×

zbMATH — the first resource for mathematics

Groups admitting ergodic actions with generalized discrete spectrum. (English) Zbl 0399.22005
The class of groups admitting an effective ergodic action with generalized discrete spectrum is a natural generalization of the class of maximally almost periodic groups. H. Freudenthal has given a complete characterization of the connected maximally almost periodic groups, and here we give a complete characterization of the almost connected groups admitting an effective ergodic action with generalized discrete spectrum. Namely, we show that an almost connected group is in this class if and only if it is type R. It is known that this is equivalent to the group being of polynomial growth, and for Lie groups is just the condition that all eigenvalues of the adjoint representation lie on the unit circle.
Reviewer: Calvin C. Moore

MSC:
22D40 Ergodic theory on groups
28D15 General groups of measure-preserving transformations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Auslander, L., Green, L., Hahn, F.: Flows on homogeneous spaces. Annals of Math. Studies, No. 53, Princeton 1963 · Zbl 0106.36802
[2] Auslander, L., Green, L.:G-induced flows, Amer. J. Math.,88, 43-60 (1966) · Zbl 0149.19903
[3] Auslander, L.: An exposition of the structure of solvmanifolds. Bull. Amer. Math. Soc.,79, 227-285 (1973) · Zbl 0265.22016
[4] Effros, E.G.: Transformation groups andC *-algebras. Annals of Math.,81, 38-55 (1965) · Zbl 0152.33203
[5] Freudenthal, H.: Topologische Gruppen mit genügend vielen fastperiodischen Funktionen. Annals of Math.,37, 57-77 (1936) · JFM 62.0437.02
[6] Furstenberg, H.: The structure of distal flows. Amer. J. Math.,85, 477-515 (1963) · Zbl 0199.27202
[7] Guivarc’h, Y.: Croissance polynomiale et périodes des fonctions harmonique. Bull. Math. Soc. France,101, 333-379 (1973)
[8] Hochschild, G.: The Structure of Lie Groups. San Francisco: Holden-Day 1965 · Zbl 0131.02702
[9] Iwasawa, K.: Some types of topological groups. Ann. of Math.,50, 507-558 (1949) · Zbl 0034.01803
[10] Jenkins, J.W.: Growth of connected locally compact groups. J. Funct. Anal.12, 113-127 (1973) · Zbl 0247.43001
[11] Knapp, A.W.: Distal functions on groups. Trans. Amer. Math. Soc.,128, 1-40 (1967) · Zbl 0154.33702
[12] Kostant, B.: Quantization and unitary representations. Lectures in Modern Analysis and Applications III, Springer Lecture Notes 170 (1970) · Zbl 0223.53028
[13] Lashof, R.K.: Lie algebras in locally compact groups. Pac. J. of Math.,7, 1145-1162 (1957) · Zbl 0081.02204
[14] Mackey, G.W.: Borel structures in groups and their duals. Trans. Amer. Math. Soc.,85, 134-165 (1957) · Zbl 0082.11201
[15] Mackey, G.W.: Ergodic transformation groups with a pure point spectrum. Illinois J. Math.,8, 593-600 (1964) · Zbl 0255.22014
[16] Mackey, G.W.: Ergodic theory and virtual groups. Math. Ann.166, 187-207 (1966) · Zbl 0178.38802
[17] Mackey, G.W.: Ergodic theory and its significance for statistical mechanics and probability theory. Advances in Math.,12, 178-268 (1974) · Zbl 0326.60001
[18] Montgomery, D., Zippen, L.: Topological transformation groups. Interscience, New York 1955
[19] Moore, C.C.: Extensions and cohomology theory of locally compact groups, I. Trans. Amer. Math. Soc.,113, 40-63 (1964) · Zbl 0131.26902
[20] Moore, C.C.: Extensions and cohomology theory of locally compact groups, II. Trans. Amer. Math. Soc.,113, 64-86 (1964) · Zbl 0131.26902
[21] Moore, C.C.: Extensions and cohomology theory of locally compact groups, III. Trans. Amer. Math. Soc.,221, 1-33 (1976) · Zbl 0366.22005
[22] Moore, C.C.: Extensions and cohomology theory of locally compact groups, IV. Trans. Amer. Math. Soc.,221, 35-58 (1976) · Zbl 0366.22006
[23] Moore, C.C.: Distal affine transformation groups. Amer. J. of Math.,90, 733-751 (1968) · Zbl 0195.52604
[24] Moore, C.C., Rosenberg, J.: Groups withT 1 primitive ideal spaces. J. Funct. Anal.22, 204-224 (1976) · Zbl 0328.22014
[25] Moore, C.C., schmidt, K.: Coboundaries and homomorphisms for non-singular actions and a problem of H. Helson, Proc. Lond. Math. Soc. · Zbl 0428.28014
[26] Parry, W.: Zero entropy of distal and related transformations, in Topological Dynamics (J. Auslander and W. Gottschalk, eds.) New York: Benjamin, 1968 · Zbl 0193.51602
[27] Ramsay, A.: Virtual Groups and Group Actions. Advances in Math.,6, 253-322 (1971) · Zbl 0216.14902
[28] Ramsay, A.: Nontransitive quasi-orbits in Mackey’s analysis of group extensions. Acta Math.,137, 17-48 (1976) · Zbl 0334.22004
[29] Varadarajan, V.S.: Geometry of Quantum Theory, Vol. II. Princeton: Van Nostrand, 1970 · Zbl 0194.28802
[30] Zimmer, R.J.: Extensions of ergodic group actions. Illinois J. Math.,20, 373-409 (1976) · Zbl 0334.28015
[31] Zimmer, R.J.: Ergodic actions with generalized discrete spectrum. Illinois J. Math.,20, 555-588 (1976) · Zbl 0349.28011
[32] Zimmer, R.J.: Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Functional Anal.,27, 350-372 (1978) · Zbl 0391.28011
[33] Zimmer, R.J.: Induced and amenable ergodic actions of Lie groups. To appear, Ann, Sci. Ec. Norm. Sup. · Zbl 0401.22009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.