% LaTeX Article Template - customizing header and footer \documentstyle[12pt]{article} \input{amssymb.def} \input{amssymb.tex} % Set left margin - The default is 1 inch, so the following % command sets a 1.25-inch left margin. \setlength{\oddsidemargin}{-0.25in} % Set width of the text - What is left will be the right margin. % In this case, right margin is 8.5in - 1.25in - 6in = 1.25in. \setlength{\textwidth}{7in} % Set top margin - The default is 1 inch, so the following % command sets a 0.75-inch top margin. \setlength{\topmargin}{-0.25in} % Set height of the header \setlength{\headheight}{0.5in} % Set vertical distance between the header and the text \setlength{\headsep}{0.2in} % Set height of the text \setlength{\textheight}{8.5in} % Set vertical distance between the text and the % bottom of footer \setlength{\footskip}{0.4in} \pagestyle{myheadings} %\markright{\empty} % Set the beginning of a LaTeX document \begin{document} \def\Ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\sim$}}}{}} \def\ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\scriptscriptstyle\sim$}}}{}} \def\R{{\Bbb R}} \def\V{{\Bbb V}} \def\N{{\Bbb N}} \def\n{{\Bbb N}} \def\Q{{\Bbb Q}} \def\O{{\cal O}} \def\e{\epsilon} \def\h{{\hat{X}}} \def\B{{\cal B}} \LARGE % \font\myfont=msb10\magstep2 \title{\Huge{\bf The Mackey doctrine}} % Enter your title between curly braces \author{\Huge Greg Hjorth} % Enter your name between curly braces \date{\LARGE{1997}} % Enter your date or \today between curly braces \maketitle %\leftskip 1in %\newpage \Huge \noindent{\bf $\S$ 0 Definition} \noindent \Large{ The name of George Mackey is often associated with the view that an equivalence relation is {\it classifiable} if there is a `nice', preferably Borel, function that assigns points in a very concrete space -- $\Large{\Bbb{R}}$,$ \Bbb C,{\mathbb T}, C([0,1])$ -- as a complete invariant. } \bigskip \noindent {\bf 0.1 Definition} Let $E$ be an equivalence relation on a separable, completely metrizable (that is {\it Polish}) space $X$. $E$ is {\it concretely classifiable} if there is a Borel function \[ \theta:X\rightarrow {\Bbb R}\] such that for all $x,y\in X$ \[xEy\Leftrightarrow \theta(x)=\theta(y).\] \medskip \newpage \noindent {\bf 0.2 Remarks:} (i) It may be helpful to think of \[ \theta:X\rightarrow {\Bbb R}\] as lifting to an injection \[ \hat{\theta}:X/E\rightarrow {\Bbb R},\] and that in this sense the {\it Borel cardinality} of $X/E$ is less than or equal to the Borel cardinality of ${\Bbb R}$. (ii) Here a function is {\it Borel} if the preimage of any open set is Borel (that is to say, in the $\sigma$-algebra generated by the open sets). (iii) The existence of $\theta$ as above is equivalent to there being a countable sequence $(A_n)_{n\in{\Bbb N}}$ of $E$-invariant Borel sets such that \[xEy\] if and only if \[\{n: x\in A_n\}=\{n: y\in A_n\}.\] (iv) Another equivalent formulation is that $X/E=\{[x]_E:x\in X\}$ be a {\it standard Borel space} in the quotient Borel structure -- that is to say, there is a Polish topology on $\tau$ on $X/E$ such that the Borel sets in $(X/E,\tau)$ are exactly the images of the $E$-invariant Borel sets in $X$. \newpage \noindent {\bf $\S$1 Examples (concretely classifiable)} \medskip \noindent {\bf 1.1 Example ($n$-dimensional linear operators)} Let $H_n$ be $n$-dimensional complex Hilbert space, $U_n$ be the space of unitary operators over $H_n$, and consider the problem of classifying these objects up to conjugation: \[A_1\sim A_2 \Leftrightarrow \exists B\in U_n(BA_1B^{-1}=A_2).\] Here the list of eigenvalues $(\lambda_1,\lambda_2,...,\lambda_n)$ in increasing order with multiple eigenvectors repeated provides a complete invariant. Since there is a Borel injection from ${\Bbb C}^n$ into ${\Bbb R}$, we obtain a concrete classification. \medskip \newpage \noindent {\bf 1.2 Example (Bernoulli shifts)} Let $S=\{s_1,...,s_n\}$ be a finite alphabet, $\sigma: S^{\Bbb Z}\rightarrow S^{\Bbb Z} $ be the shift map, and for $p_1, p_2,...,p_n$ a finite sequence of positive numbers summing to 1 let $\mu$ the product measure resulting from these weights. We may choose to think of two such systems as being equivalent if there is an invertible measure preserving map that conjugates them: that is, set $(S_1, \sigma_1, \mu_1)\cong (S_2, \sigma_2, \mu_2)$ if there is a measurable preserving bijection \[\pi: (S_1)^{\Bbb Z}\rightarrow (S_2)^{\Bbb Z}\] such that \[\sigma_1=\pi^{-1}\circ \sigma_2 \circ \pi\] \[\forall A\subset (S_2)^{\Bbb Z}(\mu_2(A)=\mu_1(\pi^{-1}(A))).\] Ornstein\footnote{\large{{\it Bernoulli shifts with the same entropy are isomorphic,} ({\bf Advances in Mathematics,} vol. 4(1970), pp. 337-52)}} showed that here a single real number, the {\it entropy} of the system $(S, \sigma, \mu)$, provides a complete invariant. (Moreover, in a suitable standard Borel structure, this invariant can be calculated in a Borel fashion.) \medskip \newpage \noindent {\bf 1.3 Example (group representations)} Consider the irreducible representations of the group ${\Bbb Z}$. Given a complex Hilbert space $H$ with associated unitary group $U$ of all inner product respecting transformations, we can let Irr$({\Bbb Z}, H)$ be the space of homomorphisms \[\tau: {\Bbb Z}\rightarrow U\] where $U$ has no non-trivial invariant subspaces under $\tau[{\Bbb Z}]$. It is natural to think of $\tau_1$ and $\tau_2$ as somehow presenting equivalent representations if there is some $A\in U$ with \[\tau_1(g)=A^{-1}\circ \tau_2(g)\circ A\] for all $g\in {\Bbb Z}$. Here Irr$({\Bbb Z}, H)$ is non-empty if and only if $H$ is one dimensional. Moreover we may identify the elements of Irr$({\Bbb Z}, H)$ with characters, and thus a complete classification of these objects may be given by points in ${\Bbb T}$, and hence ${\Bbb R}$. On the other hand if $G$ is finite the space Irr$(G, H)$ will be non-empty only when $H$ is finite dimensional. Then the above equivalence relation will be induced by the a continuous action of the now compact group $U$ on the Polish space Irr$(G, H)$. In general such orbit equivalence relations are always classifiable by points in ${\Bbb R}$. More generally, it is known that whenever $G$ is a countable abelian-by-finite group the above equivalence relation on Irr$(G, H)$ is concretely classifiable. \newpage \noindent {\bf $\S$2 Non-Examples (not concretely classifiable)} \medskip \noindent {\bf 2.1 The Vitali equivalence relation} For $x, y\in {\Bbb R}$ set \[xE_v y\] if and only if \[x-y\in{\Bbb Q}.\] So certainly there is no Lebesgue measurable selector. In fact one can show the stronger result that any Lebesgue measurable ${\Bbb Q}$-invariant $\theta:{\Bbb R}\rightarrow {\Bbb R}$ is constant a.e. In particular, $E_v$ is {\it not} concretely classifiable. A similar example is given by the equivalence relation of eventual agreement on the Cantor space -- $\{0,1\}^{\Bbb N}$ in the product topology. Given $x, y:{\Bbb N}\rightarrow \{0,1\}$ set \[xE_0y \] if and only if there is some $K$ such that for all $m>K$ \[x(m)=y(m).\] Any $E_0$-invariant Borel $\theta: \{0,1\}^{\Bbb N}\rightarrow {\Bbb R}$ must be constant on a comeager set. \newpage \noindent {\bf 2.2 Group representations again} Let $G$ be a countable discrete group that it {\it not} abelian-by-finite. Let $H_{\infty}$ be a separable infinite dimensional Hilbert space and $U_{\infty}$ the unitary group on $H_{\infty}$. Again take Irr$(G, H_{\infty})$ to be space of irreducible representations $\tau:G\rightarrow U_{\infty}$ with the equivalence relation of conjugacy -- \[\tau_1\approx \tau_2\Leftrightarrow \exists A\in U_{\infty}\forall g\in G (\tau_1(g)=A^{-1}\circ \tau_2(g)\circ A).\] $\approx$ is not concretely classifiable: there is no Borel assignment of reals as complete invariants to Irr$(G,H_{\infty})/\approx$.\footnote{\large{J. Glimm, {\it Type I $C^*$-algebras,} {\bf Annals of Mathematics,} 1961, pp. 572-612; E. Thoma, {\it Eine Charakterisierung diskreter Gruppen vom Type I,} {\bf Inventiones Mathematicae} vol. 6(1960), pp. 190-6}} \bigskip \noindent {\bf 2.3 Complex Domains} J. Becker, W. Henson, and L. Rubel \footnote{\large{{\it First-order conformal invariants,} ({\bf Annals of Mathematics,} 1980, pp. 123-178)}} obtain non-classifiability by embedding the equivalence relation $E_0$ of eventual agreement on infinite sequences of 0's and 1's into conformal equivalence on complex domains. In fact if we assign ${\cal D}$, the space of open subsets of ${\Bbb C}$, with the {\it Effros standard Borel structure} -- under which it does have a natural Borel structure -- then their argument can be seen as showing that there is a Borel function \[\theta:\{0,1\}^{\Bbb N}\rightarrow {\cal D}\] such that $xE_0y$ if and only if $\theta(x)$ and $\theta(y)$ are biholomorphic. \newpage \noindent {\bf $\S$3 More Examples (puzzling cases)} \medskip \noindent {\bf 3.1 Discrete spectrum mpt's} Let $M_{\infty}$ be the space of all measure preserving invertible transformations of the unit interval. This is a {\it Polish group} -- that is to say a topological group that is Polish as a space; for instance, if $(U_n)$ enumerates the basic open subsets of $[0,1]$ we obtain a complete metric with \[d(\pi_1, \pi_2)=\sum_{n\in{\Bbb N}}2^{-n}\lambda(\pi_1(U_n)\Delta\pi_2(U_n))+ \lambda(\pi_1^{-1}(U_n)\Delta\pi_2^{-1}(U_n)).\] We may try to classify measure preserving transformations of the unit interval, or any other probability space. Here consider $\pi_1, \pi_2:[0,1]\rightarrow [0,1]$ {\it equivalent} if conjugate -- so there is some $\sigma:[0,1]\rightarrow [0,1]$ such that \[\sigma\circ\pi_1=\pi_2\circ\sigma {\rm {a.e.}}\] P. R. Halmos and J. von Neumann \footnote{\large{{\it Operator methods in classical mechanics, II} ({\bf Annals of Mathematics,} vol. 43(1942), pp. 332-50)}} showed that for {\it discrete spectrum} mpt's, we may in a Borel fashion assign a countable collection $\{c_i(\pi):i\in{\Bbb N}\}$ of complex numbers that completely describe the equivalence class of $\pi$. While conjugacy on discrete measure preserving transformations is not concretely classifiable, the Halmos-von Neumann theorem would seem to constitute some sort of weaker notion of classification. \newpage \noindent {\bf 3.2 Conformal domains} Becker-Henson-Rubel explicitly ask: is there some reasonably non-pathological way to assign to every domain $D\subset {\Bbb C}$ some countable set of complex numbers $S_D$ such that \[D\cong D'\] if and only if \[S_D=S_{D'}?\] \newpage \noindent {\bf 3.3 $C^*$ algebras and topological dynamics} T. Giordano, I.F. Putnam, and C.F. Skau \footnote{\large{{\it Topological orbit equivalence and $C^*$-crossed products} ({\bf Journal Reine und Angewandt Mathematische,} vol. 469(1995), pp. 51-111)}} consider the problem of classifying {\it minimal Cantor systems up to orbit equivalence}. Two continuous \[\varphi_1:X_1\rightarrow X_1,\] \[\varphi_2:X_2\rightarrow X_2\] which are {\it minimal} in the sense of having no non-trivial closed invariant sets and are {\it Cantor} in the sense of $X_1$, $X_2$ being compact, uncountable and completely disconnected metric spaces, are said to be {\it orbit equivalent} if there is a homeomorphism $F:X_1\rightarrow X_2$ which respects the orbits structure set wise, in that for all $x$ \[\{\varphi_2^{i}(F(x)):i\in{\Bbb Z}\}=F[\{\varphi_1^{i}(x):i\in{\Bbb Z}\}].\] This problem is in turn equivalent to classifying a certain class of $C^*$-algebras. Here they produce countable ordered abelian groups as complete invariants. \newpage \noindent {\bf 3.4 Question} For which equivalence relations can we provide a countable set of reals as a complete invariant? So given a Polish space $X$ (separable and completely metrizable), and equivalence relation $E$, when can we find a countable sequence $(f_i)_{i\in{\Bbb N}}$ of Borel functions \[f_i:X\rightarrow {\Bbb R}\] such that \[xEy\Leftrightarrow \{f_i(x):i\in{\Bbb N}\}=\{f_i(y):i\in{\Bbb N}\}?\] \medskip More generously, for which $E$ can we provide some kind of countable structure considered up to isomorphism as a complete invariant? Letting ${\cal L}$ be a countable language, we can form Mod$({\cal L})$, the space of all ${\cal L}$-structures on ${\Bbb N}$ with the topology generated by quantifier free formulas. For which equivalence relations $E$ can we find a Borel \[\theta:X\rightarrow {{\rm{ Mod}}({\cal L})}\] such that for all $x, y\in X$ \[xEy\Leftrightarrow \theta(x)\cong \theta(y)?\] \bigskip {\bf 3.5 Remark} Most mathematical objects can be at least {\it parameterized} by points in a Polish space in a reasonable fashion. \newpage \noindent{\bf $\S$4 Sufficient conditions for classifiability} \medskip \noindent {\bf 4.1 Fact} Let $G$ be a compact metrizable group acting continuously on a Polish space $X$ with induced orbit equivalence relation $E_G$. Then $E_G$ is concretely classifiable -- that is to say, there is Borel $\theta:X\rightarrow {\Bbb R}$ such that for all $x, y\in X$ \[\exists g\in G(g\cdot x=y)\Leftrightarrow \theta(x)=\theta(y).\] \bigskip \bigskip \bigskip \bigskip \noindent {\bf 4.2 Theorem} (Kechris) Let $G$ be a locally compact Polish group acting continuously on a Polish space $X$. Then there is a countable sequence of Borel functions $(f_i)_{i\in{\Bbb N}}$ such that for all $x,y\in X$ \[xE_Gy\Leftrightarrow \{f_i(x):i\in{\Bbb N}\}=\{f_i(y):i\in{\Bbb N}\}.\] \bigskip \bigskip It turns out that conformal equivalence on complex domains, or even arbitrary complex surfaces, may be reduced to an appropriately chosen locally compact group action. \newpage \noindent {\bf 4.3 Theorem} (Hjorth-Kechris) Let ${\cal D}$ be the space of all complex domains. (a) Then there is a {\it definable} (in the language of ZFC) assignment \[M\mapsto S_M\] of countable sets of reals to domains such that for all $M, N\in {\cal D}$ \[M\cong N\Leftrightarrow S_M=S_N.\] (b) The above assignment extends to a space of parameters for complex surfaces, is Borel in a natural Borel structure, and conformal equivalence is bireducible with $C({\Bbb F_2}, {\Bbb Z})/{\Bbb F_2}$. \bigskip \bigskip The description of the Borel space of parameters for complex surfaces is somewhat involved. But for the specific case of ${\cal O}({\Bbb C})$ the set of open subsets of of ${\Bbb C}$ we can take the {\it Effros standard Borel structure}, generated by the sets of the form \[\{O\in {\cal O}({\Bbb C}): U\subset O\}\] for $U\subset {\Bbb C}$ open. Here it is known that there are Polish topologies on ${\cal O}({\Bbb C})$ with this Borel structure. In this Borel structure we may obtain a countable sequence of Borel \[f_i: {\cal O}({\Bbb C})\rightarrow {\Bbb R}\] such that $O_1$ and $O_2$ are conformally equivalent if and only if \[\{f_i(O_1):i\in{\Bbb N}\}=\{f_i(O_2):i\in{\Bbb N}\}.\] \newpage \noindent {\bf $\S$5 Sufficient conditions for non-classifiability} \bigskip \noindent {\bf 5.1 Theorem} (folklore) Let $G$ be a Polish group and $X$ a Polish space. Suppose that (i) some orbit is dense; (ii) every orbit is meager (its complement includes the intersection of countably many open dense sets). Then $E_G$ is not concretely classifiable: there is no Borel (or even Baire measurable) \[\theta:X\rightarrow {\Bbb R}\] such that for all $x, y\in X$ \[\exists g\in G(g\cdot x=y)\Leftrightarrow \theta(x)= \theta(y).\] \newpage \noindent {\bf 5.2 Theorem} (Hj.) Let $G$ be a Polish group and $X$ a Polish space. Suppose that (i) some orbit is dense; (ii) every orbit is meager (its complement includes the intersection of countably many open dense sets); (iii) for all $x\in X$, {\it the local orbits of $x$} are somewhere dense; that is to say, if $V$ is an open neighborhood of $1_G$, $U$ is an open set containing $x$, and if $O(x,U,V)$ is the set of all $\hat{x}\in [x]_G$ such that there is a finite sequence $(x_i)_{i\leq k}\subset U$ such that $x_0=x$, $x_k=\hat{x}$, and each $x_{i+1}\in V\cdot x_i$, then the closure of $O(x,U,V)$ contains an open set. Then there is no Borel (or even Baire measurable) \[\theta:X\rightarrow {{\rm{ Mod}}({\cal L})}\] such that for all $x, y\in X$ \[xE_Gy\Leftrightarrow \theta(x)\cong \theta(y).\] Consequently there is no sequence $(f_i)_{i\in{\Bbb N}}$ of Borel functions \[f_i:X\rightarrow {\Bbb R}\] such that \[xEy\Leftrightarrow \{f_i(x):i\in{\Bbb N}\}=\{f_i(y):i\in{\Bbb N}\}.\] \newpage The method of proof succeeds for classes of functions considerably more complicated than Borel. By interpreting such an orbit equivalence relation into isomorphism on higher dimensional manifolds one can obtain theorems like: \bigskip \noindent {\bf 5.3 Theorem} (Hj.-Kechris) Let ${\cal M}^2$ be the space of two dimensional complex manifolds. Then (consistently) there is no definable assignment of countable sets of reals as complete invariants. \bigskip \bigskip \bigskip \bigskip \bigskip \bigskip \noindent \underline{Class of complex manifolds} \hfill \underline{Invariants} \bigskip \bigskip \noindent Compact complex surfaces \hfill points in ${\Bbb R}$ \noindent General complex surfaces \hfill countable subsets of ${\Bbb R}$ \noindent General complex manifolds \hfill ?[none of the above] \newpage \newpage \noindent {\bf 5.4 Theorem} (Hj.) Let $M_{\infty}$ be the space of invertible measure preserving transformations on the unit interval. Consider the conjugacy equivalence relation $\sim$: $\pi_1\sim \pi_2$ if there is $\sigma\in M_{\infty}$ such that \[\sigma\circ\pi_1=\pi_2\circ\sigma \: {\rm {a.e.}}\] Then there is no sequence $(f_i)_{i\in{\Bbb N}}$ of Borel functions \[f_i:M_{\infty}\rightarrow {\Bbb R}\] such that \[\pi_1\sim\pi_2\Leftrightarrow \{f_i(\pi_1):i\in{\Bbb N}\}=\{f_i(\pi_2):i\in{\Bbb N}\}.\] In fact, $\sim$ is strictly more complicated than isomorphism on countable models: there is a Borel \[\theta:{{\rm{ Mod}}({\cal L})}\rightarrow M_{\infty}\] such that for all $M,N\in {\rm{ Mod}}$ \[M\cong N\Leftrightarrow \theta(x)\sim \theta(y),\] but (for any choice of ${\cal L}$) there is no \[\theta:M_{\infty}\rightarrow {{\rm{ Mod}}({\cal L})}\] such that for all $\pi_1,\pi_2\in M_{\infty}$ \[\pi_1\sim \pi_2\Leftrightarrow \theta(x)\cong \theta(y).\] \noindent {\bf 5.5 Theorem} (Hj.) Let $G$ be a countable group that is {\it not} abelian-by-finite. Let $H_{\infty}$ be an separable infinite dimensional Hilbert space, let Irr$(G, H_{\infty})$ be the space of irreducible representations of $G$ in $H_{\infty}$. For $\tau_1\approx\tau_2$ if there is $A\in U(H_{\infty})$, the unitary group on $H_{\infty}$, that conjugates them in the sense that for all $g\in G$ \[\tau_1(g)=A^{-1}\circ \tau_2(g)\circ A.\] Then there is no sequence $(f_i)_{i\in{\Bbb N}}$ of Borel functions \[f_i:{\rm {Irr}}(G,H_{\infty})\rightarrow {\Bbb R}\] such that \[\tau_1\approx\tau_2\Leftrightarrow \{f_i(\tau_1):i\in{\Bbb N}\}=\{f_i(\tau_2):i\in{\Bbb N}\}.\] In fact there is no {\it reasonably definable} assignment of countable models considered up to isomorphism as complete invariants. \end{document}