奈良教育大学学術リポジトリNEAR
Reduction Theory on the Relative Entropy, II
著者 KAWAKAMI Satoshi
journal or
publication title
奈良教育大学紀要. 自然科学
volume 38
number 2
page range 1‑5
year 1989‑11‑25
URL http://hdl.handle.net/10105/1966
lull. Nara Univ. Educ. Vol. 38, No. 2 (Nat.). 1989
Reduction Theory on the Relative Entropy, II
Satoshi KAWAKAMI
(Department of Mathematics, Nara University of Education, Nara 630, fapan ) (Received April 26, 1989
Abstract
Let MZ)N be a pair of von Neumann algebras of finite type and L denotes the algebra generated by N and the center Z(M) of M. Then, the formula H(M¥N)‑H(M¥L)
+H(L¥N) is shown to hold, where H{中is the relative entropy in Pimsner‑Popa's sense.
This fact makes it possible to compute the relative entropy H(MIN) for any pair M⊃N of von
Neumann algebras of finite type.Introduct ion
The classical notion of conditional entropy has been extended to non‑commutative version as the relative entropy in the category of von Neumann algebras of finite type by Connes‑Stormer [ 1 1 and Pimsner‑Popa [5]. We have tried to give several formulas on the relative entropy H(MIN) for pairs M⊃N of von Neumann algebras of finite type and we developed some reduc‑
tion theory on the relative entropy in [ 2 ] and [ 3 ]蝣 However, we did not succeed in givingc0m‑
plete formulas on the relative entropy in [ 3 , The present paper is a complement of the reduc‑
tion theory in [ 3 ] but the result obtained here seems to play an important role on the relative entropy. This is because the present result makes it possible not only to give complete formulas on the relative entropy for any pair of von Neumann algebras of finite type but also to extend the notion of the relative entropy to any pair of von Neumann algebras of any type, which will be dis‑
cussed in the subsequent paper [ 4
Let M be a von Neumann algebra of finite type with a faithful normal normalized trace r and N be a von Neumann subalgebra of 〟. Pimsner‑Popa defined the relative entropy 〟(〃lⅣ) associated with the trace r and investigated its several properties and computations. Among them, the inequality H(M N)≦H(M¥L)十H(L¥N) holds in general for a subalgebra L of M including N, but it is a difficult problem to find such a condition concerning L that the equality H(M¥N)‑H(M¥L)+H(L¥N) holds. In the present paper, we show that the equality holds for the algebra generated by N and the center Z(M) of M‑ It has been already assured by the re‑
duction theory [ 3 ] that both of H(M¥L) and H(L¥N) are computable in this case. Therefore,
as a result, one may assert that the relative entropy H(M¥N) for any pair M⊃N is computable.
Satoshi Kawakami
Acomplementofthereductiontheory
Let〟beavonNeumannalgebraoffinitetypewithafixedfaithfulnormalnormalizedtrace M+denotestheconeofallpositiveelementsofMandS(M)denotesthefamilyofallfinite partitionsoftheunit1,namely,S(M)‑¥A‑(xt)te,;xt∈M+,∑x,‑l.¥I¥<+∞LLetNbea Q3=り vonNeumannsubalgebraofM‑Then,thereexiststheuniquer‑preservingnormalconditional expectationE君ofMontoJV.For∈M¥sethMfl{x)‑rvEだ,x)‑tv(x),where77isacontinuous functiondefinedbyj)(t)‑‑tlogtU>0)andt?(0)‑0.Then,therelativeentropyH{M¥N)in Pimsner‑Popa'ssense[5]isgivenby
H(M¥N)‑sup¥HA{M¥N);△∈S(M)¥,
n whereHAMIN)‑∑hl{xt)forA‑(*,)?‑,inS(M).
巧I‑=
ForafinitedimensionalsubalgebraAofthecenterZ¥M)ofM,wedenotebyLthesub‑
algebraofMgeneratedbyNandA,anddenoteallatomsofAbye,,e,,‑,en‑Then,wehave thefollowinglemma.
Lemmal.(i)H{M¥N)‑H(M¥L)+H(L¥N) n .)H(M¥L)‑∑r(ei)H{Mm¥Ne.) 卜t
n ii)H(L¥N)‑∑ME㌶(*)) り5=‖
Proof.Forany」>0,onecantake△′‑Kxj)‑,inS(M)suchthat (1)H(M¥LトE≦HAM¥L).
Setyu‑etxjU‑l,2,‑,n,j‑l,2,‑‑‑,m)andA‑(yu)u‑Then,AisalsoinS(M)and (2)mM¥L)‑HAM¥L).
mnmn lnfact,HAM¥L)‑∑∑k"
L(肌HAM¥L)‑∑k?(xj).and∑AfW‑AfU・Set nj‑¥t=¥m‑¥i‑i
Ef{xj)‑∑eiZij(zu∈N).Then,」f(i/J‑eiZi,and∑eiziJ‑ei.Wedenote(Et(yu))uinS(L)by l‑1J‑I
E{A).Then,wehave
n (3)HUL¥N)‑∑‑V(EgW) 1‑1
Infact,
HUL¥N)‑Z∑h&etzu)‑∑∑T(り(E試etiztj)(by6‑in[2,p.612]) nmnm i=U‑1f‑U=1
7imnれ
‑∑ME尿el)∑zu))‑∑‑r(E'i∑eiZiJlogEftet))
1‑1J‑11‑1J‑I
n
‑∑t{v(E宗?<)))
̀‑1
n
and note that E景」()‑‑」"(<?(). Similar arguments to the above assure that HA(LlN)≦∑叩(#?(*))
i‑I for any A′ in S(L). Therefore, we get
n
(4) H(L¥N)‑∑叩(」?(*)).
q.‑∩
こmd
(5) HUL¥N)‑H(L¥N).
By the definitions, it is easy to see that
(6) HAM¥N)‑HAMは)+HULlN).
(7) H(M¥N)≦H{M¥L)+H(L¥N).
Under these preparations, the equality (i) is obtained as follows.
H(MIN)≧HA(M¥N)
‑HAM¥L)+HB。IL¥N) [by (6)]
≧H(M¥L)+H(L¥N)‑e [by (l), (2),(5)].
Since 」>O is arbitrary, we get
H(M¥N)≧H(M L)+H(L¥N).
Both of this inequality and (7) imply the equality I
The equality (iii) is shown in (4) and the equality (ii) is described in [ 3 ]・ [Q.E.D.]
For the abelian algebra Z(M), take an increasing sequence jAin‑i ‑^ f'n'te dimensiona一 sub‑
algebras of Z(M) such that Z(M)‑ the weak closure of UAn‑ For eachォ‑1, 2, ‑, Lndenotes n‑¥
the algebra generated by N and An, and L denotes the algebra generated by TV and Z(M).
Then, it is clear that the algebra L is the weak closure of UL.
71=1
Lemma 2. H(L¥N)‑¥imH(Ln N)
nlLr)
Proof. Concerning the r‑preserving expectations, E志 oELL. holds. Then, our situation satisfies the assumption of Proposition 3.4 in [ 5 ], so that we get the desired conclusion. [Q.E.D.]
Theorem 3‑ Let M be a von Neumann algebra offinite type and N be a von Neumann subalgebra of M. For the algebra L generated by N and the center Z(M) of M, the equality H{M¥N)
‑H(M¥L)+H(L¥N) holds.
Proof. Let lLnに=‑ be the increasing sequence of subalgebras of M described as above such that L‑the weak closure of ULn. By Lemma 1, H(M¥N)‑H(M¥Ln)+H(Ln¥N). Since
71=1
Satoshi Kawakami
M⊃L⊃Ln, we see that
H{M¥N)≧H(M¥L)十H(Ln¥N).
Applying Lemma 2 to this inequality, we get
H(M¥N)≧H(Mは)+H(L¥N).
Hence, we obtain the desired equality because the reverse inequality always holds in general.
[Q.E.D.]
The reduction theory on the relative entropy [ 3 ] assures the following facts.
a) H{叫L)‑jsH(M(s仙s))du(s) where Z(M)‑L‑(S, 〟),
(b) H{L¥N)‑jTH(L[t)¥N{t))dv{t) where Z(N)‑L‑(T,
Concernig the above formulas, we note that the components M(s) are factors for /^‑almost all ∈S
and that the components N{t) are also factors forリーalmost all t∈T because Z(M)⊂Z(L) and
Z{L)^>Z(N). Moreover, we have already established the characterizations of entropy finiteness and the formulas on H(M¥N) for pairs MDiV in the case that either M or N is a factor of finite type in [3]. Therefore, one can compute the relative entropy H(M¥N) for any pair M⊃N, owing to Theorem 3‑ For an example, we get the following.
Corollary 4. Let M⊃N be a pair of von Neumann algebras offinite type with a faithful normal
normalized trace r such that the relative commutant N′nM is atomic. ¥et; i∈/￨ denotes the set of
all atoms of Z(M)and (/}; j∈J¥ denotes the set of all atoms of Z(N). Then, we have H(M N)‑∑∑五(eifJ)H(Meif] ¥ Neifj)+ r(e,/))log(rWr(/>)/ HeJS)2主
SE/jeJ
Proof. Let L denote the algebra generated by TV and Z¥M¥ Then, by Theorem 3, we get (1) H(M¥N)‑H{MIL)+H(L¥N).
The reduction theory on the relative entropy developed in [ 3 ] assures the following formulas.
(2) H(M¥L)‑∑r(ei)H(Met￨Le
ie/
(3) H(L¥N)‑∑T(石)H{Lfi¥Nfl)
JeJ
(4) H{Met ¥ Lei)‑∑¥(r(eJ‑j)/T(el))H(MeisJ¥ Lets>)+ vMeifj)/ T(ei))¥
urn
(5) H(Lf}¥ Nfl)‑∑vMeJjVrlfj))
iEI
We note that Lc,/,‑AL./> Then, the desired formula follows from the above (1ト(5) by simple
calculations and so we omit the details. [Q.E.D.J
References
1 1 Connes, A. and Stormer, E. (1975), Entropy for automorphisms of II, von Neumann algebras, Acta Math., 134, 288‑306.
2 ] Kawakami, S. and Yoshida, H. (1987), Actions of finite groups on finite von Neumann algebras and the relative entropy, J. Math. Soc. Japan, 39, 609‑626.
Kawakami, S. and Yoshida, H. (1988), Reduction theory on the relative entropy, Math. Japon., 33, 975‑990.
4 ] Kawakami, S., Some remarks on index and entropy for von Neumann subalgebras, Preprint.
5] Pimsner, M. and Popa, S. (1986). Entropy and index for sub factors, Ann. Sci.丘cole Norm. Sup., 19.
57‑106.
