数理解析研究所講究録 1291 巻 2002 年 55-7255
行列群の連続有限型因子環上への非コサイクル同値な連続個の作用

大阪教育大学

長田 まりゑ (Marie CHODA)

Osaka Kyoiku University

1. 前置きここでは、 行列のなす群に対して、 有限連続型因子環 $M$ の自己同型写像群としての、 一連の表現$\alpha_{s},$$(s\in[0,1/2])$

を、 与える。

ここでの因子環 $M$ としては、 次のようなものが現れる :i) 近似的有限次元型環 $R$$\mathrm{i}\mathrm{i})$$L(F_{n})$自由群$F_{n}$, $(n=2,3, \cdots, \infty)$ の左正則表現により生成されるノイマン環.

i) の場合には、 行列群 $G$ として $SL(n, \mathbb{Z}),$ $n\geq 3,$
積$R\cross_{\alpha_{s}}G$$n=\infty$の性質

は Kazhdan の性質$\mathrm{T}$$Sp(n, \mathbb{Z}),$ $n\geq 2$を持つ因子環となる。 ところが、 ) の場合, 特に$\mathrm{i}\mathrm{i}$の時には、 同じ群をもってきても、 この接合積は、 因子環となるが。 Kazhdan$\mathrm{T}$を持つことは、 出来ない。

全ての場合において、 これらの連続個の表現は互いに異なると$\alpha_{t}$を取れば、 接合

$s$

と

$t$

に対して

$\alpha_{s}$とは互いに非共役である。 又、 とくに、 行列群が特殊群 $SL(n, \mathbb{Z}),$ $n\geq 3$ のときには、 更にコサイクル同値に、 なることもできない連続個の表現である。

更に、 これらの表現に現れる個々の自己同型写像のエントロピー$H(\alpha_{T}^{s})$$\alpha_{s}(T)$に対する Comes-Stormerは次のような関係式をみたす。

$\log(\max\{\prod_{i=1}^{n}\mu_{i},\prod_{i=1}^{n}\mu_{i}’\})\leq H(\alpha_{T}^{s})\leq\log(\prod_{i=1}^{n}\mu_{i}\mu_{i}’)$Typeset by$A\lambda 4\mathrm{S}\mathrm{I}\mathrm{f}\mathrm{f}\mathrm{l}$56となる。ただし、 $T$ は可逆な次正方行列で、その固有値$n$$\mu_{i}=\max(|\lambda_{i}|, 1)$,$\{\lambda_{1}, \lambda_{2}, \cdots, \lambda_{n}\}$に対して、$\mu_{i}’=\max(\frac{1}{|\lambda_{i}|}, 1)$である。

$\mathrm{I}\mathrm{I}$. 行列群の連続有限型因子環上への作用

この章では、 行列群 $GL(m, \mathbb{Z}),$ $m\geq 2$ の超有限連続有限型因子環 $R$ と $n(2\leq$$n\leq\infty)$$L(F_{n})$, 個の生成元をもつ自由群の左正則表現によって与えられる因子環$F_{n}$の自己同型写像群としての作用を、 与える。

2.1. 自由積に対するシンプレチイック
先ず通常のシンプレチイック

形式の概念を拡張して、 自由積に対するシンプ形式を定義する。

レチイック

2.1.1. Let$n$be apositive integer. Let $B(a, b)$ be the symplectic form ofin the vector space$\mathbb{R}^{2n}$($\mathbb{R}$where $a=(a_{1}, \cdots, a_{n}, a_{n+1}, \cdots, a_{2n})\in \mathbb{R}^{2n}$ . Let$n$product ofand let$\mathbb{R}^{2n}$$J_{n}=(\begin{array}{ll}0 E_{n}-E_{n} 0\end{array})$. The symplectic group$B(a, b)=B(Ta,Tb)$ for all $a,$

2.1.2. We denote by$b\in \mathbb{R}^{2n}$$E_{n}$

and

$b$

,

be the identity matrix of. Then $B(a, b)=(a, Jb)$ , the natural inner$Sp(n,\mathbb{R})$is the set of amatrix$T$such that.

acopy of the additive group$A_{i}$$a$is the set of all real numbers) :$B(a, b)= \sum_{i=1}^{n}a_{i}b_{n+i}-\sum_{i=1}^{n}a_{n+i}b_{i}$the order

形式$\mathbb{Z}^{2n}$for all$i\in \mathbb{Z}$. Let$I\subset \mathbb{Z}$be asubset, and let $A(I)=i\in I*A_{i}$ be the free product group. Each $a\in A(I)$ isexpressed by the unique form that$(^{*})$$a=a_{1}a_{2}\cdots a_{p}$,$a_{i}\in A_{\iota},$$a_{i}:\neq

0$$\iota_{i}\in I$,$\iota_{1}\neq\iota_{2}\neq\cdots\neq\iota_{p}$

.

57
The

$p$

is called the length ofand the set$a$$\{\iota_{1}, \iota_{2}, \cdots, \iota_{p}\}$is called the alphabet for

the word , which we denote by $J(a)$ . Given $i\in J(a)$ , let $I(a, i)=\{j\in I:\iota j=i\}$ .
$a$

Let

$\Phi_{i}$: $A(I)arrow A_{i}$ be the homomorphism defined by$\Phi_{i}(a)=\sum_{j\in I(a,i)}a_{j}$,for $a\in A(I)$ with the reduced form (’). If $I(a, i)=\emptyset$ , we let $\Phi_{i}(a)=0$ .
Define ahomomorphism$\alpha$:$sp(n, \mathbb{Z})arrow \mathrm{A}\mathrm{u}\mathrm{t}(A(I))$$\alpha\tau(a)=T(a_{1})T(a_{2})\cdots T(a_{p})$Here$Sp(n, \mathbb{Z})$has the form

,

for

by

$T\in Sp(n, \mathbb{Z})$.

is the symplectic group whose components are integers, and $a\in A(I)$$(^{*})$.

2.1.3. Let,$B_{I}(a, b)= \sum_{i\in J(a)\cap J(b)}B(\Phi_{i}(a), \Phi_{i}(b))$$a,$ $b\in

A(I)$

.

In the case where $J(a)\cap J(b)=\emptyset$ , we let $B_{I}(a, b)=0$ . Then we have the following
proposition by calculations.

2.1.4. Proposition. The following basic properties of the symplectic form

$B(\cdot, \cdot)$in 2.1.1 are satisfied;$B_{I}(a, 1)=B_{I}(1, a)=B_{I}(a, a^{-1})=0$ ,$B_{t}(a, b)=-B\tau(b, a)$

,

$\{$

$B_{I}$(ab, ) $=B_{I}(a, c)+B_{I}(b, c)$ ,$c$$B_{t}(a, b)=B_{I}(\alpha\tau(a), \alpha\tau(b))$

where$a,$ $b,$ $c\in

and 1is the identity of the group $A(I)$ , andA(I),$ $T\in sp(n, \mathbb{Z})$$B_{I}(a, b)=0$

,

,

if

$a\in A_{i},$$b\in A_{j},$$i\neq i$.

58
22. 自由積群に対する

2-コサイクル.

22.1. Let $A(I)=*A_{i}i\in I$ be the same group as in the section 2.1. We consider thesemidirect product$G(I)=sp(n, \mathbb{Z})\mathrm{x}_{s}A(I)$,

where the product of whose elements are defined by$(S, a)(T, b)=(ST, \alpha_{T}^{-1}(a)b)$

Let $s\in[0, \pi/2]$ be irrational (mod$.2\pi$,$(S,T\in sp(n, \mathbb{Z}), a, b\in A(I))$.

). Put$\mu_{s}((S, a),$ $(T, b))=e^{\sqrt{-1}s\pi B_{I}(\alpha_{T}^{-1}(a),b)}$,$(S,T\in sp(n, \mathbb{Z}), a,b\in A(I))$

Then we have the following Proposition using the symplectic property of
2.2.2. Proposition. The$\mathrm{T}$$\mu_{s}$.

$B_{I}(\cdot, \cdot)$.

is a nomalized 2-cocycle of $G(I)\cross G(I)$ to the torus:$\mu_{s}(1,g)=\mu_{s}(g, 1)=\mu_{s}(g,g^{-1})=1$$\{$$\mu_{s}(f,g)\mu_{s}(fg, h)=\mu_{s}(g, h)\mu_{s}(f,gh)$

2.3. 自由群因子環 $L(F_{n})$ への$GL(n, \mathbb{Z})$上で、 準備したことを用いて、 自由群因子環 $L(F_{n})$ への

の作用.
の作用を、

$GL(n, \mathbb{Z})$与える。 この与え方は、 下の命題 232 で、 示すように、 特殊な場合には、 超有限連

続有限型因子環 $R$ への作用を、 同時に与えることになることを、 注意する。

2.31. Let$\lambda^{s}$$\mu_{s}$be the normalized 2-cocycle defined in 22. The left$\mu_{s}$-representationof the group $G(I)=sp(n, \mathbb{Z})\mathrm{x}_{s}A(I)$ is defined by$(\lambda^{s}(g)\xi)(h)=\mu_{s}(h^{-1},g)\xi(g^{-1}h)$Then$\lambda^{s}$is a$\mu_{s}$,$g,$ $h\in

G(I),$$\xi\in l^{2}(G(I))$-cocycle unitary representation of $G(I)$ on $l^{2}(G(I))$ :$\lambda^{s}(g)\lambda^{s}(h)=\mu_{s}(g, h)\lambda^{s}(gh)$

.

.

59
We denote by $N_{s}(I)$ the von Neumann algebra generated by$\lambda^{s}(A(I))$.

Let $L(F_{k})$ be the von Neumann algebra generated by the left regular repersen-

tation of the free group$F_{k}$

2.32. Proposition. Let

with
$|I|$

$k$

generators, $k\geq 2$ .

be the cardinality of the set(1) $If|I|=1$ , then $N_{s}(I)$ is isomorphic to the hyperfinite$I$.

$II_{1}$

factor .
$R$(2) $If|I|\geq 2$ , then the von Neumann algebra $N_{s}(I)$ is isomorphic to the free groupfactor $L(F|I|)$ .
Define the homomorphism$\alpha^{s}$:$sp(n, \mathbb{Z})arrow \mathrm{A}\mathrm{u}\mathrm{t}(N_{s}(I))$$\alpha_{T}^{s}(x)=\lambda^{s}(T, 1)x\lambda^{s}(T, 1)^{*}$,

by$(T\in sp(n, \mathbb{Z}),$.

$x\in N_{s}(I))$

Then we have$\alpha_{T}^{s}(\lambda^{s}(1, a))=\lambda^{s}(1, \alpha_{T}(a))$

Let$\eta$

be the imbedding of$GL(n,\mathbb{R})$,$(T\in sp(n, \mathbb{Z}),$

into$sp(n,\mathbb{R})$$T^{t}$

morphism$T$$\alpha^{s}\circ\eta$

Let $M$ be a

An action$\beta$.

means the transposed matrix of .

2.3.3. Definition. We define the action

and.

given by$\eta(T)=(\begin{array}{ll}T 00 (T^{t})^{-1}\end{array})$Here$a\in A(I))$$\beta$of:$\alpha^{s}$$GL(n, \mathbb{Z})arrow \mathrm{A}\mathrm{u}\mathrm{t}(N_{s}(I))$$\mathrm{I}\mathrm{I}_{1}$factor with the unique$G\mathrm{o}.\mathrm{n}M$is “outer” if$\beta_{g}$is ” mixing” if given $x,$ $y\in M$ andof$GL(n, \mathbb{Z})$on $N_{s}(I)$ by the homO-.
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\tau$. and let$G$

is an outer automorphism for all $g\in G$ ,$\epsilon>0$

there exists afinite subset $K$ ofsuch that$|\tau(y\beta_{g}(x))-\tau(x)\tau(y)|<\epsilon$

Remark that if$\beta$is mixing, then$\beta$be adiscrete group.

is ergodic.

,$\forall g\not\in K$

.

$G$

60
2.3.4. Proposition. Let(1) The action$\alpha^{s}$$G\subset GL(n, \mathbb{Z}),$$(n\geq 2)$: $Garrow Aut(N_{s}(I))$ is outer.

(2) If given finite subsets$S_{2}\subset \mathbb{Z}^{2n},$$S_{1},$$G$

contains$S_{1}\cap\{\eta(T)a;a\in S_{2}\}=\emptyset$

then$\alpha^{s}$be a non-trivial subgroup.

,

finite subset$a$$\forall T\not\in K$$K$such that,is mixing for all .
$s$

Let $\{e_{j}; 1\leq j\leq 2n\}$ be the standard basis inby $\{e(i;j);1\leq j\leq 2n\}$ . Let$S$, whose copy in$\mathbb{Z}^{2n}$$A_{i}$

we denotebe asubset of $\{1, 2, \cdots, n\}$ and let.

$A_{S}= \{a\in A(I);a=a_{1}a_{2}\cdots a_{p}, a_{k}\in\bigcup_{l\in S}\mathbb{Z}e(\iota_{k}, l), \iota_{1}\neq\cdots\neq\iota_{p}\}$2.3.5. Lemma. Let$N_{s}(I)$$G\subset GL(n, \mathbb{Z})$onto $N_{t}(I)$ such that

(1) Let us fix an integer$j$$\theta\alpha_{T}^{s}=\alpha_{T}^{t}\theta$

$0$

whose

for all $T\in G$ .

$G$

contains$matr\dot{\mathrm{u}}ces\{T_{1,l},T_{2,k;}1\leq l,$

and $T_{1,l}(p, q)=\mathrm{O}$ otherwise, and $T_{2,k}(p,p)=1$ for all$(c(a)\in \mathbb{C})$Moreover if$G$contains a matrir$T_{3}$, then there exist a pemutation$\sigma$, in the$||\cdot||_{2}$$T_{1,l}(j, l)\neq$$T_{2,k}(k,n)\neq \mathrm{O}$

$p,$

and

(1-2) Case that $n=2$ : If

$G$convergence topology.

suct that $T_{3}(j,j)=1,$

of I and an$mj\in \mathbb{Z}$$\theta(\lambda^{s}(e(i;j)))=\lambda^{t}(m_{j}e(\sigma(i),j))$$i\in I$$p,$$k\neq$otherwise, then for all $i\in I$ we have the Fourier expansion:$\theta(\lambda^{s}(e(i;j)))=\sum_{a\in A_{\mathrm{j}}}c(a)\lambda^{t}(a),$$p\neq j$$k\leq n,l\neq j,$component $T(p, q)$ satisfies that $T_{1,l}(p,p)=1$ for all$p,$ $q$$T_{2,k}(p, q)=\mathrm{O}$be an isomorphism of$\theta$with $1\leq i\leq n$ .

(1-1) Assume that $n\geq 3$ :If$n\}$be a subgroup, and let,$T_{3}(p,j)\neq \mathrm{O}$for some

so that$\forall i\in I$.

contains the above matrix $\{T_{1,l;}l\neq j\}$ , then for all

we have the Fourier expansion for $S=\{j, 5-j\}$ :$(c(a)\in \mathbb{C})$$\theta(\lambda^{s}(e(i;j)))=\sum_{a\in A_{S}}c(a)\lambda^{t}(a),$, in the$||\cdot||_{2}$convergence topology.

61
Moreover if$I,$$c\in \mathrm{T}$$G$

and ancontains the above matrix$m_{k}\in \mathbb{Z},$$T_{3}$, then there exist a permutahon$\sigma$of$(k=1,2)$ so that$\theta(\lambda^{s}(e(i;j))=c_{j}\lambda^{t}(m_{1}e(\sigma(i),j))\lambda^{t}(m_{2}e(\sigma(i), 5-j))$.

(2) Let us fix an integer $j,$ $(n+1\leq j\leq 2n)$ . Same fomulas hold if conditions are

satisfied by replacing the matrices
2.3.6. Proposition. Let

$T$

to

$(T^{t})^{-1}$$G\subset GL(n, \mathbb{Z}),$

matrices in Lemma 2.34for all . Then$j$$\alpha^{t}$.

$(n\geq 2)$$\alpha^{s}$be a subgroup which contains the: $Garrow Aut(N_{s}(I))$ is not conjugate to: $Garrow Aut(N_{t}(I))$ if $s\neq t$ .

2.3.7. Corollary. The groups$SL(n, \mathbb{Z}),$ $GL(n, \mathbb{Z}),$$(n\geq 2)$

and the foee group$F_{2}$

have a continuous family of non-conjugate mixing outer actions on the foee grooup

factor$L(F_{m})$for all $m=2,3,$$\cdots,$$\infty$.

2.4. コホモロジー類とカズダンの性質

この節では、 特に、 行列群のうちでも、 カズダンの性質$\mathrm{T}$$\mathrm{T}$を持つ群を中心に取

り扱う。

Let

$\alpha$be an ergodic action of adiscrete groupbe the unitary operators of $M$ . We denote byrepresentation$Z_{\alpha,erg}^{1}=$

{

$u$

of

$G$

on $M$ for$\alpha$such that$u:Garrow U(M)|\mathrm{A}\mathrm{d}(u_{g})\circ\alpha_{g}$Two cocycles$u_{g},$$v_{g}$$G$

on a$Z_{\alpha,erg}^{1}$$\mathrm{I}\mathrm{I}_{1}$factor $M$ , and let $U(M)$the set of 1-cocycle unitary$\mathrm{A}\mathrm{d}(u_{g})\cdot\alpha_{g}$.is also ergodic :is ergodic, $u_{g}\alpha_{g}(u_{h})=u_{gh},$are said to be cohomologous and denoted byexists aunitary $u\in M$ such that$u_{g}=uv_{g}\alpha_{g}(u^{*}).$$H_{\alpha,erg}^{1}=\mathbb{Z}_{\alpha,erg}^{1}/\sim$

Let.

$\forall g,$$h\in G,$

$u_{g}\sim v_{g}$}.

if there62241. Theorem. Let

matrices in Lemma 2.3.5for allthen the actions$\alpha^{s}$$(1\leq j\leq n)$$j,$

be a subgroup which contains the$(n\geq 2)$$G\subset GL(n, \mathbb{Z}),$

. If

$G$

has the propertyof Kazhdan,$T$: $Garrow Aut(L(F_{m}))$ gives a continuous famdy, any two of which

are not cocycle conjugate for all $m=2,3,$
2.4.2. Corollary. Each of the group$\cdots,$$\infty$$SL(n, \mathbb{Z}),$.
$n\geq 3$

and$Sp(n, \mathbb{Z}),$$n\geq 2$

has

$a$

continuous family of ergodic outer actions on the free group factor $L(F_{m}),$ $m\geq 2$such that each two of them are not cocycle conjugate.

Proof. The group
Kazhdan by$SL(n, \mathbb{Z}),$ $n\geq$[\mathrm{K}, \mathrm{D}\mathrm{K}]$

3$

and

$Sp(n, \mathbb{Z}),$$n\geq 2$

have the property, and satisfy the conditions in Theorem 2.4.1.

2.5. 上記の作用

if $|I|=1$ , where$\lambda_{s}(\eta(SL(n, \mathbb{Z}))\cross_{s}A(I))$$R$

is the hyperfinite$\mathrm{I}\mathrm{I}_{1}$

if $2\leq|I|=m\leq\infty$ . Since the action

by Proposition 234, the $M_{s}(n, I)$ is atype$\mathrm{I}\mathrm{I}_{1}$is afactor which have “$\mathrm{H}\mathrm{T}$. Then the. Moreover,factor, and$\alpha^{s}$is outerfactor.

25.1. In this section, we remark that the crossed products$SL(2, \mathbb{Z})$口$M_{s}(n, I)=N_{s}(I)\mathrm{x}_{\alpha^{s}}SL(n, \mathbb{Z})$von Neumann algebra $M_{s}(n, I)$ is generated by$M_{s}(n, I)\cong L(F_{m})\mathrm{x}_{\alpha^{\epsilon}}SL(n, \mathbb{Z})$ofに関する接合積環の性質$\alpha^{s}$

Let us consider the crossed product$M_{s}(n, I)\cong R\mathrm{x}_{\alpha^{\epsilon}}SL(n, \mathbb{Z})$$\mathrm{T}$$M_{s}(2, I)=N_{s}(I)\mathrm{x}_{\alpha^{S}}$ffee group subfactor $L(F_{m})$ ”if $|I|=m\geq 2$ .

The notion of “HT free group subfactor $L(F_{n})$ ”is amodification of HT Cartan
subalgebra in the sense of Popa ([P02]) as follows :Let $M$ be afinite von Neumann algebra, and let $B\subset M$ be avon Neumannsubalgebra. The embedding $B\subset M$ has the Property, if it has the property$T$which is anotion in von Neumann algebra text of Margulis’ property$[\mathrm{d}\mathrm{H}\mathrm{V}])$for the pair of groups, that is, there exists afinite subset$\mathrm{T}([\mathrm{M}]$, cf.

$\{x_{1}, x_{2}, \cdots, x_{n}\}$63
of $M$ and$\epsilon>0$such that if $H$ is aHilbert $M$ bimodule with $\xi\in H$ aunit vector

which satisfies that$\xi_{0}\in H$$||x_{i}\xi-\xi x_{i}||\leq\epsilon$

for all , then there exists anon zero vector$i$such that $b\xi_{0}=\xi_{0}b$ for all $b\in B$ . When $M$ is atype

factor, the$\mathrm{I}\mathrm{I}_{1}$embedding $B\subset M$ is said to have the property $H$ if it has aproperty which isageneralization of Hagerup’s compact approximation property ([P02 :Definition23]), and $B\subset M=B\mathrm{x}_{\sigma}G$ has the property$\phi_{n}$$\mathrm{H}$

if

$G$

has positive definite functions

such that$\phi_{n}(1)=1$,$\lim_{garrow\infty}\phi_{n}(g)=0,$

Furthermore, an abelian$C^{*}$$(\forall n)$,$\lim_{narrow\infty}\phi_{n}(g)=1,$-subalgebra $B$ of atype$\mathrm{I}\mathrm{I}_{1}$$(\forall g\in G)$.

factor $M$ is called aHT

Cartan subalgebra of $M$ if it satisfies the following conditions :1) $B’\cap M=B$ and $N_{M}(B)=$ {unitary $u\in M:uBu^{*}=B$ } generates $M$.

2) $B\subset M$ has the property H.
3) $B$ has avon Neumann subalgebrathat$B_{0}\subset M$$B_{0}\subset B$

such that $B_{0}’\cap M=B$ and such

has the property T.

Popa remarked about “HT hyperfinite subfactor” $R$ of the factor$R\mathrm{x}_{\sigma}$

Go in[P02 :Remark 66].
Here, we consider anotion corresponding$\mathrm{H}\mathrm{T}$Cartan subalgebra for subfactors

which is isomorphic to the free group factor $L(F_{n}),$ $n\geq 2$ . We say that asubfactor$Q$of atype$\mathrm{I}\mathrm{I}_{1}$factor $M$ is a$\mathrm{H}\mathrm{T}$free group subfactor of$M$

if it satisfies the following

conditions :1) $Q’\cap M=\mathbb{C}$ and $N_{M}(Q)$ generates M.
2) $Q\subset M$ has the property H.

3) $Q$ has avon Neumann subalgebrasuch that$Q\mathrm{o}\subset M$

has the property T.

$Q_{0}\subset Q$

such that $Q_{0}’\cap M=Q_{0}’\cap Q$ and64
2.52. Proposition. Assume that $|I|\geq 2$ . The type

$II_{1}$

factor$M_{s}(2, I)$

free group subfactor isomorphic to $L(F|I|)$ for all $s\in[0, \pi/2]$ mod.
Remark by the same proof that $R=N_{s}(I)$ is a$\mathrm{H}\mathrm{T}$$2\pi$

has a $HT$.

hyperfinite subfactor of $M$

when $|I|=1$ .
2.5.3. Remark. Assume that $n\geq 3$ . As we showed in [Ch2], if $|I|=1$ , thenthe factor $M_{s}(n, I)$ has property$\mathrm{T}$of Connes-Jones ([CJ]) because the groupof Kadhdan. If $|I|=\infty$ , then the group$\eta(SL(n, \mathbb{Z}))\cross_{S}\mathbb{Z}^{2n}$has property$\eta(SL(n, \mathbb{Z}))\cross_{s}A(I)$does not have property

and can not have property

not have property
property$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$by [$\mathrm{T}$$\mathrm{K}$$\mathrm{T}$because it has infinite generators:Theorem 2], so that the factor $M_{s}(n, I)$ doesby [CJ :Theorem 2]. We don’t know whether $M_{s}(n, I)$ hasor not, in the case where $1\neq|I|<\infty$ .

III. 作用の中に現れる個々の自己同型写像のエントロピーの値
The each automorphism of $L(F_{m}),$ $m\geq 2$ in the actions that we discussed in thesection 2is given essentially as the ffee products of those in [Ch2]. Using this fact, in
this section, we give an estimation of theautomorphism$\alpha_{T}^{s},$$(T\in GL(n, \mathbb{Z}))$$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{s}- \mathrm{S}\mathrm{t}\phi \mathrm{r}\mathrm{m}\mathrm{e}\mathrm{r}$of the tyPe$\mathrm{I}\mathrm{I}_{1}$entropy$\mathrm{B}\mathrm{C}$for eachfactor $N_{s}(I)$ . The cocycle actionsof Popa are given as the reduced actions of the free permutation
it is known that $H(\sigma)=0$ ([S1], cf. [S2,$H(\alpha_{T}^{s})$$\sigma\in \mathrm{A}\mathrm{u}\mathrm{t}(L_{\infty})$, and, D4]).

3.1. Entropy$ht_{\phi}(\alpha)$

To obtain an estimation of the values for the entropies, we need an entropy

defined in [Ch5], which$\mathrm{i}\mathrm{a}$aslight modification of Voiculescu’s topological entropy([V2], cf.[Br]). First, we review the definition and basic properties of the entropy

65
By aC’-dynamical system
algebra,$\alpha$is$\mathrm{a}*$3.LL Given a

of

$A$

Here

-automorphism of$C^{*}$, we mean that

and

$A$

-dynamical system$\phi$

is aseparable unital$A$

is an -invariant state of$\alpha$$(A, \alpha, \phi)$

, let

$\pi$

$\omega\xi$of afinite dimensional C’-algebra

be afaithful ’-representation$C$$\phi=\omega\xi\circ\pi$-algebra is denoted by $S(\psi)$ . For afinite subset$\psi$$\omega\subset A$

on afinite dimensional, and a $\delta>0$ , put: $(\rho, \eta, C)\in CPA(A, B(H))$

and $||\eta\circ\rho(a)-\pi(a)||<\delta||a||$ , for all$scp_{\phi}(\pi;\omega, \delta)$

is defined to be$\infty$$ht_{\phi}( \pi;\alpha,\omega)=\sup_{\delta>0}ht_{\phi}(\pi;\alpha,\omega, \delta)$$ht_{\phi}( \pi;\alpha)=\sup_{\omega}ht_{\phi}(\pi;\alpha,\omega)$

$\pi$

’

.
$(A, \alpha, \phi)$

has afiithfifl: $Aarrow B(H)$ and acydic unit vector $\xi\in H$ such that

we can prove that the vmlue
3.1.3. Remark. Aunital$scp_{\phi}(\pi;\omega, \delta)$$C^{*}$$\phi=\omega\xi\circ\pi$does not depend of the choice of-algebraA is exact if and only if for some$C^{*}$$\pi$$\omega\subset A$

there exist afinite dimensional$C^{*}$,.

-algebrathere exists an embedding $\iota:Aarrow B$ which is nuclear, that is, for arbitrary

and for every finite set}.

.

3.1.2. Remark. In the case where aC’-dynamical system-representation$a\in\omega$

if no such approximation exists. Let$ht_{\phi}( \pi;\alpha,\omega, \delta)=\varlimsup_{\mathrm{N}arrow\infty}\frac{1}{\mathrm{N}}scp_{\phi}(\pi;\cup\alpha^{i}(\omega), \delta)N-1i=0$$B$.

and unital completely positive maps

. The von Neumann entropy of astate$scp \phi(\pi;\omega, \delta)=\inf\{S(\omega\xi\circ\eta)$

$*$

‘-

is the vector state $<\cdot\xi,\xi>$ . Let $CPA(A, B(H))$ be the set of all triplets$ACB(H)\vec{\rho}\vec{\eta}$

The

$C$

.

$A$

on aHilbert space $H$ , and let $\xi\in H$ be aunit vector such that$(\rho, \eta, C)$$C^{*}$$(A, \alpha, \phi)$$\epsilon>0$-algebra $C$ andunital completely positive maps $A\vec{\rho}CB\vec{\eta}$ such that $||\iota(a)-\eta 0\rho(a)||<\epsilon||a||$66
for all$a\in\omega$. ([Kir :Theorem 4.1], [Was]). Let$(H\phi, \pi\phi, \xi\phi)$be the GNS(Gelfand-Naimark-Segal representation)-triplet associated with astate
algebra$A$. Consider the completely positive extension$\iota(A)arrow B(H_{\phi})$

to

$B$

([Ar]), then

$ACB(H\phi)\vec{\rho}\vec{\rho 0\eta}$that the approximation approach for3.1.4. Definition. Let

the GNS-representation$(A, \alpha, \phi)$$\pi\phi$$scp\phi(\omega;\delta)$

be a$C^{*}$

and$\beta$of an exact C’-of the map$\pi_{\phi}\circ\iota^{-1}$implies the nuclearity of$\pi_{\phi}$:(so

is reasonable).

-dynamical system, whereis faithful. We define the entropy

essential when we discuss the entropy of the ffee product$\alpha$$\rho$$\phi$$\alpha*\beta$$ht_{\phi}(\alpha)$$A$

is exact andby$ht_{\phi}(\pi_{\phi}, \alpha)$.

of two automorphisms.

3.1.5. Remark. In the case where$A$

is nuclear, this entropy coincides with thatdefined in in [Ch4], and in the form of$scp\phi(\omega;\delta)$, we only need triplets

where $C$ is afinite dimensional C’-algebra, and $AC$

$A\vec{\rho}\vec{\eta}$$(\rho, \eta, C)$,

are unital completely

positive maps.

3.1.6. Proposition. Let
and$\phi$$(A, \alpha, \phi)$be a$C^{*}$-dynamical system such thathas the faithfdl GNS-representation. Ihen thetween the Connes-Namhofer- Thining entropy

topological entropy$ht(\alpha)$

If A

$ia$

:

abelian, then we have$h_{\phi}(\alpha)=ht_{\phi}(\alpha)$.

is exacttakes the value be-

and the Brown- Voiculescu’s$h_{\phi}(\alpha)$$h_{\phi}(\alpha)\leq ht_{\phi}(\alpha)\leq ht(\alpha)$$ht\phi(\alpha)$

$A$

.

67
3.17. Theorem. For each $i\in I$ , letbe a state of$\phi_{i}$$A_{i}$$A_{i}$

be a unital exact C’-algebra, and let

whose $GNS$-representation$\pi_{i}$is faithful. Let

$A$

and

$\phi$

be the-algebra and the state given by the reduced free product construction :$C^{*}$$(A, \phi)=*(A_{i}, \phi_{i})i\in I^{\cdot}$

If$\alpha_{i}\in \mathrm{A}\mathrm{u}\mathrm{t}(A_{i})$satisfies$\alpha=i\in I*\alpha_{i}\in \mathrm{A}\mathrm{u}\mathrm{t}(A)$$\phi_{i}\circ\alpha_{i}=\phi_{i}$for allpresemes the state$\phi$$i\in I$

, then free product automorphism

and$ht_{\phi}( \alpha)=\sup_{i\in I}ht_{\phi}(:\alpha_{i})$.

Remark. About the topological entropy,

3.1.8.

$\mathrm{B}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{n}- \mathrm{D}\mathrm{y}\mathrm{k}\mathrm{e}\mathrm{m}\mathrm{a}rightarrow \mathrm{S}\mathrm{h}\mathrm{l}\mathrm{y}\mathrm{a}\mathrm{k}\mathrm{h}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{k}\mathrm{o}$proved in [BDS :Theorem 57] that$ht_{\phi}( \alpha)=\sup_{i\in I}ht_{\phi_{i}}(\alpha_{i})$under the same conditions as in Theorem 3.17.

3.2. Now we discuss on the$\alpha_{T}^{s}$$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{s}- \mathrm{S}\mathrm{t}\phi \mathrm{r}\mathrm{m}\mathrm{e}\mathrm{r}$of $N_{s}(I)(T\in GL(n, \mathbb{Z}))$ . Here$H(\cdot)$is the action of$\alpha^{s}$in 233, and $N_{s}(I)$ is the hyperfinite

entropy$\mathrm{I}\mathrm{I}_{1}$factor$R$

for each automorphism$GL(n, \mathbb{Z})$on $N_{s}(I)$ defined

if $|I|=1$ and is the ffee groupfactor $L(F_{m})$ if $2\leq|I|=m\leq\infty$ .
32.1. We denote byrespect to the$C_{r}^{*}(A(I), \mu_{s})$by$\beta_{T}^{s}$$A(I)$$\mu_{s}$$C_{r}^{*}(A(I), \mu_{s})$-representation$\lambda^{s}$

the C’-algebra generated by

of $G(I)$ in 23.1, and bywith

the tracial state ofgiven by $\tau(\lambda^{s}(g))=0,$ $(e\neq g\in G(I))$ . For a $T\in GL(n, \mathbb{Z})$ , we denotethe automorphism of$C_{r}^{*}(A(I), \mu_{s})$defined in 2.12. The C’-algebra$N_{s}(I)$$\tau$$\lambda^{s}(A(I))$, and the automorphism$\alpha_{T}^{s}$induced ffom the automorphism$C_{r}^{*}(A(I), \mu_{s})$$\alpha_{\eta(T)}$ofis weakly dense in the factor

of $N_{s}(I)$ is the extension of$\beta_{T}^{s}$

.

67

68
Proposition. Let $n\geq 2$ , and let $T\in GL(n, \mathbb{Z})$ . For all $s\in[0, \pi/2]$

322.

irrational mod.

entropy$h_{\tau}(\cdot)$$2\pi$, the $Connes- St\phi rmer$ entropy$H(\cdot)$, Connes-Namhofer-Thining, Brown-Voiculescu’s topological entropy$ht(\cdot)$and$ht_{\tau}(\cdot)$

satisfy that.

$H( \alpha_{T}^{s})=h_{\tau}(\beta_{T}^{s})\leq ht_{\tau}(\beta_{T}^{s})\leq ht(\beta_{T}^{s})\leq\log(\prod_{i=1}\mu_{i}\mu_{i}’)$

Here $\mu_{i}=\max(|\lambda_{i}|, 1)$ andof the matrix$\mu_{i}’=\max(\frac{1}{|\lambda.|}., 1)$$T\in GL(n, \mathbb{Z})$, for the eigenvalue list$\{\lambda_{1}, \lambda_{2}, \cdots, \lambda_{n}\}$.

In par icular if $T\in SL(n, \mathbb{Z})$ , then$\log(\max\{\prod_{i=1}\mu_{i},\prod_{i=1}\mu_{i}’\})\leq H(\alpha_{T}^{s})=h_{\tau}(\beta_{T}^{s})\leq ht_{\tau}(\beta_{T}^{s})\leq ht(\beta_{T}^{s})\leq\log(\prod_{i=1}\mu_{i}\mu_{i}’)$3.2.3. Remark. We gave actions$\{\alpha^{s}\}_{s}$.

in Section 2, in order to obtain anon

cocycle conjugate continuous family of actions on the free group factors. However,ffom apoint of view of entropy theory, it would be interesting to treat the action
such that we can get the exact value of the entropy for each automorphism appearing

the action. As an example of such an action on the ffee group factors, we have thefollowings :Let
by$\tau_{i}$$I\subset \mathbb{Z}$, and let

$A_{i}$

the tracial state ofbe the coPy of the the group$C_{i}$

taking 0for

product $(C, \tau)=i\in I*(Ci, \tau i).$ Let$\pi_{i}(C_{i})$, where$\pi_{i}$$M_{i}$$\pi_{\tau}(C)$. ThenCorollary 5.3], where $m=|I|$ . Let

such that, then$\tau\cdot\gamma=\tau$$\tau_{i}\cdot\gamma_{i,T}=\tau_{i}$$g\neq 1$-algebra$C^{*}(\mathbb{Z}^{n})$. Denote. Consider the reduced freebe the von Neumann algebra generated byis the GNS-representation by

algebra generated by$T\in SL(n, \mathbb{Z})$$g\in \mathbb{Z}^{n},$$C^{*}$$M$$\tau_{i}$, and let $M$ be the von Neumann

is isomorphic to $L(F_{m})$ by Dykema [D2 :$\gamma i,T$be the automorphism of, and we have the automorphism$C_{i}$$\gamma\tau=*\gamma i,Ti\in I$(cf. [Ch3], [BD]). By the proof of Proposition 322,

$h_{\tau}( \gamma\tau)\leq ht_{\tau}(\gamma\tau)=ht_{\tau}.\cdot(\gamma_{\mathrm{i}},\tau)=h_{\tau}\dot{.}(\gamma_{i},\tau)=\log(\prod\mu_{i})$$i=1$induced by,

of $C$

69
where $\{\mu_{i}; 1\leq i\leq n\}$ are the same as in Proposition 322. We denote by$\hat{\gamma}_{i},\tau)$

the extension of(resp.

$\gamma\tau$$\hat{\gamma}_{i},\tau$) to $M$ (resp.

$M_{i}$$\hat{\gamma}\tau$(resp.

). Then.

$\log(\prod_{i=1}^{n}\mu_{i})=H(\hat{\gamma}_{i,T})\leq H(\hat{\gamma}\tau)=h_{\tau}(\gamma_{T})$Thus we have the action$\hat{\gamma}$of$SL(n, \mathbb{Z})$on $L(F_{m})$ such that

$H( \hat{\gamma}\tau)=\log(\prod_{i=1}^{n}\mu_{i})$,$\forall T\in SL(n,\mathbb{Z})$.

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