Celebratio Mathematica — Moore — Complete Bibliography

[1] 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

[2] C. C. Moore: “On the Frobenius reciprocity theorem for locally compact groups,” Pacific J. Math. 12 (1962), pp. 359–365. MR 141737 Zbl 0121.03802 article

[3] C. C. Moore: “The degree of randomness in a stationary time series,” Ann. Math. Statist. 34 : 4 (1963), pp. 1253–1258. MR 159405 Zbl 0121.36302 article

[4] C. C. Moore: “Compactifications of symmetric spaces,” Am. J. Math. 86 : 1 (January 1964), pp. 201–218. MR 161942 Zbl 0156.03202 article

Furstenberg in a recent work [1963] has introduced the notions of a boundary and of boundary subgroups of a connected Lie group. If \( G \) is a semi-simple group with finite center and \( K \) a maximal compact subgroup, then these boundaries can be used to compactify the Riemannian symmetric space \( G/K \). In this paper we shall study in detail the structure of these boundaries and the resulting compactifications. Also we shall show that this construction leads to essentially the same compactifications which Satake [1960] has studied. Some of our results are in Satake’s paper, but we need them in modified forms. Hermann has also studied these boundaries and has obtained many interesting results [1961, 1962], using more geometrical methods. There is some overlap in the results.

[5] C. C. Moore: “Compactifications of symmetric spaces, II: The Cartan domains,” Am. J. Math. 86 : 2 (April 1964), pp. 358–378. MR 161943 article

If \( G \) is a semi-simple Lie group with finite center and \( K \) a maximal compact subgroup, the space \( G/K \) is a symmetric Riemiannian space of non-compact type. In a previous paper [1964], we have investigated certain compactifications of \( G/K \) which are obtained by embedding \( G/K \) into a space of probability measures on a boundary of \( G \). We have shown that these compactifications, introduced by Furstenberg [1963], are the same as Satake’s compactifications [1960]. Satake remarks (p. 104) that if \( G/K \) is a classical Cartan domain [Cartan 1935], then one can verify case by case that one of these compactifications is homeomorphic to the closure of the Cartan domain when it is viewed as a bounded domain in complex Euclidean space. One of our main results will be to give an intrinsic proof of this theorem. In the course of the proof we will be able to construct the desired homeomorphism in a very natural way using the properties of Furstenberg’s compactifications.

Section 2 is devoted to the proof of an algebraic result concerning the roots of a Cartan domain which is needed later. This theorem has been conjectured by Bott and Korányi. Although one could verify this result also by checking it case by case, it seems to be of interest to have an intrinsic proof. In Section 3 we prove the theorem mentioned above. Lemma 3.1 on the structure of the Shilov boundary of the domain has been proved by Korányi and Wolf [1965]. In Section 4 we shall discuss the boundary components of a Cartan domain as defined by Pyatetskiĭ-Shapiro [1961]. We will give intrinsic proof of his main theorems and some additional information which we can obtain from our point of view.

[6] C. C. Moore: “Extensions and low dimensional cohomology theory of locally compact groups, I & II,” Trans. Amer. Math. Soc. 113 (1964), pp. 40–63; 64–86. MR 171880 Zbl 0131.26902 article

[7] C. C. Moore: “Extensions and low dimensional cohomology theory of locally compact groups, II,” Trans. Am. Math. Soc. 113 (1964), pp. 64–86. Parts III and IV (both 1976) had slightly different titles. article

In a previous paper [1964], we have defined a sequence of cohomology groups \( H^r(G,A) \) defined when \( G \) is a locally compact group and \( A \) is an abelian locally compact group on which \( G \) acts continuously as a group of automorphisms. The structure of these cohomology groups, particularly the two dimensional groups, was discussed in this paper; here we shall discuss the structure of these cohomology low dimensional groups under more general assumptions on \( G \). Explicitly we shall be concerned with the class of locally compact groups \( G \) which have the property that \( G/G_0 \) is compact (where \( G_0 \) is the connected component of the identity in \( G \)). For simplicity we shall call such groups almost connected. When \( G \) is in fact connected, then we can obtain more explicit information by use of a principal series of normal subgroups in \( G \) discussed by Iwasawa [1949], in conjunction with a spectral sequence. Also we shall introduce the notion of splitting groups and use them to discuss the problem of topologizing the two dimensional group \( H^2(G,A) \) of extensions of \( G \) by \( A \).

[8] C. C. Moore: “Decomposition of unitary representations defined by discrete subgroups of nilpotent groups,” Ann. of Math. (2) 82 (1965), pp. 146–182. MR 181701 Zbl 0139.30702 article

[9] C. C. Moore: “Ergodicity of flows on homogeneous spaces,” Amer. J. Math. 88 (1966), pp. 154–178. MR 193188 Zbl 0148.37902 article

[10] L. Auslander and C. C. Moore: Unitary representations of solvable Lie groups. Mem. Amer. Math. Soc. 62. Amer. Math. Soc. (Providence, RI), 1966. MR 207910 Zbl 0204.14202 book

[11] C. C. Moore: “Invariant measures on product spaces,” pp. 447–459 in Proceedings of the fifth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 21 June–18 July 1965 and 27 December 1965–7 January 1966), vol. II, Part 2: Probability theory. Edited by L. M. Le Cam and J. Neyman. University of California Press (Berkeley, CA), 1967. MR 227366 Zbl 0208.06801 incollection

Lucien Marie Le Cam	Related
Jerzy Neyman	Related

[12] C. C. Moore: “Distal affine transformation groups,” Am. J. Math. 90 : 3 (July 1968), pp. 733–751. MR 232887 Zbl 0195.52604 article

[13] C. C. Moore: “Group extensions of \( p \)-adic and adelic linear groups,” Inst. Hautes Études Sci. Publ. Math. 35 (1968), pp. 157–222. MR 244258 Zbl 0159.03203 article

[14] C. C. Moore: “Restrictions of unitary representations to subgroups and ergodic theory: Group extensions and group cohomology,” pp. 1–35 in Group representations in mathematics and physics (Seattle, WA, summer 1969). Edited by V. Bargmann. Lecture Notes in Physics 6. Springer (Berlin), 1970. MR 279232 Zbl 0223.22020 incollection

[15] L. Auslander and B. Kostant: “Polarization and unitary representations of solvable Lie groups,” Invent. Math. 14 (1971), pp. 255–354. MR 293012 Zbl 0233.22005 article

[16] C. C. Moore and J. A. Wolf: “Totally real representations and real function spaces,” Pac. J. Math. 38 : 2 (April 1971), pp. 537–542. MR 310130 Zbl 0227.43008 article

[17] C. C. Moore: “Groups with finite dimensional irreducible representations,” Trans. Amer. Math. Soc. 166 (1972), pp. 401–410. MR 302817 Zbl 0236.22010 article

[18] Harmonic analysis on homogeneous spaces (Williamstown, MA, 31 July–18 August 1972). Edited by C. C. Moore. Proceedings of Symposia in Pure Mathematics 26. American Mathematical Society (Providence, RI), 1973. MR 327965 Zbl 0272.00009 book

[19] C. C. Moore and J. A. Wolf: “Square integrable representations of nilpotent groups,” Trans. Amer. Math. Soc. 185 (1973), pp. 445–462. MR 338267 Zbl 0302.43014 article

[20] C. C. Moore: “Representations of solvable and nilpotent groups and harmonic analysis on nil and solvmanifolds,” pp. 3–44 in Harmonic analysis on homogeneous spaces (Williamstown, MA, 31 July–8 August 1972). Edited by C. C. Moore. Proceedings of Symposia in Pure Mathematics 26. American Mathematical Society (Providence, RI), 1973. MR 385001 Zbl 0292.22015 incollection

[21] 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

[22] C. C. Moore and J. Rosenberg: “Comments on a paper of I. D. Brown and Y. Guivarc’h: ‘Espaces de Poisson des groupes de Lie’” [Comments on a paper of I. D. Brown and Y. Guivarc’h: ‘Poisson spaces of Lie groups’], Ann. Sci. École Norm. Sup. (4) 8 : 3 (1975), pp. 379–381. MR 427539 Zbl 0312.22009 article

Yves Guivarc'h	Related
Jonathan Micah Rosenberg	Related
Ian Douglas Brown	Related

[23] M. Duflo and C. C. Moore: “On the regular representation of a nonunimodular locally compact group,” J. Functional Analysis 21 : 2 (1976), pp. 209–243. MR 393335 Zbl 0317.43013 article

[24] C. C. Moore: “Group extensions and cohomology for locally compact groups,” Trans. Amer. Math. Soc. 221 : 1 (1976), pp. 1–33; 35–58. MR 414775 Zbl 0366.22006 article

[25] C. C. Moore and J. Rosenberg: “Groups with \( T_{1} \) primitive ideal spaces,” J. Functional Analysis 22 : 3 (1976), pp. 204–224. MR 419675 Zbl 0328.22014 article

[26] C. C. Moore and J. Repka: “A reciprocity theorem for tensor products of group representations,” Proc. Am. Math. Soc. 64 : 2 (June 1977), pp. 361–364. MR 450455 Zbl 0336.22005 article

[27] C. C. Moore: “Square integrable primary representations,” Pac. J. Math. 70 : 2 (October 1977), pp. 413–427. MR 507220 Zbl 0382.22004 article

If \( \pi \) is a unitary representation of a locally compact group \( G \), a weight \( \phi \) on the von Neumann algebra \( R(\pi) \) generated by \( \pi \) is called semi-invariant if it is transformed into scalar multiples of itself by the action of \( G \). If \( \pi \) is primary we show such objects are essentially unique if one specifies the scaling factor (Schur’s lemma). We then study square integrable primary representations and show that a number of possible different definitions are equivalent. We show that any such representation \( \pi \) has a semi-invariant weight scaling by the modular function, and this object is seen to be the proper generalization of the formal degree of \( \pi \). We formulate and prove generalized Schur orthogonality relations for \( \pi \). Finally we specialize to the semi-finite case and identify the formal degree as an operator affiliated to \( R(\pi) \). For irreducible \( \pi \) the results reduce to those of Duflo and the author, and Phillips.

[28] 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

[29] 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 \( \mathbf{\mathrm{A}}_1 \), \( \mathbf{\mathrm{A}}_2 \).

[30] 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

[31] C. C. Moore: “Approximately finite von Neumann algebras,” Am. Math. Mon. 85 : 8 (October 1978), pp. 657–659. MR 508228 Zbl 0392.46041 article

[32] C. C. Moore and R. J. Zimmer: “Groups admitting ergodic actions with generalized discrete spectrum,” Invent. Math. 51 : 2 (1979), pp. 171–188. MR 528022 Zbl 0399.22005 article

[33] R. E. Howe and C. C. Moore: “Asymptotic properties of unitary representations,” J. Functional Analysis 32 : 1 (1979), pp. 72–96. MR 533220 Zbl 0404.22015 article

[34] 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

[35] C. C. Moore: “Amenable subgroups of semisimple groups and proximal flows,” Israel J. Math. 34 : 1–2 (March 1979), pp. 121–138. MR 571400 Zbl 0431.22014 article

We present a classification of maximal amenable subgroups of a semi-simple group \( G \). The result is that modulo a technical connectivity condition, there are precisely \( 2^l \) conjugacy classes of such subgroups of \( G \) and we shall describe them explicitly. Here \( l \) is the split rank of the group \( G \). These groups are the isotropy groups of the action of \( G \) on the Satake–Furstenberg compactification of the associated symmetric space and our results give necessary and sufficient conditions for a subgroup to have a fixed point in this compactification. We also study the action of \( G \) on the set of all measures on its maximal boundary. One consequence of this is a proof that the algebraic hull of an amenable subgroup of a linear group is amenable.

[36] 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

[37] C. C. Moore and K. Schmidt: “Coboundaries and homomorphisms for nonsingular actions and a problem of H. Helson,” Proc. London Math. Soc. (3) 40 : 3 (May 1980), pp. 443–475. MR 572015 Zbl 0428.28014 article

Let \( f \) be a one-cocycle for a non-singular action of a locally compact group \( G \) on a standard measure space \( (Y,\mu) \) with values in a locally compact abelian group \( A \). If \( \chi\in\hat{A} \), \( \chi(f) \) is a one-cocycle with values in the circle group \( \mathrm{\mathbf{T}} \). We investigate the question of when one can conclude that \( f \) is a coboundary, given that \( \chi(f) \) is a coboundary for some ‘sufficiently large’ set of \( \chi \). We obtain very precise results, extending previous work of Hamachi, Oka, and Osikawa. We also show that if \( G \) is measure-preserving, and \( \hat{A} \) is connected, \( f \) is a coboundary if and only if it is ‘bounded’ in a certain natural sense. We then investigate a companion problem posed by H. Helson to determine whether one can conclude that \( f \) is cohomologous to a homomorphism of \( G \) into \( A \) if the same is known to be true for all or for sufficiently many \( \chi(f) \)’s. We show that the answer is affirmative in some cases of interest but is negative in general, and, in fact, compute the defect. The counter-examples produced are examples of non-singular integer actions which have certain prescribed uncountable groups of eigenfunctions; and hence may be of some independent interest.

Klaus Schmidt	Related
Henry Berge Helson	Related

[38] C. C. Moore: “The Mautner phenomenon for general unitary representations,” Pacific J. Math. 86 : 1 (1980), pp. 155–169. MR 586875 Zbl 0446.22014 article

[39] J. Brezin and C. C. Moore: “Flows on homogeneous spaces: A new look,” Am. J. Math. 103 : 3 (June 1981), pp. 571–613. MR 618325 Zbl 0506.22008 article

[40] C. C. Moore: “Ergodic theory and von Neumann algebras,” pp. 179–226 in Operator algebras and applications, part 2. Edited by R. V. Kadison. Proceedings of Symposia in Pure Mathematics 38. American Mathematical Society (Providence, RI), 1982. MR 679505 Zbl 0524.46045 incollection

[41] C. C. Moore: “Cocompact subgroups of semisimple Lie groups,” J. Reine Angew. Math. 1984 : 350 (1984), pp. 173–177. MR 743540 Zbl 0525.22017 article

[42] C. C. Moore: “Mathematical Sciences Research Institute, Berkeley, California,” Math. Intell. 6 : 1 (March 1984), pp. 59–64. Zbl 0538.01021 article

[43] Operator algebras and their connections with topology and ergodic theory (Buşteni, Romania, 29 August–9 September 1983). Edited by H. Araki, C. C. Moore, Ş. Strătilă, and D. Voiculescu. Lecture Notes in Mathematics 1132. Springer (Berlin), 1985. with the assistance of G. Arsene. MR 799557 Zbl 0562.00005 book

Dan-Virgil Voiculescu	Related
Grigore Arsene	Related
Şerban Valentin Strătilă	Related
Huzihiro Araki	Related

[44] 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

[45] C. C. Moore: “Exponential decay of correlation coefficients for geodesic flows,” pp. 163–181 in Group representations, ergodic theory, operator algebras, and mathematical physics (Berkeley, Calif., 1984). Edited by C. C. Moore. Math. Sci. Res. Inst. Publ. 6. Springer (New York), 1987. MR 880376 Zbl 0625.58023 incollection

[46] C. C. Moore and C. Schochet: Global analysis on foliated spaces. Mathematical Sciences Research Institute Publications 9. Springer (New York), 1988. With appendices by Moore, Schochet, S. Hurder and Robert J. Zimmer. A second edition was published in 2006. MR 918974 Zbl 0648.58034 book

Claude L. Schochet	Related
Steven Edmond Hurder	Related
Robert Jeffrey Zimmer	Related

[47] S. Mac Lane, R. H. Herman, and C. C. Moore: “Comments,” Am. Math. Monthly 101 : 8 (October 1994), pp. 714, 809. MR 1542574 article

Richard Howard Herman	Related
S. Mac Lane	Related	Volume

[48] F. A. Grünbaum and C. C. Moore: “The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets,” Acta Cryst. Sect. A 51 : 3 (1995), pp. 310–323. MR 1331196 Zbl 1176.82034 article

The three-dimensional configuration of crystallized structures is obtained by reading off partial information about the Fourier transform of such structures from diffraction data obtained with an X-ray source. We consider a discrete version of this problem and discuss the extent to which ‘intensity only’ measurements as well as ‘higher-order invariants’ can be used to settle the reconstruction problem. This discrete version is an extension of the study undertaken by Patterson in terms of ‘cyclotomic sets’, corresponding to arrangements of equal atoms that can occupy positions on a circle subdivided into \( N \) equally spaced markings. This model comes about when the usual three-dimensional Fourier transform is replaced by a one-dimensional discrete Fourier transform. The model in this paper considers molecules made up of atoms with possibly different (integer-valued) atomic numbers. It is shown that information of order six suffices to determine a structure uniquely.

[49] L. Corwin and C. C. Moore: “\( L^p \) matrix coefficients for nilpotent Lie groups,” Rocky Mt. J. Math. 26 : 2 (1996), pp. 523–544. MR 1406494 Zbl 0866.22011 article

[50] article S. S. Chern, T. Kailath, B. Kostant, C. C. Moore, and A. Tsao: “Louis Auslander (1928–1997),” Notices Am. Math. Soc. 45 : 3 (1998), pp. 390–395. MR 1606428 Zbl 0908.01018

S. S. Chern	Related	Volume
Thomas Kailath	Related
Louis Auslander	Related
Anna Tsao	Related
Bertram Kostant	Related

[51] C. C. Moore and C. L. Schochet: Global analysis on foliated spaces, 2nd edition. Mathematical Sciences Research Institute Publications 9. Cambridge University Press (New York), 2006. Republication of 1988 original. MR 2202625 Zbl 1091.58015 book

[52] C. C. Moore: Mathematics at Berkeley: A history. A. K. Peters (Wellesley, MA), 2007. MR 2289685 Zbl 1111.01009 book

[53] 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

[54] 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

[55]J. Schwartz, M. Gerstenhaber, P. Bateman, J. T. Tate, A. Magid, G. D. Mostow, W. Ferrer Santos, C. Moore, B. Kostant, G. M. Bergman, M. Moskowitz, and N. Nahlus: “Gerhard Hochschild (1915–2010),” Notices Am. Math. Soc. 58 : 8 (2011), pp. 1078–1099. Ferrer Santos and Moskowitz were coordinating editors. MR 2856143 Zbl 1225.01083 article

Murray Gerstenhaber	Related
Gerhard Paul Hochschild	Related
Andy R. Magid	Related
Martin Allen Moskowitz	Related
George Daniel Mostow	Related
Nazih Nahlus	Related
George M. Bergman	Related
Walter Ricardo Ferrer Santos	Related
James Oliver Schwartz	Related
John Torrence Tate, Jr.	Related
Paul T. Bateman	Related	Volume
Bertram Kostant	Related

[56] T. Austin and C. C. Moore: “Continuity properties of measurable group cohomology,” Math. Ann. 356 : 3 (2013), pp. 885–937. MR 3063901 Zbl 1330.22005 article

[57] C. C. Moore: “Ergodic theorem, ergodic theory, and statistical mechanics,” Proc. Natl. Acad. Sci. USA 112 : 7 (2015), pp. 1907–1911. MR 3324732 Zbl 1355.37001 article

This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundamental problem of the subject — namely, the rationale for the hypothesis that time averages can be set equal to phase averages. The evolution of this problem is traced from the origins of statistical mechanics and Boltzman’s ergodic hypothesis to the Ehrenfests’ quasi-ergodic hypothesis, and then to the ergodic theorems. We discuss communications between von Neumann and Birkhoff in the Fall of 1931 leading up to the publication of these papers and related issues of priority. These ergodic theorems initiated a new field of mathematical-research called ergodic theory that has thrived ever since, and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics.