product B o N of B by the corner endomorphism in [R, I2], which is dened by n using the canonical property of n : Further, the canonical extension ^ n (in the sense of [Ch2, Ch3]) of n to hb; n i is obtained. The *-endomorphism ^ n of hb; n i is dened by a modication of the automorphism Ad u of A o n Z and has the property like the canonical extension in the sense of [I1, HS]. In the case of C - algebras, the extension algebra hb; n i is the Cuntz algebra O n and ^ n is nothing P but Cuntz's canonical inner endomorphism of O n dened by n (x) = i=1 S ixs i ; (x 2 O n) for generators fs 1 ; : : : ; S n g of O n : In the case of von Neumann algebras, hb; n i is the unique injective type III 1=n factor and ^ n is Longo's canonical endomorphism for the subfactor of hb; n i; which appears naturally in the construction of the extension algebra hb; n i by the canonical *-endomorphism n ([Ch3]). In x4, we show that ht() = log n = ht( n ): Applying to Connes-Narnhofer-Thirring's entropy h () relative to the unique log n-kms state of O n ; we have h () = log n = h ( n ): This relation implies the same relation for Longo's canonical endomorphism. Thus the canonical extension of the non-commutative Bernoulli shift has the same entropy with the original one in the case of *-endomorphisms too. The author thanks F. Hiai for his interest in this work and encouragement during the preparation.

3 ENTROPY OF CUNTZ'S CANONICAL ENDOMORPHISM Preliminaries Let H 0 be a Hilbert space of dimension n < 1: Put H i = H 0 ; i 2 Z : For two integers i and j with i < j; we put H [i;j] = H i H i+1 H j : Let f(i) : i = 1; :::; ng be an orthonormal basis of H 0 : The emmbedding H [i;j],! H [i?1;j+1] is given by 2 H [i;j]! (1) (1) 2 H [i?1;j+1] : We denote by H i the inductive limit of fh [i;i+j] : j = 0; 1; :::g and by H the inductive limit of the incleasing sequence fh i : i = 0;?1; :::g: Given k; l 2 Z k < l; let W n [k;l] = f = ( k; : : : ; l ) : i 2 f1; : : : ; ng; (k i l)g: Let 2 W n and 2 W n : We put [k;l] [l+1;m] Further, let = ( k ; : : : ; l ; l+1 ; : : : ; m ): W n 0 = f0g; W n [0;1] = [1 k=0 W n [0;k] and W n 1 = [ 1 k=0 W n [?k;k] : The shift : i 2 Z! i + 1 induces the mapping on W1 n ; which we denote by the same notation : For 2 W n ; we put [k;l] () = ( k ) ( l ) 2 H [k;l] : Then f() : 2 W n [k;l] g is an orthonormal basis in H [k;l]: Let A 0 = B(H 0 ) and fe(i; j) : i; j = 1; :::; ng be the matrix unit of A 0 with respect to the orthonormal basis f(i) : i = 1; :::; ng: We denote the trace (1/n)Tr of A 0 by 0 : Put A i = A 0 ; (i 2 Z) and i = 0 : For two integers i < j; let A [i;j] = A i A i+1 A j : For ; 2 W n ; we put [k;l] e(; ) = e( k ; k ) e( l ; l ) 2 A [k;l] : Then fe(; ) : ; 2 W n [k;l] g is a matrix units of A [k;l]: 2.2. We apply the entropy of Connes-Narnhofer-Thirring and Voiculescu's topological entropy to both of automorphisms and unital *-endomorphisms on C -algebras. To x notations, we recall the denition of the topological entropy. Let B be a nuclear C -algebra with unity. Let CAP (B) be triples (; ; C); where C is a nite dimensional C -algebra, and : B! C and : C! B are unital completely positive maps. Let be the set of nite subsets of B: For an! 2 ; put rcp(!; ) = inffrank C : (; ; C) 2 CAP (B); k (a)? ak < ; a 2 Bg;

4 238 MARIE CHODA where rank C means the dimension of a maximal abelian self-adjoint subalgebra of C: For a unital *-endomorphism of B; put 1 ht(;! ; ) = lim N!1 N log rcp?! [ (!) [ [ N?1 (!); and ht(;!) = sup ht(;!; ): >0 Then the topological entropy ht() of is dened by ht() = sup ht(;!):!2 Assume that there exists an increasing sequence (! j ) j2n of nite subsets of B such that the linear span of [ j2n! j is dense in B: Even in the case of *-endomorphisms which are not automorphisms, by the obvious analogoues of [V, Proposition 4.3], ht() is obtained as the following form which we use later: ht() = sup j2n ht(;! j): Let be a state of B with = : The essential relation between ht() and Connes-Narnhofer-Thirring entropy h () is by [V, Proposition 4.6] h () ht(): 3. Entropy of Ad u for non-commutative Bernoulli shift. In this section, we only state results without proof. We remark that these are proved by similar methods as in x4 and x Let n(2 n < 1) be an integer. Let A i ; i (i 2 Z) be as in x2.1 and let A be the innite C -tensor product A = N i2z A i: We denote the unique tracial state of A by : The non-commutative Bernoulli shift n is the automorphism of the C -algebra A induced by the shift : i(2 Z)! i + 1: Let u be the implimenting unitary in the reduced C -crossed product A o n Z for n : Let E be the conditional expectation of A o n Z onto A with E(u j ) = 0; (j 6= 0): Then E is a tracial state of A o n Z which is invariant under Ad u : We denote by the same notation n the extension of n to the hypernite II 1 factor N i2z (A i; i ) o n Z : Theorem 3.2. Under the above notations, ht( n ) = ht(ad u ) = h E (Ad u ) = h ( n ) = log n = H( n ) = H(Ad u ):

5 ENTROPY OF CUNTZ'S CANONICAL ENDOMORPHISM Entropy of Cuntz's canonical endomorphism. In this section, we apply the denition of Connes-Narnhofer-Thirring entropy and Voiculescu's topological entropy for unital *-endomorphisms of nuclear C -algebras. All facts for automorphisms, which we need here, work for unital *-endomorphisms by the analogues of denitions and proofs in [CNT] and [V]. Let n (2 n < P 1) be an integer. Given n isometries fs i g on a Hilbert n space such that i=1 S is i = 1; Cuntz dened the Cuntz algebra O n as the C -algebra generated by fs i g i ([Cu1]). So called Cuntz's canonical endomorphism of O n is dened by (x) = nx i=1 S i xs i ; (x 2 O n): The O n has exactly one log n-kms state ([OP]). In this section we compute Voiculescu's topological entropy of and Connes-Narnhofer-Thiring's entropy h (): Applying to the factor generated by (O n ), we get the entropy of Longo's canonical endomorphism To compute the entropy of ; we recall some of the representation for the Cuntz algebra O n as a crossed product in [Cu1], (cf., [Ch2, I2, P, R]). Let A i ; i ; (i 2 Z) and e(i; j); (i; j 2 N ) be the same as in x2.1: For a j 2 Z; A j is given as the innite tensor product: Dene embeddings A j = 1O i=j A i : A j,! A j?1,! A j?2,! by x 2 A j! e j?1 (1; 1)x 2 A j?1 ; where e j?1 (i; l) is a copy of e(i; l) in A j?1 : The inductive limit of this sequence is denoted by A: Since the embedding A j,! A j?1 and the embedding H i,! H i?1 in x2.1 are compatible, we can consider A acting faithfully on H: The automorphism of A is induced by the shift : i(2 Z)! i + 1: Then the crossed product A o Z acts faithfully on the Hilbert space K = X i2z M u i H; where u is the implimenting unitary in A o Z for the automorphism of A: Let p be the unit of A 0 A o Z and put We remark u j p = w j : w = up:

6 240 MARIE CHODA Then Cuntz algebra O n is reresented as p(a o Z)p; which is the C subalgebra C (A 0 ; w) of (A o Z) generated by fa 0 ; wg: There exists a conditional expectation E of C (A 0 ; w) onto A 0 with E(w j ) = 0 for all j = 1; 2; : : : : The unique tracial state of A 0 is extended to the state of C (A 0 ; w) by = E: Then is the unique log n-kms state of C (A 0 ; w) ([OP]) Since j (p)(h) = H j ; j 2 Z; the algebra p(a o Z)p is acting faithfully X on M pk = u i H?i : i2z The restriction j A0 of to A 0 is the one sided non commutative Bernoulli shift. Cuntz's canonical inner endomorphism of O n is nothing but the extension of j A0 to the Cuntz algebra C (A 0 ; w) which maps a! (a); (a 2 A 0 ); and w! vw; where ([Cu2], cf. [Ch2]). v = nx j=1 e((j; 1); (1; j)) 2 A [0;1] ; 4.3. Let k; m 2 N : We dene K(k; m) = kx l=?k M u l H [?l;?l+m] and we denote the orthogonal projection of K onto K(k; m) by Q(k; m): The set fu j () :?k j k; 2 W n g is an orthonomal basis of [?j;?j+m] K(k; m): We denote by E((j; ); (l; )) the partial isometry in B(K(k; m)) such that E((j; ); (l; )) : u l ()! u j (); Then the set 2 W n [?j;?j+m] ; 2 W n [?l;?l+m] : E(k; m) n = E((j; ); (l; )) :?k j; l k; 2 W n [?j;?j+m] ; 2 W n [?l;?l+m] is a matrix units of B(K(k; m)): o

7 ENTROPY OF CUNTZ'S CANONICAL ENDOMORPHISM Let k; m 2 N : We dene the completely positive unital linear map by ' k;m : p(a o Z)p! B(K(k; m)) ' k;m (x) = Q(k; m)xq(k; m)j K(k;m) ; x 2 p(a o Z)p: We remark that if e(; )w j 6= 0 for in W n ; (b j); then [0;b] = (1; : : : ; 1; j ; : : : ; b ) and () 2 H [j;b] : For two integers a and b with a < b; we let n! a;b = e(; )w j : 0 j a and ; 2 W n [0;b] Let e(; )w j 2! a;b for a; b 2 N ; (a < b) and e(; )w j 6= 0: Since?l (p)() = () for u l () 2 K(k; m), we have that if k a and m b then where ' k;m (e(; )w j ) = We remark that k?j X l=?k E((j + l;?(j+l) () l ); (l;?(j+l) () l )); l = (1; : : : ; 1) 2 W n [?(j+l)+b+1;?(j+l)+m] ; l = (1; : : : ; 1) 2 W n [?l+b+1;?l+m] : (?(j+l) ()) 2 H [?l;?l+b+1] ; so that E((j + l;?(j+l) () l ); (l;?(j+l) () l )) 2 E(k; m): 4.5. We dene the linear map by k;m : B(K(k; m))! p(a o Z)p k;m(e((j; ); (l; ))) = 1 2k + 1 puj e(; )u l p; for E((j; ); (l; )) 2 E(k; m): Let T j ; (j 2 Z) be the unitary operator on K dened by Then we have T j (u i ()) = u i+j (?j ()); i 2 Z; 2 W n 1 : w? lim rx r!! i=?r T i E((j; ); (l; ))T i = u j e(; )u l o :

8 242 MARIE CHODA for any E((j; ); (l; )) 2 B(K(k; m)): Here! is a nontrivial ultralter on N : Hence k;m is a unital completely positive map from B(K(k; m)) to p(a o Z)p: Since u j p = w j ; we have k;m ' k;m (e(; )w j ) = 2k? j + 1 2k + 1 e(; )wj ; for all e(; )w j 2! a;b ; a k and b m: Theorem 4.6. Let be Cuntz's canonical inner endomorphism of O n : Then ht() = log n: Proof. Let e(; )w j 2! a;b : Then we have, for a k and b m; by x4.5 k k;m ' k;m (e(; )w j )? e(; )w j k = and we have for an i 2 N Here and i (e(; )w j ) = i (e(; )) = nx s=1 j 2k + 1 ke(; )wj k nx s=1 e( s ; s )w j e( s i (); s j )w j : s = (1; : : : ; 1; s i?1 ; 1; : : : ; 1) 2 W n [0;j+i?1] ; s = (1; : : : ; 1; s) 2 W n [0;j+i?1] s = (1; : : : ; 1; s i?1 ) 2 W n [0;i?1] ; Hence for k a and m b + i we have a 2k + 1 j = ( j+i ; : : : ; b ) 2 W n [j+i;b+i] : k k;m ' k;m ( i (e(; )w j ))? i (e(; )w j )k Therefore, we have for N 2 N rcp N[ i=0 i! a;b [ (! a;b ) : an 2k + 1 :! an rank B(K(k; N + b + 1)) 2k + 1 = (2k + 1)n N+b+1 ; where (! a;b ) = fx ; x 2! a;b g: This implies that for all integers a; b with a < b; ht ;! a;b [ (! a;b ) ; an 2k + 1 lim N!1 1 N log (2k + 1)n N+b+1 = log n:

9 ENTROPY OF CUNTZ'S CANONICAL ENDOMORPHISM 243 Increasing k; we have ht(;! a;b [ (! a;b ) ) log n; for all a; b 2 N with a < b: Put! a =! a;2a [ (! a;2a ) ; for a 2 N : Then the set f! a : a 2 N g is an increasing sequence of nite subsets of p(a o Z)p and the linear span of S f! a : a 2 N g is dense in p(a o Z)p: Hence ht() = sup a2n ht(;! a) log n: On the other hand, the restriction ja 0 of to A 0 is j A0 = n j A0 and h ( n j A0 ) = h ( n ) = log n: Since there exists a conditional expectation of O n onto A 0 and n j A0 = ; log n = h ( n j A0 ) ht(j A0 ) ht() log n by the version for unital *-endomorphisms of [V, Proposition 4.4]. Therefore, ht() = log n: Corollary 4.7. Let be the unique log n-kms state of O n : Then h () = log n: Proof. Let be the unique tracial state of A 0 and E be the conditional expectation of p(a o Z)p onto A 0 ; then = E: Hence = : This relation implies, by the endomorphism version of [V, Proppsition 4.6], Therefore h () = log n: log n = h (ja 0 ) h () ht() = log n: 5. Entropy of Longo's canonical endomorphism. In this section we apply the result in x4 to Longo's canonical endomorphism. We use the same notations as in x Let i be the tracial state of A i ; for i 2 N and let ~A = 1O i=0; (A i ; i ): The ~ A has the canonical trace N 1 i=0; i; which we denote by : The shift ja 0 is extended to the *-endomorphism of the hypernite II 1 factor ~ A: The is canonical in the sense of [Ch3]. Hence we have the extension algebra ~M = h ~ A; i; which is the injective type III1=n factor generated by ~ A and an isometry W: Then is extended to the canonical *-endomorphism? of ~ M and?(a) = (a); a 2 ~ A; and?(w ) = (vw ): The? is Longo's canonical endomorphism for the inclusion ~ N ~ M [Ch3, Theorem 6.10]. Here the subfactor ~ N is obtained naturally in the step of

10 244 MARIE CHODA constructing M: ~ The factor M ~ is the von Neumann algebra generated by (ha 0 ; j A0 i) and the C -algebra ha 0 ; j A0 i is O n : Hence? is the extension of to M: ~ Since is -preserving, as an application of 4.7 Corollary, we have the following by [CNT, Theorem VII.2]: Corollary 5.2. Let M be the von Neumann algera generated by (O n ) and let? be the extension of Cuntz's canonical endomorphism of O n to M: Then? is Longo's canonical endomorphism and h (?) = log n: References [Ch1] M. Choda, Entropy for *-endomorphisms and relative entropy for subalgebras, J. Operator Theory, 25 (1991), [Ch2], Canonical *-endomorphisms and simple C -algebras, J. Operator Theory, 33 (1995), [Ch3], Extension algebras of II 1 factors via *-endomorphisms, Int. J. Math., 5 (1994), [CNT] A. Connes, H. Narnhofer and W. Thirring, Dynamical entropy of C algebras and von Neumann algebras, Commun. Math. Phys., 112 (1987), [CS] A. Connes and E. Strmer, Entropy of II 1 von Neumann algebras, Acta Math., 134 (1975), [Cu1] J. Cuntz, Simple C -algebras generated by isometries, Commun. Math. Phys., 57 (1977), [Cu2] [HS], Regular actions of Hopf algebras on the C -algebra generated by a Hilbert space, in `Operator algebras, mathematical physics, and low dimensional topology' (Istanbul, 1991), , Res, Notes Math., 5, Wellesley, MA, (1993). U. Haagerup and E. Strmer, Equivalence of normal states on von Neumann algebras and ow of weights, Adv. Math., 83 (1990), [Hu] T. Hudetz, Topological entropy for appropriately approximated C -algebras, J. Math. Phys., 35 (1994), [I1] [I2] [L1] M. Izumi, Canonical extension on factors, in Subfactors, H. Araki et al. (eds.), World Scientic, Singapore, (1994), , Subalgebras of innite C -algebras with nite indices, II, Cuntz-Krieger algebras, preprint. R. Longo, Index of subfactors and statistics of quantum elds, I, Commun. Math. Phys., 126 (1989), [L2], Duality for Hopf algebras and for subfactors, Commun. Math. Phys., 159 (1994), [LR] [OP] R. Longo and J.E. Roberts, A theory of dimension, preprint. D. Olesen and G.K. Pedersen, Some C -dynamical systems with a single KMS state, Math. Scand., 42 (1978),

11 ENTROPY OF CUNTZ'S CANONICAL ENDOMORPHISM 245 [P] [R] W. Paschke, The crossed product of a C -algebra by an endomorphism, Proc. Amer. Math. Soc., 80 (1980), M. Rrdam, Classication of certain innite simple C -algebras, J. Funct. Anal., 131 (1995), [S] E. Strmer, Entropy in operator algebras, preprint in University of Oslo, (1994). [T] [V] K. Thomsen, Topological entropy for endomorphisms of local C -algebras, Comm. Math. Phys., 164 (1994), D. Voiculescu, Dynamical approximation entropies and topological entropy in operator algebras, Comm. Math. Phys., 170 (1995), Received July 1, 1997 and revised October 2, Osaka Kyoiku University Kashiwara , Japan address: marie@cc.osaka-kyoiku.ac.jp This paper is available via