Celebratio Mathematica

C. C. Moore: Contributions to the theory of locally compact groups and their representations. Ph.D. thesis, Harvard University, 1960. Advised by G. W. Mackey. MR 2939140 phdthesis

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

Group representations, ergodic theory, operator algebras, and mathematical physics: Proceedings of a conference in honor of George W. Mackey (Berkeley, CA, 21–23 May 1984). Edited by C. Moore. Mathematical Sciences Research Institute Publications 6. Springer (New York), 1987. MR 880370 Zbl 0602.00003 book

Group representations, ergodic theory, and mathematical physics: A tribute to George W. Mackey. Edited by R. S. Doran, C. C. Moore, and R. J. Zimmer. Contemporary Mathematics 449. 2008. MR 2385373 Zbl 1131.22001 book

George Whitelaw Mackey	Related
Robert Stuart Doran	Related
Robert Jeffrey Zimmer	Related

C. C. Moore: “Virtual groups 45 years later,” pp. 263–300 in Group representations, ergodic theory, and mathematical physics: A tribute to George W. Mackey (New Orleans, 7–8 January 2007). Edited by R. S. Doran, C. C. Moore, and R. J. Zimmer. Contemporary Mathematics 449. American Mathematical Society (Providence, RI), 2008. MR 2391808 Zbl 1158.22004 incollection

In 1961 George Mackey introduced the concept of a virtual group as an equivalence class under similarity of ergodic measured groupoids, and he developed this circle of ideas in subsequent papers in 1963 and 1966. The goal here is first to explain these ideas, place them in a larger context, and then to show how they have influenced and helped to shape developments in four different but related areas of research over the following 45 years. These areas include first the general area and the connections between ergodic group actions, von Neumann algebras, measurable group theory and rigidity theorems. Then we turn to the second area concerning topological groupoids, \( C^* \)-algebras, \( K \)-theory and cyclic homology, or as it is now termed non-commutative geometry. We briefly discuss some aspects of Lie groupoids, and finally we shall turn attention to the fourth area of Borel equivalence relations seen as a part of descriptive set theory. In each case we trace the influence that Mackey’s ideas have had in shaping each of these four areas of research.

George Whitelaw Mackey	Related
Robert Stuart Doran	Related
Robert Jeffrey Zimmer	Related

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