数理解析研究所講究録 1354 巻 2004 年 1-9

ENTROPY FOR AUTOMORPHISMS
OF THE CROSSED PRODUCTS
エルゴ-ド理論におけるエントロピーから非可換力学系でのエントロピーへ
大阪教育大学長日まりゑ (Marie CHODA)

Department of Mathematics, Osaka Kyoiku University

Asahigaoka, Kashiwara, Osaka, 582-8582, Japanmarie@cc.osaka-kyoiku.ac.jp.ac.jp$\mathrm{m}\pi \mathrm{i}\mathrm{e}@\mathrm{c}\mathrm{c}.\mathrm{o}\mathrm{s}\mathrm{a}\mathrm{k}\mathrm{a}-\mathrm{k}\mathrm{y}\mathrm{o}\mathrm{i}\mathrm{k}\mathrm{u}$Abstract. Let$G\subset \mathrm{A}\mathrm{u}\mathrm{t}(A)$be a discrete group which is exact, that is, admits an

amenable action on some compact space. Then the entropy of an automorphism of
the algebraA$\mathrm{x}$$G$. This$A$$\mathrm{i}_{8}$does not change by the canonical extension to the crossed productshown for the topological entropy of an exact C’-algebra

the dynamical entropy of an AFD von Neumann algebra

to the case of transformations on$\mathrm{L}\mathrm{e}\mathrm{b}\mathrm{e}8\mathrm{g}\mathrm{u}\mathrm{e}$

$A$

$A$

and for. These have applications

spaces.

1. INTRODUCTION
The notion of the Kolmogoroff-Sinai entropy in the ergodic theory was brought

into the theory of finite von Neumann algebras by Connes-Stormer ([10]),commutative extension. Replacing the finite$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{a}\mathrm{e}$a non-

to a state , it was extended to$\phi$

general von Neumann algebras and to C’-algebras by Connes-Narnhofer-Thirring([9]). In this paper we call the Connes-Stormer entropy the-entropy and the$\mathrm{C}\mathrm{S}$Connes-Narnhofer-Thirring entropy the CNT-entropy. We denote byentropy and the by$h\phi(\cdot)$$H(\cdot)$

the CS-

the CNT-entropy

In the ergodic theory, we are given a probability space $(X, \mu)$ together with asure preserving nonsingular transformation
Neumann algebra$\alpha\tau$of$L^{\infty}(X,\mu)$$L^{\infty}(X,\mu)$with the$T$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\tau_{\mu}$of $X$ . Then we have the abelian voninduced by$\mu$and the automorphism

induced by . In this setting, the Connes-Stormer entropy

with respect to the$T$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\tau_{\mu}$$\mathrm{m}\mathrm{e}\mathrm{a}$$H(\alpha\tau)$

is nothing but the Kolmogoroff-Sinai entropy $h(T)$ .

The noncommutative algebra $M$ is given from this dynamical ststem $(X,\mu,T)$ by

taking the crossed product$M=L^{\infty}(X, \mathrm{u})$naturally to the automorphismof$\tau_{\mu}$$\mathrm{x}_{\alpha}$

Z. The automorphism$\alpha\tau$is extendedof $M$, and it preserves the natural extension$\overline{\alpha\tau}$$\overline{\tau_{\mu}}$. As a logical consequence, the following question was suggested by Stormer in([17]) : Do we have$H(\overline{\alpha\tau})=h(T)$?

The first positive answer is due to Voiculescu. He showed that
$H(\overline{\alpha\tau})=h(T)=\log n$

for the ergodic measure preserving Bernoull transformation $T$ on the space $(X, \mu)$ ,where $X$ is the product space 1,$\{$$\mu_{n}^{\otimes \mathrm{Z}}$. Here$\mu_{n}\mathrm{i}_{8}$,$\cdots$$n\}^{\mathrm{Z}}$and the measure$\mu \mathrm{i}_{8}$the product$\mathrm{m}\mathrm{e}\mathrm{a}8\mathrm{u}\mathrm{r}\mathrm{e}$the equal weights probability measure on the set $\{1, \cdots, n\}$ .

It was an application of the$\mathrm{r}\mathrm{e}8\mathrm{U}\mathrm{l}\mathrm{t}$on his topological entropy$ht(\cdot)$

introduced in

the paper ([20]) for automorphisms of nuclm C’-algebras. After then, Brown [3]extended the notion to automorphisms of more large class ofexact$\mathrm{C}^{*}- \mathrm{a}1\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}8$abelian von Neumann algebra$\mu$$\mathrm{C}^{*}$-algebras, that is,.

Let us replace the integer groupstate , or a$\mathrm{C}^{*}$-algebra$A$

with respect to an action$\mathrm{Z}$to a discrete group$L^{\infty}(X,\mu)$$\mathrm{C}^{*}-$, and let us replace theto a general von Neumann algebra $M$ with. Then we have the von Neumann crossed product$\alpha$of$G$$\mathrm{a}$$M\mathrm{x}_{\alpha}G$

on $M$ with$\mu\circ\alpha_{g}=\mu$

and also we have the

$G$

,

for

crossed product$A$$\mathrm{a}\mathbb{I}$$g\in G$with$\mathrm{x}_{\alpha}G$$\mathrm{r}\mathrm{e}8\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}$to an action$\alpha$

of

$G$

on .
$A$

In the case of von Neumann algebras, the state$M\mathrm{x}_{\alpha}G$$\mu$which is -invariant. If an automorphism$\overline{\theta}$$\alpha_{g}fi=\theta\alpha_{g}$,$\theta$

has the natural extension$\overline{\mu}$toof $M$ satisfies thatfor all $g\in G$

and$\mu\circ\theta=\mu$,then 0 can be canonically extended to the automorphism
following problem naturaly arises :$h_{\mu}(\theta)=ht_{\overline{\mu}}(\overline{\theta})$?

$\overline{\theta}$of$M\mathrm{x}_{a}G$, and the

$\theta$

Similarly in the case of C’-algebras, if an automorphism 0 of$A$satisfies

for all $g\in G,$ then it is canonically extended to the automorphism of A$\alpha_{g}\theta=\theta\alpha_{g}$$\overline{\theta}$$\mathrm{x}_{\alpha}G$, andthe following problem naturally arises :?

$ht(\theta)=ht(\overline{\theta})$

When

$G$

is amenable, known results for these two problems are as follows:

Theorem. [6, 11, 13]. Assume that$G$

is an amenable discrete countable group.

(1) $[6, 11]$ . Let $A$ be a unital exact C’ algebra andan automorphism of$A$such that$\alpha\theta=\theta\alpha gg$$\alpha$an action of$G$

on A. If$\theta\dot{l}S$

for all $g\in G$ , then$h_{\phi}(\theta)=h_{\overline{\phi}}(\overline{\theta})$.

(2) [13]. Let $M$ be an approximately finite-dimensional von Neumann algebra witha no rmal state $ and at an action of

If is an automorphism of$\theta$$M$$G$

on $M$ withsuch that$\phi\cdot$$\phi 0\theta=\phi$$\alpha_{g}=6$

and

for all $g\in G.$
for all $g\in G,$$\alpha_{g}\theta=\theta\alpha_{g}$then$h_{\phi}(\theta)=h_{\overline{\phi}}(\overline{\theta})$where$\overline{\phi}$is the canonical extension of$\phi$

to,$M\mathrm{x}_{a}G$.

There are a large class of interesting non amenable discrete groups suchgroups$F_{n}$, $n\geq 2$ and$\mathrm{d}\mathrm{i}_{8}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{e}$subgroups of connected Lie groups, etc..

$\mathrm{a}\mathrm{e}$ffaeHowevereach of these nonamenable groups has an amenable action on some compact space([1, 2, 15]).

A discrete group
if

$G$

$G$

has an amenable action on some compact space if and onlyis exact in the sense of Kirchberg and Wassermann ([14]), that is, its reducedgroup C’-algebra$\mathrm{C}_{f}^{*}(G)$is exact. This is first proved by Ozawa in [15].

Here, we report our results which show that the amenability of $G$ is not always

necessary and it is replaced to more large class of groups, that is, exact$\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}_{8}$.

2. BASIC NOTATIONS AND TERMINOLOGIES.
Proo of the main results are given using partly some
Here we only denote some basic notations and terminologies.

$\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{d}_{8}$$\mathrm{f}\mathrm{s}$in [4, 5, 6, 7, 13].

2.1. Approximation property and exactness. Here ourseparable. Let $M$ be a von Neumann algebra (resp. unital$\mathrm{C}^{*}$$\mathrm{C}^{*}$-algebras are all-algebra). Then $M$ iscalled approximately finite dimensional if there exists an increasing sequence$(N_{k})_{k}$of finite dimensional subalgebras such that JINk is weakly (resp. norm) dense in$M$.

This approximation property is extended in the case of C’-algebras in $[14, 21]$$\mathrm{a}\mathrm{e}$exactness.

A$\mathrm{C}^{*}$-algebra is exact if there exists a representationand triplets$(\varphi_{\iota},B_{\iota}, lj_{\iota})_{\iota}$of finite dimensional algebras$p$,:$Aarrow B_{\iota}$,$\psi_{\iota}$

:

$\pi$

$B_{\iota}$

of

$A$

on a Hilbert space $H$, conpletely positive maps

$B_{\iota}\vdash*B(H)$such that$||\pi(a)-\psi_{\iota}\cdot\varphi_{\iota}(a)|$I$arrow$$0$

for all $a\in A.$

A discrete groupis called exact if the C’-algebra$G$$\mathrm{C}_{f}^{*}(G)$generated by the left

regular representation is exact.

is defined for an automorphism

2.2. Entropy. Topological entropy$ht(\theta)$

exact C’-algebra.

is defined for an automorphism

entropy$\mathrm{C}\mathrm{S}$$H(\alpha)$

Neumann algebra $M$ with a finite

is defined for an$\phi\cdot$$\theta=\phi$such that$\mathrm{r}$$\phi$

representation$\overline{\theta}$a.

such that$\theta=\phi$$\pi \mathrm{p}(A)$of $M$ naturally$\mathrm{f}\mathrm{i}\mathrm{n}\cdot \mathrm{t}\mathrm{e}$$\tau$

such that$\phi\cdot$$\alpha=\tau$of a unital C’-algebra[$\tau\cdot$. Let$\mathrm{c}*$$M$$=\mathrm{r}$and CNT-entropy

$A$

and

$\mathrm{C}\mathrm{N}\mathrm{T}$. Then such a as

If$\theta$$\phi$$h_{\phi}(\theta)$$\phi\cdot$$\theta=\phi$of anof a finite vonentropywith a state$H(\alpha)$$\phi$$h\phi(\theta)$

such that

depends on

ako depends on a state

be the von Neumann algebra generated by the$\mathrm{G}\mathrm{N}\mathrm{S}$is extended to the autonorphism

is a tracial state, then the natural$\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}8\mathrm{i}\mathrm{o}\mathrm{n}\overline{\phi}\mathrm{i}8$a$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$ofvon Neumann algebra $M$ and$H(\overline{\theta})=h_{\phi}(\theta)$

If$\tau\cdot$. They are both called dynamical entropies. CS-entropy

a finite traoe$\phi$$\mathrm{a}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}_{8}\mathrm{m}\theta$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\alpha$$\theta$$\theta=\phi$, then topological entropy$ht(\theta)$

and.

$\mathrm{C}\mathrm{N}\mathrm{T}$relation :$h_{\phi}(\theta)\leq ht(\theta)$.

entropy$h\phi(\theta)$

has the following

5

We refer these [3, 10, 9, 20]

2.2.1. Topological entropy. We refer $[3, 20]$ for definitions and notations about the
topological entropy.

2.2.2. Dynamical entropy. We refer $[10, 9]$ for definitions and notations about the
topological entropy.

2.3. Crossed product. Let A be a unutal
gebra), G a discrete countable group and$\mathrm{C}’$$\alpha$-algebra (resp. von Neumann al-

be an action of G on A, that is ahomomorphism from G to the automorphism group$\mathrm{A}\mathrm{u}\mathrm{t}(A)$of A. We may assume

that A is acting on a Hilbert space H faithfully. The crossed product A$\mathrm{C}^{*}$$\mathrm{x}_{\alpha}$

G is the-subalgebra (re8p. von Neumann subalgebra) of$B(l^{2}(G,H))\cong B(l^{2}(G))\otimes B(H)$generated by$\pi_{\alpha}(A)$and$\lambda_{G}$, where$\pi_{\alpha}(a)\xi(g)=\alpha_{\mathit{9}^{-1}}(a)\xi(g)$,(a $\in A,g\in G,\xi\in l^{2}(G,H))$

and$\lambda_{g}\xi(h)=\xi(g^{-1}h)$$\mathrm{E}\mathrm{s}\mathrm{s}$(a $\in A,g\in G, \xi\in l^{2}(G, H))$ .

,entially, we use the following representation as in [3, 4, 6, 7, 13, 19] :$\pi_{\alpha}(a)\lambda_{g}=\sum_{t\in G}e_{t,g^{-1}t}\otimes\alpha_{t}^{-1}(a)$where (

Since
onto

$e_{e,t}\}_{s,tfG}$$G$,$(a\in 4,g\in G)$ ,is the standard matrix units in $B(l^{2}(G))$ .

is discrete, there exists always the conditional expectation $E$ of$\pi_{\alpha}(A)$$A$$\mathrm{x}_{\alpha}G$such that$E(\lambda_{g})=0$

for$\mathrm{a}\mathbb{I}$$g\in G$

except the unit. If

denote the state
If$\theta\in \mathrm{A}\mathrm{u}\mathrm{t}(A)$$\phi\circ E$

by$\overline{\phi}$$\phi$

commutes with$\overline{\theta}(r_{\alpha}(a)\lambda g)$$\overline{\theta}$

$A$

with

$\phi\circ\alpha_{g}=\phi$and call it the canonical extension of$\mathrm{a}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}_{8}\mathrm{m}\overline{\theta}\in \mathrm{A}\mathrm{u}\mathrm{t}(A\mathrm{x}_{\alpha}G)$

We call theis a state of$\alpha_{g}$for$\phi$$\mathrm{a}\mathrm{H}$to A$g\in G$$\mathrm{x}_{\alpha}G$

, we

.

for all $g\in G,$ then there exists always ansuch that$=ra(\theta(a)):*g$

the canonical extension of .
$\theta$,$(a\in A,g\in G)$.

2.4. Amenability. The notion of amenablity for groups is generalized to amenability of actions of groups, that is, a group admits an amenable action on some compact

space (cf. [1, 2, 5, 14]).
For example, the discriptions in [1] and [5] are as follows :2.4.1. Amenable action, ([5]) :
Let

$G$

be a countable discrete group, and let$\alpha^{G}$

be the action : $Garrow$ Aut(l”(G))

given by$\alpha_{g}^{G}(x)(h)=x(g^{-1}h)$Let$l^{1}(G, l^{\infty}(G))$$T:Garrow l^{\infty}(G)$,$(x\in l^{\infty}(G), g, h\in G)$.

be the closure of the linear space of finitely supported functions

with respect to the norm$||T|\mathrm{h}$$=||$$1$ $|T(g1$ $||l^{oo}(G)$ .
$g$

Let us put$s.T(g)=\alpha_{s}^{G}(T(s^{-1}g))$

The action(1)$T_{n}$$\alpha^{G}$,$(s,g\in G)$

is amenable if there exist functions

is nonnegative (i.e. Tn )$\{\mathrm{g}$$\geq 0$.

$T_{n}\in l^{1}(G, l^{\infty}(G))$

such that, $(g\in G))$ ,(2) finitely supported,(3) $\sum_{g}T_{n}(g)=1_{l^{\infty}(G)}$ and(4) $||s.T\mathrm{n}-Tn||_{1}arrow 0$ for all $s\in G.$

2.4.2. Amenable at infinity. ([1]) :A group is amenable at infinity if and only if there exists a sequence$G$nonnegative functions on$G\mathrm{x}G$$(g_{n})_{n\geq 1}$

with support in a tube such thata) for each yt and each 8,$\sum_{t}g_{n}(s,t)=1,$b) uniformly on tubes,$\lim_{u}\sum_{\in G}n|g_{n}($” $u)-g_{n}(t, u)|=0.$

Here, a tube means the set $\{(s, t) : s^{-1}t\in F\}$ for some finite suset $F$ of $G$ .

of2.4.3. Equivalence. These two notions of 3.4.1 and 3.4.2 are equivalent. In fact, let$(T_{n}(t))(s)=g_{n}(s^{-1}, s^{-1}t)$for$\mathrm{a}\mathbb{I}$$s$, $t\in G,$ then conditions in one side are implied by the other side.

A group$G$is exact if

admits an amenable action on a compact space ([15])$G$

which is equivalent to thatto that2.4.4.

$\alpha^{G}$$G$

is amenable at infinity ([1]) and also it is equivalentamenable.

$\mathrm{i}$$\infty p\dot{|}cal$

examples of exact groups.

(1) Amenable groups.

(2) Free groups.

(3) Discrete subgroups of connected Lie groups.
(4) Subgroups, extensions, free products of the above groups.
(5) Quotients by classical amenable groups

3. MAIN RESULTS
Our results are followings :3.1. Theorem. Let
group, andIf$\alpha$$A$be a unital exact C’-algebra,

an action of$\beta\in Aut(A\mathrm{x}_{\alpha}G)$

$G$

on

satisfies

$A$

$G$

an exact discrete countable.

$\beta(\lambda_{g})=\lambda_{g}$for all $g\in G$ and$\beta(\pi_{\alpha}(A))=\pi_{\alpha}(A)$,then$ht(\beta)=ht(\beta|_{\pi_{\alpha}(A)})$

Herethat A$\pi_{\alpha}$

is the representation of$\mathrm{x}_{\alpha}G$is generated by$A$.

and A is the unitary representation of$\{\mathrm{n}\{\mathrm{A}), \lambda_{G}\}$

$G$

such

.

3.2. Remark. We have more general result on the topological entropy. In fact, by
replacing the condition that$\beta(\lambda_{g})=$$\mathrm{X}_{g}$for aU$g\in G$

to the condition that$\beta(\lambda_{G})=\lambda_{G}$we have a similar result in [7]. This gives an application to the proof of the main
theorem in [12].

$\epsilon$

3.3. Theorem. Let $M$ be an approirnately finite-dimensional von Neumann algebra with a normal state ,$\phi$$G$$G$

an exact discrete countable group, and$\alpha$an action of

on $M$ withfor all $g\in G.$$\phi\cdot\alpha_{g}=\phi$

If is an automorphism of$\theta$$M$such that$\phi\circ\theta=\phi$andfor all $g\in G,$$\alpha_{g}\theta=\theta\alpha_{g}$then$h_{\phi}(\theta)=h_{\overline{\phi}}(\overline{\theta})$where$\overline{\phi}$is the canonical extension of$\phi$

to,$M\mathrm{x}_{\alpha}G$.

3.4. Proof. Proofs for these results are in [8]. In [8], we adopt as exactness of the

group

$G$

the amenability of the canonical action$\alpha^{G}$in [8].

$\square$

REFERENCES[1] C. Anantharaman-Delaroche,

Amenability and exactness for dynamical systems and their$C^{*}-$

algebras, Trans. Amer. Math. Soc, 354 (2002), 4153-4178.
[2] C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographs of L’Enseignement

Mathematique, 36 Geneva, 2000.
[3] N. Brown, Topological entmpy in exact$C^{*}$-algebms, Math. Ann., 314 (1999), 347-367.

[4] N. Brown and M. Choda, Approximation entropies in crossed products with an application to free$sh|.fl\sigma$, Pacific J. Math., 198 (2001), 331- 346.

[5] N. Brown and E. Germain, Dual entropy in discrete groups with amenable actions, Erg. Th. andDynam. Sys., 22 (2002), 711-728.

[6] M. Choda, Entropy of Crossed Products and Entropy of Free Products, J. Operator Theory, 48 (2002),

356-367.
[7] M. Choda, Dynamical entropy for automorphisms on exact C’-algebras, J. Funct. Anal., 198 (2003),

481-498.
[8] M. Choda, Bntroy of automorphisms arising from the dynamical rytitems through discrete groups

with amenable$act|.\circ n\iota$, Preprint.

[9] A. Connes H. Narnhofer and W. Thirring, Dynamical entropy of$C^{\mathrm{s}}$algebras and von Neumann

algebras, Coramun. Math. Phys., 112 (1987), 691-719.

[10] A. Connes and E. Stormer, entropy of II1 von Neumann algebras, Acta Math., 134 (1975), 289346.
[11] K.J. Dykema and D.Shlyakhtenko, Exactness of Cuntz-Pimsner $C’$-algebtea, Proc. Edinburgh Math.

soc., 44 (2001), 425-444.

[12] V. Ya. Golodets and M.S.Boyko, Dynamical entropy for a das’ of algebraic origin automorphisms,

Mat. Fiz. Anal. Geom., 8 (2001), 385-391.

$\epsilon$[13] V. Ya. Golodets and S.V. Neshveyev, Entropy of automorphisms of$II_{1}$-factors arising fmm the dy-

namical systems theory, J. Punct. Anal., 181 (2001), 14-28.
[14] E. Kirchberg and S. Wassermann, Exact groups and continuous bundles of C’-algebras, Math. Ann.,

315 (1999), 169-203.
[15] N. Ozawa, Amenable actions and exactness for discrete groups, C.
I Math.,C. R. Acad. Sci. Paris Ser. 1Math.,$aet*\cdot ons$

SS0 (2000), 691-695.
[16] J. Peters, Entropy on discrete akliun
abelian groups, Adv. Math., SS (1979), 1-13.

[[17] E. Stormer,, Entropy in operator algebras,, Asterisque No. 232 (1995), 211-230.
$1\eta$$\mathrm{S}\mathrm{t}\emptyset \mathrm{m}\mathrm{e}\mathrm{r}$$\mathit{0}\mu mtor$$algebm\epsilon$$\mathrm{A}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}_{8}\mathrm{q}\mathrm{u}\mathrm{e}$mm\mathrm{u}tat|.\mathrm{t}/edynamcal$[18] E. Stormer,, A sumey
C’-algebras,
survey of $non\omega
entropy. Clasrificatl.on
dynamical entmpy.
noncommutative
Classification of nuclear C.-dgebms.
$\mathrm{S}\mathrm{t}\emptyset \mathrm{r}\mathrm{m}\mathrm{e}\mathrm{r}$Entropy |.n
Bntmpyalgebras,, 147-198, EncyclopaedIa
in opemur
operator $algebm\sigma$
Encyclopaedia Math. Sci., 126, Springer, Berlin, 2002.
[19] A.M. Sinclair and R.R. Smith, The completely
pmpehy for discrete crossed
cornpletdy bounded approximation property$appr\sigma ae|.m\mathrm{n}t|.on$$\alpha n\epsilon s\epsilon d$

products, Indiana Univ. Math. J., 46 (1997), 1311-1322.
pmducls,$\mathrm{I}\mathrm{n}\mathrm{d}\dot{\mathrm{t}}\mathrm{a}\mathrm{n}\mathrm{a}$lgebms$ , Com[20] D. Voiculescu, Dynamical approximation entropies and topalogial
algebras,
topological entropy in operator $\iotaPhys., 170 (1995), 249-281.
mun. Math. Phy8.,249281.
[21] S. Wassermann, Exact C’-algebratt-algebms and related topics, Seoul National University Lecture Notes Series,$C^{*}$

19 (1994).