Conjugate Pairs of Subfactors and Entropy for Automorphisms
aa r X i v : . [ m a t h . OA ] F e b Conjugate Pairs of Subfactors and Entropy forAutomorphisms
Marie ChodaNovember 1, 2018
Osaka Kyoiku University, Asahigaoka, Kashiwara 582-8582, [email protected]
Abstract
Based on the fact that, for a subfactor N of a II factor M, the firstnon-trivial Jones index is 2 and then M is decomposed as the crossedproduct of N by an outer action of Z , we study pairs { N, uN u ∗ } froma view point of entropy for two subalgebras of M with a connectionto the entropy for automorphisms, where the inclusion of II factors N ⊂ M is given as M is the crossed product of N by a finite group ofouter automorphisms and u is a unitary in M. keywords : Subfactor, entropy, conjugate.Mathematics Subject Classification 2000: 46L55; 46L35 For two von Neumann subalgebras A and B of a finite von Neumann algebra M, in the previous paper [4] we gave a modified constant h ( A | B ) of theConnes-Størmer relative entropy H ( A | B ) in [7] (cf. [18] ). The aim was tosee the entropy for unistochastic matrices from the viewpoint of the operatoralgebras and we showed among others that h ( D | uDu ∗ ) = H ( b ( u )) , where D is the algebra of the diagonal matrices in the n × n complex matrices M n ( C ) , and where H ( b ( u )) is the entropy in [21] for the unistochastic matrix b ( u )induced by a unitary u in M n ( C ) , and in general, it does not holds that H ( D | uDu ∗ ) = H ( b ( u )) (see, for example [17] ).1n this paper, we replace the type I n factor M n ( C ) to a II factor M .The above relation in [4] (cf. [15]) suggests us that if A and B are maximalabelian subalgebras of a II factor M , then h ( A | B ) is not necessarily finite.In order to discuss on a subalgebra A of M with h ( A | uAu ∗ ) < ∞ for allunitaries u in M , we pick up here a subfactor N ⊂ M with the Jones index[ M : N ] < ∞ ([11]) (cf. [8]).Let M be a type II factor, and let N be a subfactor of M such that[ M : N ] = 2 which is the simplest, nontrivial unique subfactor of M upto conjugacy. Then M is decomposed into the crossed product of N by anouter automorphism with the period 2. Based on this fact, we study the setof the values h ( N | uN u ∗ ) for the inclusion of factor-subfactor N ⊂ M, with aconnection to the inner automorphisms Adu, where M is given as the crossedproduct N ⋊ α G of a type II factor N by a finite group G with respect toan outer action α and u is a unitary in M. First, for two von Neumann subalgebras A and B of a finite von Neumannsubalgebra M , we show, in Corollary 2.2.3 below, that if E A E B = E B E A (which is called the commuting square condition in the sense of [9]) then H ( A | B ) = h ( A | B ) , where E A is the conditional expectation of M onto A. We give an extended notion H N ( Adu ) of H ( b ( u )) to the inner automor-phisms Adu in Definition 3.1.1, and we show that h ( N | uN u ∗ ) ≤ H N ( Adu )in Theorem 3.1.4.The inner conjugacy class of N is rich from the view point of the valuesof h ( N | uN u ∗ ) , that is, in the special case of G = Z , keeping the fact that h ( N | uN u ∗ ) ≤ H ( M | uN u ∗ ) = log 2 for all unitary u ∈ M in mind, we havethat { h ( N | uN u ∗ ) : u ∈ M a unitary } = [0 , log 2] in Theorem 3.2.3. In this section, we summarize, for future reference, notations, terminologiesand basic facts.Let M be a finite von Neumann algebra, and let τ be a faithful normaltracial state. For each von Neumann subalgebra A , there is a unique τ preserving conditional expectation E A : M → A. .1 Connes-Størmer relative entropy Let S be the set of all finite families ( x i ) i of positive elements in M with1 = P i x i . Let A and B be two von Neumann subalgebras of M. The relativeentropy H ( A | B ) is defined by Connes and Størmer([7]) as H ( A | B ) = sup ( x i ) ∈ S X i ( τ ηE B ( x i ) − τ ηE A ( x i )) . Here, η is the function defined by η ( t ) = − t log t, (0 < t ≤
1) and η (0) = 0 . Let φ be a normal state on M. Let Φ be the set of all finite families ( φ i ) i of positive linear functionals on M with φ = P i φ i . The relative entropy H φ ( A | B ) of A and B with respect to φ is given by Connes ([6]) as H φ ( A | B ) = sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S ( φ i | B , φ | B ))and if φ = τ then H τ ( A | B ) = H ( A | B ) . Here S ( φ, ψ ) is the relative entropyfor positive linear functionals φ and ψ on M (cf. [14, 16]). We modified in [4] the Connes-Størmer relative entropy for a pair of subal-gebras as follows :Let A and B be two von Neumann subalgebras of M. Let S be the setof all finite families ( x i ) of positive elements in M with 1 = P i x i . The conditional relative entropy h ( A | B ) of A and B corresponding H ( A | B ) isgiven as h ( A | B ) = sup ( x i ) ∈ S X i ( τ ηE B ( E A ( x i )) − τ ηE A ( x i )) . Let S ( A ) ⊂ S be the set of all finite families ( x i ) of positive elements in A with 1 = P i x i . Then it is clear that h ( A | B ) = sup ( x i ) ∈ S ( A ) X i ( τ ηE B ( x i ) − τ η ( x i )) . S ′ ( A ) ⊂ S ( A ) be the set of all finite families ( x ′ i ) with each x ′ i a scalarmultiple of a projection in A. Then h ( A | B ) = sup ( x ′ i ) ∈ S ′ ( A ) X i ( τ ηE B ( x ′ i ) − τ η ( x ′ i )) . The conditional relative entropy of A and B with respect to φ correspond-ing H φ ( A | B ) is given as h φ ( A | B ) = sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S (( φ i ◦ E A ) | B , ( φ ◦ E A ) | B ) . If we let Φ( A ) ⊂ Φ be the set of all finite families ( φ ′ i ) in Φ with each φ ′ i = φ ′ i ◦ E A , then h φ ( A | B ) = sup ( φ ′ i ) ∈ Φ( A ) X i ( S ( φ ′ i | A , φ | A ) − S ( φ ′ i | B , ( φ ◦ E A ) | B ) . We give conditions for that h φ ( A | B ) = H φ ( A | B ) in Corollary 2.2.4 be-low, and show relations for h φ ( A | B ) , H φ ( A | B ) , H φ ( A ) and h φ ( A ) in Theorem2.2.2, where h φ ( A ) is given by modifing H φ ( A ) in [6] for a von Neumannsubalgebra A of M (cf. [14]), that is h φ ( A ) = sup ( φ i ) ∈ Φ( A ) X i ( η ( φ i (1)) + S ( φ i | A , φ | A ))and H φ ( A ) = sup ( φ i ) ∈ Φ X i ( η ( φ i (1)) + S ( φ i | A , φ | A )) . Cleary, we have that 0 ≤ h φ ( A ) ≤ H φ ( A ) . In the case of φ is the trace τ,h φ ( A ) = h ( A ) , which is given as h ( A ) = sup ( x i ) ∈ S ( A ) X i ( η ( τ ( x i )) − τ η ( x i )) . We need the following lemma in order to prove Theorem 2.2.2, in whichwe show relations among H φ ( A ) , h φ ( A ) , H φ ( A | B ) and H φ ( A | B ) : Lemma 2.2.1.
Let A and B be von Neumann subalgebras of a finite vonNeumann algebra M , and let ψ, φ be positive linear functionals on M. If E A E B = E B E A , then S (( ψ ◦ E A ) | B , ( φ ◦ E A ) | B ) = S (( ψ ◦ E A ) | A ∩ B , ( φ ◦ E A ) | A ∩ B ) . roof. Relative entropy for positive linear functionals ψ, φ on a unital C ∗ -algebra C is expressed in [13] by S ( ψ, φ ) = sup n ∈ N sup x { ψ (1) log n − Z ∞ /n (cid:18) ψ ( y ( t ) ∗ y ( t )) + 1 t φ ( x ( t ) x ( t ) ∗ ) (cid:19) dtt } where x ( t ) : ( n , ∞ ) → C is a step function with finite range, and y ( t ) =1 − x ( t ) . Let x ( t ) : ( n , ∞ ) → B be a step function with finite range, then E A ( x ( t )) :( n , ∞ ) → A ∩ B is a step function with finite range because of that E A E B = E B E A . Since E A ( x ) ∗ E A ( x ) ≤ E A ( x ∗ x ) for all x ∈ M, we have that ψ (1) log n − Z ∞ /n (cid:18) ψ ◦ E A ( y ( t ) ∗ y ( t )) + 1 t φ ◦ E A ( x ( t ) x ( t ) ∗ ) (cid:19) dtt ≤ ψ (1) log n − Z ∞ /n (cid:18) ψ ( E A ( y ( t )) ∗ E A ( y ( t ))) + 1 t φ ( E A ( x ( t )) E A ( x ( t ) ∗ )) (cid:19) dtt ≤ S (( ψ ◦ E A ) | A ∩ B , ( φ ◦ E A ) | A ∩ B ) . This implies that S (( ψ ◦ E A ) | B , ( φ ◦ E A ) | B ) ≤ S (( ψ ◦ E A ) | A ∩ B , ( φ ◦ E A ) | A ∩ B ) . Since the opposite inequality is clear, we have the equality.
Theorem 2.2.2.
Let M be a finite von Neumann algebra with a normalfaithful tracial state τ. Let φ be a normal state of M, and let
A, B be vonNeumann subalgebras of M. Then (1) h φ ( A | B ) ≤ h φ ( A | C ) = h φ ( A ) . (2) Assume that E A E B = E B E A . Then h φ ( A | B ) = h φ ( A | A ∩ B ) . Hence, if A ∩ B = C , then h φ ( A | B ) = h φ ( A ) . (3) If E A E B = E B E A and if φ = φ ◦ E A , then H φ ( A | B ) = H φ ( A | A ∩ B ) . Especially, if A ∩ B = C , then H φ ( A | B ) = H φ ( A ) . (4) If B ⊂ A, then H φ ( A | B ) = h φ ( A | B ) . Especially, H φ ( A ) = h φ ( A ) . roof. (1) Let φ, ψ be normal states of M, and let ( φ i ) i ∈ Φ . Then ( φ ◦ E A ) | B and φ i (1) ( φ i ◦ E A ) | B are states of B so that S ( 1 φ i (1) ( φ i ◦ E A ) | B , ( φ ◦ E A ) | B ) ≥ S ( 1 φ i (1) ( φ i ◦ E A ) | B , ( φ ◦ E A ) | B )= 1 φ i (1) S ( φ i ◦ E A | B , ( φ ◦ E A ) | B ) − φ i (1) η ( 1 φ i (1) )= 1 φ i (1) S ( φ i ◦ E A | B , ( φ ◦ E A ) | B ) − log( φ i (1)) . Hence − S (( φ i ◦ E A ) | B , ( φ ◦ E A ) | B ) ≤ η ( φ i (1)) , and if B = C then the equality holds because φ i (1) φ i | C = φ | C . These implythat h φ ( A | B )= sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S (( φ i ◦ E A ) | B , ( φ ◦ E A ) | B )) ≤ sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) + η ( φ i (1)))= h φ ( A ) , and the equality holds if B = C . (2) Assume that E A E B = E B E A , then by lemma 2.2.1, we have that h φ ( A | B )= sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S (( φ i ◦ E A ) | B , ( φ ◦ E A ) | B ))= sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S (( φ i ◦ E A ) | A ∩ B , ( φ ◦ E A ) | A ∩ B ))= h φ ( A | A ∩ B ) . Since h φ ( A | B ) is decreasing in B, it implies that h φ ( A | B ) = h φ ( A | A ∩ B ) . A ∩ B = C
1, then h φ ( A | B ) = h φ ( A | C
1) = h φ ( A ) . (3) Assume that E A E B = E B E A and that φ ◦ E A = φ. Let ( φ i ) i ∈ Φ , then( φ i ◦ E A ) i ∈ Φ and we have that H φ ( A | B )= sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S ( φ i | B , φ | B )) ≥ sup ( φ i ) ∈ Φ X i ( S (( φ i ◦ E A ) | A , φ | A ) − S (( φ i ◦ E A ) | B , φ | B ))= sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S (( φ i ◦ E A ) | B , ( φ ◦ E A ) | B ))= sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S (( φ i ◦ E A ) | A ∩ B , ( φ ◦ E A ) | A ∩ B ))= sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S ( φ i | A ∩ B , φ | A ∩ B ))= H φ ( A | A ∩ B ) . In general, H φ ( A | A ∩ B ) ≥ H φ ( A | B ) so that H φ ( A | B ) = H φ ( A | A ∩ B ) . Especially, H ( A | B ) = H ( A | A ∩ B ) (cf. [22]), and if A ∩ B = C , then H φ ( A | A ∩ B ) = H φ ( A | C
1) = H φ ( A ) so that H φ ( A | B ) = H φ ( A ) . (4) If B ⊂ A, then H φ ( A | B )= sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S ( φ i | B , φ | B ))= sup ( φ i ) ∈ Φ X i ( S ( φ i | A , φ | A ) − S (( φ i ◦ E A ) | B , ( φ ◦ E A ) | B ))= h φ ( A | B ) . By combining with (1), we have H φ ( A ) = h φ ( A ) . orollary 2.2.3. Assume that E A E B = E B E A . Then H ( A | B ) = h ( A | B ) .Moreover, if φ = φ ◦ E A , then H φ ( A | B ) = h φ ( A | B ) . Proof.
Assume that E A E B = E B E A and that φ = φ ◦ E A . Then by usingTheorem 2.2.2 (3), (4) and (2), we have that H φ ( A | B ) = H φ ( A | A ∩ B ) = h φ ( A | A ∩ B ) = h φ ( A | B ) . Since E A is the τ -conditional expectation, τ = τ ◦ E A , hence H ( A | B ) = h ( A | B ) . Connes-Størmer defined the entropy H ( α ) for a trace preserving automor-phism α of a finite von Neumann algebra in [7]. The definition is arivable fora trace preserving *-endomorphism too.For a trace preserving *-endomorphism σ of a finite von Neumann algebra N, it was shown a relation between the entropy H ( σ ) for σ and the relativeentropy H ( N | σ ( N )) in the papers [1, 2, 3, 10, 20] (cf. [14]). The relationis, roughly speaking, that H ( σ ) = 12 H ( N | σ ( N ))under a certain condition. Such a *-endomorphism σ can be extended offtento an automorphism α of a finite von Neumann algebra M which contains N as a von Neumann subalgebra. Some examples of such endomorphisms ap-peared in a relation to Jones index theory of subfactors. In [2], we studied anice class of such a *-endomorphism σ of a type II factor N which is extend-able to an automorphism α of the big type II factor M obtained by the basicconstruction from N ⊃ σ ( N ) . We called such a σ basic *-endomorphism and8howed that H ( α ) = H ( N | σ ( N )) . Since σ ( N ) ⊂ N, we have by Theorem2.2.2 (4) that H ( N | σ ( N )) = h ( N | σ ( N )) so that H ( α ) = 12 H ( N | σ ( N )) = 12 h ( N | σ ( N )) = 12 h ( N | α ( N )) . This means that for an automorphism α of a II factor M we may be ableto choose a subfactor N ⊂ M such that the entropy for α is given from h ( N | α ( N )) . Our study in this section is motivated by these results. The above auto-morphism α arising from a *-endomorphism as is outer. Here, we discuss byreplacing the α to inner automorphisms Adu and the entropy H ( α ) to theentropy H N ( Adu ) defined below.
Let N be a type II factor with the canonical trace τ and let G be a finitegroup. Let α be an outer action of G on N, so that for all g ∈ G, g = 1 if α g ( x ) a = ax for all x ∈ N, then a = 0 . Hereafter, we let M be the crossedproduct of N by G with respect to α : M = N ⋊ α G. We identify N with the von Neumann subalgebra embedded in M, and denoteby v the unitary representation of G in M such that every v g is a unitary in M with α g ( x ) = v g xv ∗ g , ( x ∈ N, g ∈ G ) . Then every x ∈ M is written by the Fourier expansion x = X g ∈ G x g v g , ( x g ∈ N )and x g = E N ( xv ∗ g ) . A u ∈ M is a unitary if and only if X g ∈ G u hg α h ( u ∗ g ) = δ h, and X g ∈ G α − g ( u ∗ g u gh ) = δ h, , where we denote the identity of G by 1 . This imply that P g ∈ G τ ( u g u ∗ g ) = 1 , and we can put as the followings : 9 efinition 3.1.1. The entropy of the inner automorphism
Adu of M withrespect to N is given by H N ( Adu ) = X g ∈ G ητ ( u g u ∗ g ) . Comment 3.1.2.
Each x ∈ M is representated as the matrix x = ( x ( g, h )) gh indexed by the elements of G. Here x ( g, h ) ∈ N for all g, h in G, and x ( g, h ) = α − g ( E N ( xv ∗ h )) = α − g ( x h ) . The entropy H ( b ( u )) defined in [21] is written as H ( b ( u )) = 1 n X i,j η ( | u ( i, j ) | ) , when b ( u ) is the unistochastic matrix induced by a unitary u = ( b ( i, j )) ij in M n ( C ) . A matrix representation for an x in M n ( C ) is depend on thediagonal matrix algebra. In that sense, we consider the notion of H N ( Adu )corresponds to the notion of the entropy for a unistochastic matrix.
Lemma 3.1.3. (1)
If Ad u and Ad w are conjugate, then H N ( Adu ) = H N ( Adw ) . (2) If θ = Adu for some unitary u ∈ M, then H N ( θ − ) = H N ( θ ) . Proof. (1) Assume
Adu = θAdwθ − for some automorphism θ of M. Then θ ( w ) = λu for some complex number λ with | λ | = 1 and so ητ ( w g w ∗ g ) = ητ ( u g u ∗ g ) which implies that H N ( Adu ) = H N ( Adw ) . (2) Let w ∈ M be a unitary with Adw = θ − , then w = γu ∗ for some γ ∈ T . For the expression that w = P g w g v g , we have that w g = γα g ( u ∗ g − )for all g ∈ G so that H N ( θ − ) = X g ητ ( w g w ∗ g ) = X g ητ ( u g u ∗ g ) = H N ( θ ) . The h ( N | uN u ∗ ) is bounded by H N ( Adu ) as follows :
Theorem 3.1.4.
Assume that N is a type II factor, G is a finite groupand M = N ⋊ α G with respect to the outer action α. Then for each unitary u ∈ M, we have that h ( N | uN u ∗ ) ≤ H N ( Adu ) = X g ∈ G ητ ( u g u ∗ g ) ≤ log | G | , | G | is the cardinarity of G. Proof.
Let ( λ i p i ) i ∈ I ∈ S ′ ( N ) be a finite partition of the unity, that is, X i ∈ I λ i p i = 1where ( λ i ) i ∈ I are positive numbers and ( p i ) i ∈ I are projections in N. For agiven ε, choose an ǫ > | G | η ( ǫ ) < min { ε, /e } . There existmutually orthogonal projections ( q i,k ) k ⊂ N and nonnegative numbers ( α gi,k ) k which satisfy that p i = X k q i,k and 0 ≤ q i,k u g u ∗ g q i,k − α gi,k q i,k ≤ ǫq i,k . This is possible by the induction method of the spectral decompositions for( p i u g u ∗ g p i ) i ∈ I,g ∈ G , (see for example. [18, Proof of 4.3 Lemma]). In fact, letting G = { g , · · · , g m } and by the spectral decomposition for p i u g u ∗ g p i ∈ p i N p i , we have mutually orthogonal projections ( q i,k ) k ⊂ p i N p i and nonnegativenumbers ( α i,k ) k p i = X k q i,k and 0 ≤ q i,k u g u ∗ g q i,k − α i,k q i,k ≤ ǫq i,k . Next by the consideration for q i,k u g u ∗ g q i,k , we have a partition ( q i,k ,k ) k of q i,k and ( α i,k ,k ) k . Put α g j i,k = α ji,k , ··· ,k j and q i,k = q mi,k , ··· ,k m . Then thesesatisfy the desired conditions.Since η ( x + y ) ≤ η ( x ) + η ( y ) , ( x, y ∈ N ) , η is increasing on [0 , /e ] , andthe family ( q i,k ) k is mutually orthogonal, we have for ǫ with ǫ ≤ /eτ η ( X k q i,k u g u ∗ g q i,k ) ≤ τ η ( X k ( q i,k u g u ∗ g q i,k − α gi,k q i,k )) + τ η ( X k α gi,k q i,k ) ≤ η ( ǫ ) τ ( p i ) + X k η ( α gi,k ) τ ( q i,k ) . Hence, by the condition that P i,k λ i τ ( q i,k ) = 1 , the operator concavity of η X i,g λ i τ η ( X k q i,k u g u ∗ g q i,k ) ≤ X i,k,g λ i η ( α gi,k ) τ ( q i,k ) + | G | η ( ǫ )= X i,k λ i τ ( q i,k ) X g η ( α gi,k ) + | G | η ( ǫ ) ≤ X g η ( X i,k λ i τ ( q i,k ) α gi,k ) + | G | η ( ǫ ) . Remark that for all g ∈ Gτ ( u g u ∗ g ) − X i,k λ i α gi,k τ ( q i,k ) = X i,k λ i τ ( q i,k u g u ∗ g q i,k − α gi,k q i,k )and that 0 ≤ X i,k λ i τ ( q i,k u g u ∗ g q i,k − α gi,k q i,k ) ≤ ǫ. Then we have that0 ≤ τ ( u g u ∗ g ) − X i,k λ i τ ( q i,k ) α gi,k ≤ ǫ, ( g ∈ G )so that X g η ( X i,k λ i τ ( q i,k ) α gi,k ) ≤ X g ητ ( u g u ∗ g ) + | G | η ( ǫ ) . Here we used the following inequality in [14, (2.8)] | η ( s ) − η ( t ) | ≤ η ( s − t ) for 0 ≤ s − t ≤ . Remark that P g u g u ∗ g = 1 and that for all i the projections ( q i,k ) k is mutuallyorthogonal. Hence by using the following fact that τ η ( u ∗ g q i,k u g ) = τ η ( q i,k u g u ∗ g q i,k ) ,
12e have that X i,k ( τ ηE N ( u ∗ λ i q i,k u ) − τ η ( λ i q i,k ))= X i,k ( τ η ( X g ∈ G α g ( u ∗ g λ i q i,k u g ) − η ( λ i ) τ ( q i,k ))) ≤ X i,k X g ( τ η ( u ∗ g λ i q i,k u g ) − η ( λ i ) τ ( q i,k ))= X i,k,g ( η ( λ i ) τ ( u ∗ g q i,k u g ) + λ i τ η ( u ∗ g q i,k u g ) − X i,k η ( λ i ) τ ( q i,k )= X i ( η ( λ i ) τ ( p i X g u g u ∗ g ) + X i,k,g λ i τ η ( u ∗ g q i,k u g ) − X i η ( λ i ) τ ( p i )= X i,k,g λ i τ η ( q i,k u g u ∗ g q i,k )= X i,g λ i τ η ( X k q i,k u g u ∗ g q i,k ) ≤ X g ητ ( u g u ∗ g ) + 2 | G | η ( ǫ ) . Thus h ( N | uN u ∗ ) = sup ( λ i p i ) ∈ S ′ ( N ) X i ( τ ηE N ( u ∗ λ i p i u ) − τ η ( λ i p i ))= sup ( λ i q ik ) i,k X i,k ( τ ηE N ( u ∗ λ i q i,k u ) − τ η ( λ i q i,k )) ≤ X g ητ ( u g u ∗ g ) . Since η is a concave function, Theorem 3.1.4 implies the following : Corollary 3.1.5.
Assume that
N, G, u be as in Theorem 3.1.4 and that h ( N | uN u ∗ ) = log | G | . Then τ ( u g u ∗ g ) = | G | for all g ∈ G. Remark and Example 3.1.6.
Let A and B be subalgebras of a type II factor M. Then h ( A | B ) ≤ H ( A | B ) ≤ H ( M | B ) , and if B is a subfactor with13 ′ ∩ M = C then H ( M | B ) = log[ M : B ] by [18] so that h ( A | B ) ≤ log[ M : B ].Størmer says that relative entropy can be viewed as a measure of distancebetween two subalgebras, which in the noncommutative case also measurestheir sizes and relative position.Here, we give an example, which shows that h ( A | B ) measures relativeposition and that some small size subalgebra A G can take the maximal valueof h ( A | B ) , although the entropy h ( A | B ) is increasing in A. Assume that the finite group G in Theorem 3.1.4 is abelian. Let B = uN u ∗ . By taking the inner automorphism
Adu ∗ , we may consider M as thecrossed product of B by G so that x ∈ M has a unique expansion x = P g ∈ G x g v g , ( x g ∈ B ) . Let A G be the von Neumann algebra generated by theunitary group v G , (that is, | G | dimensional abelian algebra). Then h ( A G | B ) = log | G | = H ( M | B ) = log[ M : B ] . In fact, it is clear that h ( A | B ) ≤ H ( M | B ) = log[ M : B ] = log | G | . Toshow the opposite iniquality, let ˆ G be the character group of G. Given χ ∈ ˆ G, let p χ = 1 | G | X g ∈ G χ g v g . Then { p χ ; χ ∈ ˆ G } is a family of mutually orthogonal projections in A G with P χ ∈ ˆ G p χ = 1 . Hence h ( A G | B ) ≥ X χ ∈ ˆ G ( τ ηE B ( p χ )) = X χ ∈ ˆ G η ( 1 | G | ) = | ˆ G | | G | log( | G | ) = log | G | , and we have that h ( A G | B ) = log | G | . In the next section, we show that inner conjugacy classes of subfactors N of type II factor can take the maximum value of h ( N | uN u ∗ ) . .2 Case of G = Z n Here, we assume that the group G in 3.1 is a finite cyclic group Z n , that is, M is the crossed product N ⋊ α Z n of a II -factor N by the group generatedby an automorphism α on N such that α n is the identity and α i is outerfor i = 1 , · · · , n − . Such an automorphism α is called a minimal periodicautomorphism (cf. [5]). Let γ be a primitive n -th root. Connes showed in the proof for the char-acterization of minimal periodic automorphisms ([5, Cor. 2.7]) that if α isminimal periodic, then there exists a set of matrix units { e ij } ni,j =1 in N suchthat α ( e ij ) = γ i − j e ij , ( i, j = 1 , · · · , n ) . Let w = P i e i +1 ,i . Then w is a unitary in N which satisfies that w j = X i e i + j,i , w i ∗ e jj w i = e j − i,j − i and α ( w ) = γw. The following indicates that the inner congugacy class of N can take themax L h ( N | L ) , where L is a subfactor of M with [ M : L ] = n. Theorem 3.2.2.
Let N ⊂ M be the above. Then there exists a unitaryoperator u in M which satisfies the following properties : (1) h ( N | uN u ∗ ) = H N (Ad u ) = log n. (2) The conditional expectations E N and E uNu ∗ commute.Proof. Let w ∈ N be the unitary operator in 3.2.1. Let v ∈ M be a unitaryin M implimenting α, that is, α ( x ) = vxv ∗ for all x ∈ N. We put u = 1 √ n X i w i v i − . Then u is a unitary and √ nE N ( u ) = w. Since H N ( u ) = log n , we have byTheorem 3.1.4 that h ( N | uN u ∗ ) ≤ log n.
15s a finite partition of the unity, we choose { p j : p j = e jj , j = 1 , · · · , n } . Then h ( N | uN u ∗ ) ≥ X j τ ηE uNu ∗ ( p j ) − τ η ( p j ) = X j τ ηE N ( u ∗ p j u )= X j τ η ( X i α − i ( w i ∗ p j w i n )) = X j τ η ( P i p j − i n ))= log n Hence h ( N | uN u ∗ ) = log n. (2) To show that E uNu ∗ E N = E N E uNu ∗ , remark that for all a ∈ N,E uNu ∗ ( av k ) = 1 n X j ( X i,l γ k +2 ki − ( j + l ) k − jl w j + l − k − i α j + l − k − i ( a ) w i − l ) v j . Assume that k = 0 . Then E N ( av k ) = 0 . On the other hand, E N E uNu ∗ ( av k ) = 1 n γ k w − k α − ( X i X l γ ki − lk w l − i α l − i ( a ) w i − l )and X i X l γ ki − lk w l − i α l − i ( a ) w i − l = X j ( X i γ k ( i − j ) w j α j ( a ) w ∗ j ) = 0 . Therefore, E N E uNu ∗ ( av k ) = 0 = E uNu ∗ E N ( av k ) . for all a ∈ N and k = 1 , · · · , n −
1. Also for each a ∈ N, we have that E uNu ∗ E N ( a )= 1 n X j ( X i,l γ jl w j + l − i α j + l − i ( a ) w i − l ) v j = 1 n X l X i w i α i ( a ) w ∗ i = 1 n X i w i α i ( a ) w ∗ i = E N ( 1 n X i w i α i ( a ) w ∗ i )= E N E uNu ∗ ( a ) . E uNu ∗ E N ( x ) = E N E uNu ∗ ( x ) for all x ∈ M. At the last, in the case of G = Z , we show a result corresponding one in[4] for maximal abelian subalgebras of the type I n factos M n ( C ). Theorem 3.2.3.
Let N be a type II factor and let M be the crossed product N ⋊ α Z by an outer automorphism α with the period 2. For the unitary w ∈ N in 3.2.1, let u ( λ ) = √ λw + √ − λv, (0 ≤ λ ≤ . Then u ( λ ) is a unitary in M which satisfies the followings : (1) h ( N | u ( λ ) N u ( λ ) ∗ ) = H N ( u ) = η ( λ ) + η (1 − λ ) . and { h ( N | u ( λ ) N u ( λ ) ∗ ) : λ ∈ [0 , } = [0 , log 2] . (2) N ⊂ M ∪ ∪ N ∩ u ( λ ) N u ( λ ) ∗ ⊂ u ( λ ) N u ( λ ) ∗ is a commuting square in the sense of [9] if and only if λ = . (3) h ( N | u ( 12 ) N u ( 12 ) ∗ ) = H N ( Adu ( 12 )) = max u h ( N | uN u ∗ ) = log 2where u is a unitary in M. roof. (1) It is clear that H N ( Adu ( λ )) = η ( λ ) + η (1 − λ ). Hence by Theorem3.1.4, we have h ( N | u ( λ ) N u ( λ ) ∗ ) ≤ η ( λ ) + η (1 − λ ) . We remark that for each x ∈ N,E u ( λ ) Nu ( λ ) ∗ ( x ) = u ( λ ) E N ( u ( λ ) ∗ xu ( λ )) u ( λ ) ∗ = λw ∗ xw + (1 − λ ) α ( x ) . Let { e ij } i,j =1 , ⊂ N be the set of matrix units for α in 3.2.1. Then E u ( λ ) Nu ( λ ) ∗ ( e ii ) = λw ∗ e ii w + (1 − λ ) α ( e ii ) = λe i +1 ,i +1 + (1 − λ ) e ii , (mod 2) . Hence, we have that for each i = 1 , ,τ η ( E u ( λ ) Nu ( λ ) ∗ ( e ii )) = 12 ( η ( λ ) + η (1 − λ )) , so that h ( N | u ( λ ) N u ( λ ) ∗ ) ≥ η ( λ ) + η (1 − λ ) . This implies that h ( N | u ( λ ) N u ( λ ) ∗ ) = η ( λ ) + η (1 − λ ) . (2) First we remember the following ; the diagram is a commuting squarein the sense of [9] means that E N E uNu ∗ = E uNu ∗ E N . Let x ∈ N. Since α ( w ) = − w ∗ , we have that E N E u ( λ ) Nu ( λ ) ∗ ( x ) = λ x + 2 λ (1 − λ ) wα ( x ) w ∗ + (1 − λ ) x and E uNu ∗ E N ( x )= λ x + 2 λ (1 − λ ) wα ( x ) w ∗ + (1 − λ ) x + p λ (1 − λ )(2 λ − xwv Hence E N E u ( λ ) Nu ( λ ) ∗ ( x ) = E uNu ∗ E N ( x ) for all x ∈ N if and only if λ = 1 / . Similarly, for all x ∈ N,E N E u (1 / Nu (1 / ∗ ( xv ) = wα ( x ) + x w ∗ + x α ( w ) + α ( w ∗ x ) = 0and E u (1 / Nu (1 / ∗ E N ( xv ) = 0 . N is a subfactor of M with [ M : N ] = 2 , we have that h ( N | uN u ∗ ) ≤ H ( N | uN u ∗ ) ≤ H ( M | N ) = log 2so that h ( N | u ( 12 ) N u ( 12 ) ∗ ) = log 2 = max u h ( N | uN u ∗ ) . References [1] M. Choda, Entropy for *-endomorphisms and relative entropy for subal-gebras,
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