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These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

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\rightheadtext{The intrinsic invariant of an AFD factor}

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\topmatter
\title The intrinsic invariant of an approximately
finite dimensional factor and the cocycle conjugacy
of discrete amenable group actions \endtitle

%\bigskip
%\centerline{\bf THE INTRINSIC INVARIANT OF AN APPROXIMATELY}
%\centerline{\bf FINITE DIMENSIONAL FACTOR}
%\centerline{\bf AND THE COCYCLE CONJUGACY OF DISCRETE}
%\centerline{\bf AMENABLE GROUP ACTIONS}
%\vskip .6in
%\centerline {\bf ANNOUNCEMENT}
%\medskip\centerline{\bf May,  1995}
%\vskip .6in

\author Yoshikazu Katayama,
 Colin E. Sutherland,
 Masamichi Takesaki \endauthor

%\author Colin E. Sutherland \endauthor

%\author Masamichi Takesaki \endauthor

%\communicatedby{}

\address Yoshikazu Katayama, Department of Mathematics,
Osaka Kyoiku University, Osaka, Japan. \newline\hbox{\hskip\parindent}
{\it E-mail address}:{\rm\ F61021\@sinet.adjp}
\endaddress

\address Colin E. Sutherland, Department of Mathematics,
University of New South\newline
Wales, Kensington, NSW, Australia. \newline\hbox{\hskip\parindent}
{\it E-mail address}:
{\rm\ colins\@solution.maths.unsw.edu.au} \endaddress

\address Masamichi Takesaki, Department of Mathematics,
University of California, Los Angeles, Califnornia 90024-1555. \newline
\hbox{\hskip\parindent}{\it E-mail address}:{\rm\ mt\@math.ucla.edu} \endaddress

%\email  F61021\@sinet.adjp \endemail
%\email colins\@solution.maths.unsw.edu.au \endemail
%\email  mt\@math.ucla.edu \endemail

\cyear{1995}
\cvolyear{1995}
\cvol{1}
\cvolno{1}

\date May 17, 1995 \enddate

%\centerline{\bf Yoshikazu Katayama}
%\centerline {Department of Mathematics}
%\centerline {Osaka Kyoiku University}
%\centerline {Osaka, Japan}
%\bigskip
%\centerline {\bf Colin E. Sutherland}
%\centerline {Department of Mathematics}
%\centerline {University of New South Wales}
%\centerline {Kensington, NSW, Australia}
%\bigskip
%\centerline {\bf Masamichi Takesaki}
%\centerline {Department of Mathematics}
%\centerline {University of California}
%\centerline {Los Angeles, Califnornia 90024-1555}

\subjclass 46L40 \endsubjclass

\thanks  This research is supported in part by NSF Grant
DMS92-06984 and
DMS95-00882, and also supported by the Australian Research Council Grant
\endthanks

%\footnote""{{\it 1980 Mathematics Subject Classification
%(1985 Revision).} 46L40.  This research is supported in part by NSF Grant
%DMS92-06984 and
%DMS95-00882, and also supported by the Australian Research Council Grant.}

\abstract
{We announce in this article that i) to each approximately finite dimensional
factor $\r$ of any type there corresponds canonically a group cohomological
invariant, to be called the {\bf intrinsic invariant} of $\r$ and denoted
$\Theta(\r)$, on which $\Aut(\r)$ acts canonically; ii) when a group $G$
acts on $\r$ via $\a: G \ \mapsto \ \Aut(\r)$, the pull back of Orb($\Theta
(\r)$), the orbit of $\Theta(\r)$ under $\Aut(\r)$,by $\a$ is a cocycle
conjugacy invariant of $\a$; iii) if $G$ is a discrete countable amenable
group, then the pair of the module, mod($\a$), and the above pull back is a
complete invariant for the cocycle conjugacy class of $\a$.  This result
settles the open problem of the general cocycle conjugacy classification of
discrete amenable group actions on an AFD factor of type \threeone, and
unifies known results for other types.}
\endabstract

\endtopmatter

\document

\advance\pageno by 42

%\newpage
\centerline {\bf Introduction}
\bigskip
The celebrated work of Connes, \cite{1,3}, surveyed in \cite{2}, Ocneanu's
analysis, \cite {12},
and the previous work of Kawahigashi, and the second and third authors,
\cite{11}, reveal the
beatiful structure of the automorphism group of an approximately finite
dimensional, or AFD, factor $\r$.  In this note, we announce that it is
possible to describe
the structure of Aut($\r$) independently of the type of $\r$.  This structure
of $\Aut(\r)$ enables us to define a cohomological invariant which we call
the {\bf intrinsic
invariant} of $\r$.  If a group G acts on $\r$ via  $\a$, then the pull
back of the orbit of the
intrinsic invariant of $\r$ under the natural action of the group Aut($\r$)
gives a cocyle
conjugacy invariant which is complete if G is a countable discrete amenable
group.  This
completes the cocylce conjugacy classification of discrete amenable group
actions on an AFD
factor $\r$ including the type \threeone \ case.
\bigskip
\centerline{\bf Intrinsic Invariant and Main Theorem}
\bigskip
Let $\r$ be an AFD factor, and let Aut($\r$) and Int($\r$) be the group of
automorphisms
and the group of inner automorphisms respectively.  Let
$$
\e: \a \in \text{\rm Aut}(\r) \longmapsto \dot \a \in \text{\rm Out}(\r)
= \text{\rm Aut}(\r)/\text{\rm Int}(\r)
$$
be the canonical quotient map and set
$$
\text{\rm Cnt}(\r) = \e^{-1} \text{\rm{(Center of Out}($\r$))}.
$$

As $\text{\rm Int}(\r)$ is not closed in $\text{\rm Aut}(\r)$, we
need to consider its closure $\overline{\text{\rm Int}}(\r)$ and the
quotient group $\text{\rm Mod}(\r) = \text{\rm Aut}(\r)/\overline{\text{\rm
Int}}(\r)$.  We denote the canonical quotient map by $\text{\rm mod}: \a \in
\text{\rm Aut}(\r) \longmapsto \text{\rm mod}(\a) \in \text{\rm Mod}(\r)$.
The map mod
will be called the $module$ and the image $\text{\rm mod}(\a)$ of $\a \in
\text{\rm Aut}(\r)$ in $\text{\rm Mod}(\r)$ the $module$ of $\a$.  Two more
groups
and a map are evidently associated with $\r$: the unitary group $\U(\r)$, its
center
$\bold T$ which is the one dimensional torus group of complex numbers of
modulus one, and the adjoint map:
$$
\text{\rm Ad}: u \in \U(\r)\longmapsto \text{\rm Ad}(u) \in \text{\rm Int}(\r).
$$
Of course, $\T$ is the kernel of the map $\text{\rm Ad}$.  Finally, we need to
consider the flow of weights $F(\r)$ and its cohomology groups: $B^1(F(\r)),
 \ Z^1(F(\r))$ and $H^1(F(\r))$.  It is known,\cite{5}, that $\text{\rm Mod}
(\!\r\!)\! =\! \text{\rm Aut}(\!\r\!)\!/\overline{\text{\rm Int}}(\!\r\!)$ is
canonically identified with $\text{\rm Aut}(F(\r))$.  By the work of Wong,
\cite{15}, the short exact sequence:
$$
1 \longrightarrow \Intb(\r) \longrightarrow \Aut(\r) \longrightarrow \Mod(\r)
\longrightarrow 1
$$
splits, but not canonically. By \cite{11; Theorem 1} and \cite{5},
there exists a canonical isomorphism from
$H^1(F(\r))$ onto the Center of $\text{\rm Out}(\r)$, which will be denoted by
$\delta$.  These groups and maps are related as described in the
following commutative diagrams of exact sequences:
$$
\align
& \qquad\quad \text{\rm Cnt}(\r) \qquad \longrightarrow \qquad \text{\rm
H}^1(F(\r))\\
& \nearrow \hskip 1in \qquad\qquad\qquad\qquad\qquad \searrow \\
1\quad \longrightarrow \quad \text{\rm Int}(\r) & \hskip .6in \downarrow
\hskip 1.2in \downarrow \ \delta \hskip .8in 1. \\
& \searrow \hskip 1in \qquad\qquad\qquad\qquad\qquad \nearrow \\
& \qquad \quad \text{\rm Aut}(\r) \qquad \overset \varepsilon \to
\longrightarrow \hskip .3in \text{\rm Out}(\r) \\
\endalign
$$
The sequence involving $\Cnt(\r)$ forms part of an exact square, as follows:
$$
\align
& \ 1 \hskip .9in 1 \hskip 1in 1 \\
& \downarrow \hskip .9in  \downarrow \hskip 1in \downarrow \\
1 \ \ \longrightarrow \ \ & {\text{\rm\bf T}} \ \ \ \ \ \ \ \longrightarrow \
{\Cal U}(F(\r)) \ \ \overset \partial \to \longrightarrow \ \text{\rm
B}^1(F(\r)) \ \
\longrightarrow \hskip .15in 1 \\
& \downarrow \hskip .9in \downarrow \hskip 1in \downarrow \\
1 \ \ \longrightarrow \ \ & {\Cal U}(\r) \hskip .16in \longrightarrow
\hskip .15in
\widetilde{\Cal U}(\r)\hskip .1in \longrightarrow \ \ \ \text{\rm
Z}^1(F(\r))) \hskip .08in
\longrightarrow \ \ 1 \\
& \downarrow \hskip .9in \downarrow \hskip 1in \downarrow \\
1 \ \ \longrightarrow \ \ & \text{\rm Int}(\r) \ \ \longrightarrow
\text{\rm Cnt}(\r) \ \ \underset \delta^{-1}\circ\varepsilon \to
\longrightarrow \ \text {\rm H}^1(F(\r)) \ \ \ \longrightarrow \ \ 1 \\
& \downarrow \hskip .9in \downarrow \hskip 1in \downarrow \\
& \ 1 \hskip .85in \ 1 \hskip 1in 1 \\
\endalign
$$
Here, $\widetilde{\U}(\r)$ is the semi direct product of $\U(\r)$ by the
extended modular
action of $\text{\rm Z}^1(F(\r))$ as in \cite{14}.  Except for the lower
right corner
$\text{\rm H}^1(F(\r))$, all groups are Polish and all maps are continuous.
As the above
square of exact sequences is canonical, $\text{\rm Aut}(\r)$ acts on the
square, i.e.
the above sequare is an equivariant square under the action of $\text{\rm
Aut}(\r)$.
Let $\nu$ denote the map $\delta^{-1}\circ \varepsilon$, called the
$modular$ $invariant$.
The middle vertical $\text{\rm Aut}(\r)$ equivariant exact sequence of the
above exact
square:
$$
1 \ \ \longrightarrow \ \ {\Cal U}(F(\r)) \ \ \longrightarrow
\widetilde{\Cal U}(\r)
 \ \ \longrightarrow \text{\rm Cnt}(\r) \ \ \longrightarrow\ \ 1 \
$$
\newline
gives rise to a cohomological invariant, called the $characteristic$
$invariant$ \newline
$\chi \in \Lambda(\text{\rm Aut}(\r), \text{\rm Cnt}(\r),{\Cal
U}(F(\r)))$.  Thus we have
the triplet:
$$\align
(\mod, \chi, \nu) \in \Hom(\Aut(\r), \Aut(F(\r)) &\times
\Lambda(\text{\Aut}(\r),
\text{\rm Cnt}(\r), {\Cal U}(F(\r))) \\ &\times\ \text{\rm
Hom}_{\text{\Aut}(\r)}
(\text{\rm Cnt}(\r), \text{\rm H}^1(F(\r))),
\endalign$$
consiting of the action mod of $\Aut(\r)$ on $F(\r)$, the characteristic
invariant and the
$\text{\rm Aut}(\r)$-equivariant homomorphism $\nu$,
which will be called the $intrinsic$ $invariant$ of the AFD factor $\r$ and
denoted by
$\Theta(\r)$.  Naturally, $\text{\rm Aut}(\r)$ acts on $\text{\rm
Hom}(\Aut(\r)\hskip-2pt,$ $\Aut(F(\r)))$,
$\Lambda(\text{\rm Aut}(\r), \text{\rm Cnt}(\r), \U(F(\r))))$ and
$\text{\rm Hom}_{\text{\rm Aut}(\r)}(\text{\rm Cnt}(\r)), \text{\rm
H}^1(F(\r)))$.
Let $\text{\rm Orb}(\Theta(\r))$ denote the orbit of $\Theta(\r)$ under the
actionof
$\text{\rm Aut}(\r)$.

We are now at the position to state the main result:
\proclaim{Theorem 1}
Let $\r$ be an approximately finite dimensional separable factor and $G$ be
a countable
discrete amenable group.  If $\a$ is an action of $G$ on $\r$, then the
pull back
$\a^*(\text{\rm Orb}(\Theta(\r))$ of the orbit of the intrinsic invariant
is a complete
invariant of the cocylce conjugacy class of $\a$.  More precisely, the
inverse image,
$N(\a) = \a^{-1}(\text{\rm Cnt}(\r))$, of $\text{\rm Cnt}(\r)$ under $\a$,
the action
$\text{\rm mod} \circ \a$ of G on $F(\r)$, the characteristic invariant
$\chi(\a) \in
\Lambda(G, N(\a), {\Cal U}(F(\r)))$ of $\a$ which is obtained as the
pull back of
$\Theta(\r)$ and $\nu_{\a} \in \text{\rm Hom}_G(N(\a), {\Cal U}(F(\r)))$
determine
the cocycle conjugacy class of $\a$.
\endproclaim
It should be noted that this one theorem applies to all AFD factors of {\bf
any type}.  Of course,
the type of the carrier factor $\r$ affects on the nature of these
invariants.   We list the
the invariants in each type as follows:

\noindent
${\text{\rm Type I}_n}, n \in {\text{\rm {\bf N}}}$:
$$
\Int(\r) = \Cnt(\r) = \Aut(\r) \ \ is\  \ compact
$$
$$
{\text{\rm Mod}}(\r) = 1
$$
$$
G = N(\a), \ \chi_{\a} \in \text{\rm H}^2(G, \T), \nu_{\a} = 1.
$$
\noindent
${\text{\rm Type}}\ \text{\rm I}_{\infty}$:
$$
\Aut(\r) = \Cnt(\r) = \Int(\r) \ \ is \ \ not \ \ compact.
$$
$$
\text{\rm Mod}(\r) = 1
$$
$$
G = N(\a), \ \chi_{\a} \in \text{\rm H}^2(G, \T), \nu_{\a} = 1.
$$
\text{\rm Type} \twoone:
$$
\Int(\r) = \Cnt(\r),\  \overline{\Int}(\r) = \Aut(\r),\  \text{\rm Mod}(\r) = 1.
$$
Therefore the characteristic invariant $\chi_{\a}$ $\in {\Lambda}(G, N(\a),
\T)$ alone determines
the cocylcle conjugacy of the action $\a$ of $G$.
\newline
\noindent
$\text{\rm Type}\ \text{\rm \two}_{\infty}$:
$$
\Int(\r) = \Cnt(\r),\quad\ F(\r) = \{L^{\infty}(\R), \text{\it Translation}
\},\qquad
\text{\rm H}^1(F(\r)) = 1,
$$
$$
\text{\rm Mod}(\r) = \R^*_+, \qquad \Aut(\r) = \Intb(\r) \rtimes \R^*_+
$$
$$
\widetilde{\U}(\r) = \U(\r) \times\U(L^{\infty}(\R))/\T.
$$
Type \threezero:\hskip 1in \text{\rm All invariants are non-trivial in general.}
$$
\Aut(\r) = \Intb(\r) \rtimes \Aut(F(\r)) \  \quad \text{\rm by} \quad
\text{\rm Wong, [23]};
$$
$$
\widetilde{\U}(\r) = \U(\r) \rtimes \Z^1(F(\r))   \quad \text{\rm by} \quad
[22].
$$
\newline
\noindent
\text{\rm Type} $\three_\lambda, 0 < \lambda < 1$:
$$
\eqalign{
& \F(\r) = \{L^{\infty}(\R/(-\text{\rm log}(\lambda)\Z), \text{\it
Translation} \}, \cr
& \text{\rm H}^1(F(\r)) = \R/T\Z \quad\text{\it with}\quad T = -
2\pi/\text{log}\lambda;}
$$
$$
\Cnt(\r) = \Int(\r)\sigma(\R)\  \text{\rm where} \ \sigma(\R) = \ \text{\rm
Modular Automorphism
Group};
$$
$$
\widetilde{\U}(\r) = (\U(\r) \rtimes Z^1(F(\r)) , \quad \Aut(\r) =
\Intb(\r) \rtimes
\R/(\log\lambda)\Z;
$$
and
$$
 Z^1(F(\r)) = (\U(L^{\infty}(\R/(-\text{\rm log}(\lambda)\Z)/\T) \rtimes \T.
$$
\newline
\noindent
\text{\rm Type} \threeone:

$$
\F(\r) = \{\C, \text{\it Trivial action of $\R$} \}, \Intb(\r) = \Aut(\r),
\ \Mod(\r) = 1 {;}
$$
$$
\widetilde{\U}(\r) = \U(\r) \rtimes \R, \qquad \text{\rm H}^1(F(\r)) = \R,
\qquad \Cnt(\r) =
\Int(\r) \rtimes\R.
$$
It is interesting to note that the structure of the invariants in the type
\threeone  case
is simplest among type \three \ cases yet the proof is the hardest.
Special cases of the result
in the \threeone case have been established in \cite{11}.  The general case
will appear in
\cite{10}.  We state it here as an independent result:

\proclaim{Corollary 2}
If $\r$ is an AFD factor of type \threeone, then with $N =\a^{-1}(\text{\rm
Cnt}(\r))$ the pair
$$
(\chi_{\a}, \nu_{\a}) \in \Lambda(G, N, \T) \times\text{\rm Hom}_G(N, \R)
$$
is a complete invariant for the cocycle conjugacy class of the action $\a$
of a countable
discrete amenable group G on $\r$.  Every element $(\chi, \nu) \in
\Lambda(G, N. \T) \times
\Hom_G(N, \R)$ arises as the invariant of an action of G on $\r$.
\endproclaim



%\centerline{\bf References}\medskip

\Refs
\widestnumber\key{[15]}

\ref\key 1\by Connes, A.\paper Outerconjugacy classes of
automorphisms of factors\jour Ann\. Sci\. \'Ecole Norm\.
Sup\.\vol 8\yr 1975\pages 383-419\endref

%\noindent [1]  A. Connes,
%{\it Outer conjugacy classes of
%automorphisms of factors},
%Ann\. Sci\. \'Ecole Norm\. Sup\.
%{\bf 8}  (1975),  383--419.

\ref\key 2\by Connes, A.\paper On the classification of von
Neumann algebras and their automorphisms\jour Symposia Math\.
\vol XX\yr 1976\pages 435-478\endref

%\noindent [2]  A. Connes,
%{\it On the classification of von Neumann
%algebras and their automorphisms}
%Symposia Math\. XX (1976), 435--478.

\ref\key 3\by Conne, A.\paper Periodic automorphisms
of the hyperfinite factor of type \twoone\jour Acta Sci\. Math\.
\vol 39\yr 1977\pages 39-66\endref

%\noindent [3]   A. Connes,
%{\it Periodic automorphisms
%of the hyperfinite factor of type \twoone},
%Acta Sci\. Math\. {\bf 39}  (1977),  39--66.

\ref\key 4\by Connes, A.\paper Factors of type \threeone, property
$L'_\la$ and closure of inner automorphisms\jour J. Operator Theory
\vol 14\yr 1985\pages 189-211\endref

%\noindent [4]  A. Connes,
%{\it Factors of type \threeone, property
%$L'_\la$ and closure of inner automorphisms},
%J. Operator Theory {\bf 14}  (1985), 189--211.

\ref\key 5\by Connes, A. \& Takesaki, M\paper The flow of weights on factors
of type \three\jour Tohoku Math\. J.\vol 29\yr 1977\pages 473-555\endref

%\noindent [5]  A. Connes \& M. Takesaki,
%{\it The flow of weights on factors of type \three},
%Tohoku Math\. J. {\bf 29} (1977), 473--555.

\ref\key 6\by Haagerup, U.\paper Connes' bicentralizer problem
and uniqueness of the injective factor of type
\threeone\jour Acta Math\.\vol 158\yr 1987\pages 95-147\endref


%\noindent [6]   U. Haagerup,
%{\it Connes bicentralizer problem
%and uniqueness of the injective factor of type
%\threeone},  Acta Math\. {\bf 158}  (1987),
%95--147.

\ref\key 7\by Haagerup, U. \& St\o rmer, E.\paper
Pointwise inner automorphisms of von Neumann
algebras {\rm with an appendix by C. Sutherland}
\jour J.\ Funct.\ Anal.\vol 92\yr 1990\pages 177-201\endref

%\noindent [7]   U. Haagerup \& E. St\o rmer,
%{\it Pointwise inner automorphisms of von Neumann
%algebras} with an appendix by C. Sutherland,
%J.\ Funct.\ Anal., {bf 92} (1990),
%177-201.

\ref\key 8\by Jones, V. F. R.\paper Actions of finite groups
on the hyperfinite type \twoone\ factor\jour Mem\. Amer\. Math\. Soc\.
\vol 237\yr 1980\endref

%\noindent [8]   V. F. R. Jones,
%{\it Actions of finite groups
%on the hyperfinite type \twoone\ factor},
%Mem\. Amer\. Math\. Soc\.
%{\bf 237}  (1980).

\ref\key 9\by Jones, V. F. R. \& Takesaki, M.\paper Actions of
compact abelian groups on semifinite injective
factors\jour Acta Math\.\vol 153\yr 1984\pages 213-258\endref

%\noindent [9]   V. F. R. Jones  \& M. Takesaki,
%{\it Actions of
%compact abelian groups on semifinite injective
%factors},  Acta Math\. {\bf 153} (1984),  213--258.

\ref\key 10\by Katayama, Y., Sutherland, C. E. \& Takesaki, M.
\paper The intrinsic invariant of an approximately finite dimensional factor
and the cocycle
conjugacy of discrete amenable group actions {\rm , to appear}\endref

%\noindent [10] Y. Katayama,  C. E. Sutherland \& M. Takesaki,
%{\it The intrinsic invariant of an approximately finite dimensional factor
%and the cocycle
%conjugacy of discrete amenable group actions}, to appear.

\ref\key 11\by Kawahigashi, Y., Sutherland, C. E. \& Takesaki, M.
\paper The structure of the automorphism group of an injective factor and the
cocycle conjugacy
of discrete abelian group actions\jour Acta Math\.\vol 169
\yr 1992\pages 105-130\endref

%\noindent [11]  Y. Kawahigashi, C. E. Sutherland \& M. Takesaki,
%{\it The structure of the automorphism group of an injective factor and the
%cocycle conjugacy
%of discrete abelian group actions}, Acta Math\.{\bf 169} (1992), 105-130.

\ref\key 12\by Ocneanu, A.\book Actions of discrete
amenable groups on factors\publ Lecture Notes in Math\.\vol 1138
\publaddr Springer, Berlin\yr 1985\endref

%\noindent [12]   A. Ocneanu,
%Actions of discrete
%amenable groups on factors,''
%Lecture Notesin Math\. No\. 1138,
%Springer, Berlin,  1985.

\ref\key 13\by Sutherland, C. E. \& Takesaki, M.
\paper Actions of
discrete amenable groups and groupoids on von Neumann
algebras\jour RIMS Kyoto Univ\.\vol 21\yr 1985\pages 1087-1120\endref

%\noindent [13]   C. E. Sutherland \& M. Takesaki,
%{\it Actions of
%discrete amenable groups and groupoids on von Neumann
%algebras},  Publ\.  RIMS Kyoto Univ\.
%{\bf 21}  (1985),  1087--1120.

\ref\key 14\by Sutherland, C. E. \& Takesaki, M.\paper
Actions of discrete amenable groups on
injective factors of type \threel, $\la\neq1$\jour
Pacific J. Math\.\vol 137\yr 1989\pages 405-444\endref

%\noindent [14]   C. E. Sutherland \& M. Takesaki,
%{\it Actions of discrete amenable groups on
%injective factors of type \threel, $\la\neq1$},
%Pacific J. Math\., {\bf 137}  (1989),  405--444.

\ref\key 15\by Wong, S. Y. R.\paper On the dictionary between
ergodic transformations, Krieger factors and
ergodic flows\jour Thesis, Univ. Newsouth
Wales\yr 1986\pages
72 + v\endref

%\noindent [15]  S. Y. R. Wong,
%{\it On the dictionary between ergodic transformations, Krieger factors and
%ergodic flows},
%Thesis, Univ. Newsouth Wales, (1986), pp. 72 + v.

\endRefs

\enddocument





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