Celebratio Mathematica

J. Feldman and C. C. Moore: “Ergodic equivalence relations, cohomology, and von Neumann algebras,” Bull. Am. Math. Soc. 81 : 5 (September 1975), pp. 921–924. MR 425075 Zbl 0317.22002 article

J. Feldman and C. C. Moore: “Ergodic equivalence relations, cohomology, and von Neumann algebras, I,” Trans. Am. Math. Soc. 234 : 2 (1977), pp. 289–324. These papers are dedicated to George Mackey on his 60th birthday. MR 578656 Zbl 0369.22009 article

Let \( (X,\mathcal{B}) \) be a standard Borel space, \( R\subset X\times X \) an equivalence relation \( \in\mathcal{B}\times\mathcal{B} \). Assume each equivalence class is countable.

There exists a countable group \( G \) of Borel isomorphisms of \( (X,\mathcal{B}) \) so that \[ R = \{(x,gx):g\in G\} .\]

\( G \) is far from unique. However, notions like invariance and quasi-invariance and \( R-N \) derivatives of measures depend only on \( R \), not the choice of \( G \). We develop some of the ideas of Dye [1959, 1963] and Krieger [1969a, 1969b, 1970a, 1971, 1970b] in a fashion explicitly avoiding any choice of \( G \); we also show the connection with virtual groups. A notion of “module over \( R \)” is defined, and we axiomatize and develop a cohomology theory for \( R \) with coefficients in such a module. Surprising application (contained in Theorem 7): let \( \alpha,\beta \) be rationally independent irrationals on the circle \( \mathbb{T} \), and \( f \) Borel: \( \mathbb{T}\to\mathbb{T} \). Then there exists Borel \( g,h: \mathbb{T}\to\mathbb{T} \) with \[ f(x) = (g(ax)/g(x))(h(\beta x)/h(x)) \quad\text{a.e.} \] The notion of “skew product action” is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the \( R-N \) derivative of an ergodic transformation: we define, for a cocycle \( c \) on \( R \) with values in the group \( A \), a subgroup of \( A \) depending only on the cohomology class of \( c \), and in Theorem 8 identify this with another subgroup, the “normalized proper range” of \( c \), defined in terms of the skew action. See also Schmidt [1974].

George Whitelaw Mackey	Related
Jacob Feldman	Related

J. Feldman and C. C. Moore: “Ergodic equivalence relations, cohomology, and von Neumann algebras, II,” Trans. Am. Math. Soc. 234 : 2 (1977), pp. 325–359. MR 578730 Zbl 0369.22010 article

Let \( R \) be a Borel equivalence relation with countable equivalence classes, on the standard Borel space \( (X,\mathcal{A},\mu) \). Let \( \sigma \) be a 2-cohomology class on \( R \) with values in the torus \( \mathbb{T} \). We construct a factor von Neumann algebra \( \mathrm{\mathbf{M}}(R,\sigma) \), generalizing the group-measure space construction of Murray and von Neumann [1936] and previous generalizations by W. Krieger [1970] and G. Zeller-Meier [1968].

Very roughly, \( \mathrm{\mathbf{M}}(R,\sigma) \) is a sort of twisted matrix algebra whose elements are matrices \( (a_{x,y}) \), where \( (x,y)\in R \). The main result, Theorem 1, is the axiomatization of such factors; any factor \( \mathbf{\mathrm{M}} \) with a regular MASA subalgebra \( \mathbf{\mathrm{A}} \), and possessing a conditional expectation from \( \mathbf{\mathrm{M}} \) onto \( \mathbf{\mathrm{A}} \), and isomorphic to \( \mathbf{M}(R,\sigma) \) in such a manner that \( \mathbf{\mathrm{M}} \) becomes the “diagonal matrices”; \( (R,\sigma) \) is uniquely determined by \( \mathbf{\mathrm{M}} \) and \( \mathbf{\mathrm{A}} \). A number of results are proved, linking invariants and automorphisms of the system \( (\mathbf{\mathrm{M}},\mathbf{\mathrm{A}}) \) with those of \( (R,\sigma) \). These generalize results of Singer [1955] and of Connes [1973]. Finally, several results are given which contain fragmentary information about what happens with a single \( \mathbf{\mathrm{M}} \) but two different subalgebras \( \mathbf{\mathrm{A}}_1 \), \( \mathbf{\mathrm{A}}_2 \).

J. Feldman, P. Hahn, and C. C. Moore: “Orbit structure and countable sections for actions of continuous groups,” Adv. Math. 28 : 3 (June 1978), pp. 186–230. MR 492061 Zbl 0392.28023 article

It is shown that if a second countable locally compact group \( G \) acts nonsingularly on an analytic measure space \( (S,\mu) \), then there is a Borel subset \( E\subset S \) such that \( EG \) is conull in \( S \) and each \( sG\cap E \) is countable. It follows that the measure groupoid constructed from the equivalence relation \( s\sim sg \) on \( E \) may be simply described in terms of the measure groupoid made from the action of some countable group. Some simplifications are made in Mackey’s theory of measure groupoids. A natural notion of “approximate finiteness” (\( AF \)) is introduced for nonsingular actions of \( G \), and results are developed parallel to those for countable groups; several classes of examples arising naturally are shown to be \( AF \). Results on “skew product” group actions are obtained, generalizing the countable case, and partially answering a question of Mackey. We also show that a group-measure space factor obtained from a continuous group action is isomorphic (as a von Neumann algebra) to one obtained from a discrete group action.

Peter Florin Hahn	Related
Jacob Feldman	Related

J. Feldman, P. Hahn, and C. C. Moore: “Sections for group actions, and some applications,” pp. 167–174 in Algèbres d’opérateurs et leurs applications en physique mathématique [Operator algebras and their applications in mathematical physics] (Marseille, 20–24 June 1977). Edited by A. Connes, D. Kastler, and D. W. Robinson. Colloques Internationaux du Centre National de la Recherche Scientifique 274. CNRS (Paris), 1979. MR 560632 Zbl 0429.28019 incollection

Alain Connes	Related
Jacob Feldman	Related
Peter Florin Hahn	Related
Daniel Kastler	Related
Derek W. Robinson	Related

J. Feldman, D. J. Rudolph, and C. C. Moore: “Affine extensions of a Bernoulli shift,” Trans. Am. Math. Soc. 257 : 1 (1980), pp. 171–191. MR 549160 Zbl 0427.28016 article

For any automorphism \( \phi \) of a compact metric group \( G \), and any \( a > 0 \), we show the existence of a free finite measure-preserving (m.p.) action of the twisted product \( Z\times^{\phi}G \) whose restriction to \( Z \) is Bernoulli with entropy \( a+h(\phi) \), \( h(\phi) \) being the entropy of \( \phi \) on \( G \) with Haar measure.
A classification is given of all free finite m.p. actions of \( Z\times^{\phi}G \) such that the action of \( Z \) on the \( \sigma \)-algebra of invariant sets of \( G \) is a Bernoulli action.
The classification of (b) is extended to “quasifree” actions: those for which the isotropy subgroups are in a single conjugacy class within \( G \). An existence result like that of (a) holds in this case, provided certain necessary and sufficient algebraic conditions are satisfied; similarly, an isomorphism theorem for such actions holds, under certain necessary and sufficient conditions.
If \( G \) is a Lie group, then all actions of \( Z\times^{\phi}G \) are quasifree; if \( G \) is also connected, then the second set of additional algebraic conditions alluded to in (c) is always satisfied, while the first will be satisfied only in an obvious case.
Examples are given where the isomorphism theorem fails: by violation of the algebraic conditions in the quasifree case, for other reasons in the non-quasifree case.

Daniel Jay Rudolph	Related
Jacob Feldman	Related

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