Abstract
For locally compact groups G, the author introduced a notion of [IA] groups that is if there exists an inner invariant mean on G. This concept generalizes the concept of amenability for locally compact groups and the concept of inner amenability for discrete groups, hence gives a new classification of locally compact groups. The purpose of this paper is to characterize [IA] groups by generalizing the well-known Følner's conditions. Our main result is the following equivalence: A locally compact group G is [IA] if and only if G admits measurable subsets having finite non-zero Haar measure which are not substantially modified by conjugations of the gorup.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Charles A. Akemann, Operator algebras associated with fuchsian groups, Houston J. Math., 7, 1981, 295–301.
Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1, 1957, 509–544.
Mahlon M. Day, Convolutions, means, and spectra, Illinois J. Math., 8, 1964, 100–111.
Jean Dieudonné, Sur le produit de composition (II), J. Math. Pures Appl., 39, 1960, 275–292.
Erling Følner, On groups with full Banach mean value, Math. Scand., 3, 1955, 243–254.
Frederick P. Greenleaf, Invariant Means on Topological Groups, Van Nostrand, New York, 1969.
Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Vol. 1, 2nd ed., Spring-Verlag, Berlin, 1979.
Viktor Losert and Harald Rindler, Conjugation invariant means, Colloq. Math. 51, 1987, 221–225.
Jean-Paul Pier, Quasi-invariant interieure sur les groupe Localement compacts, Actualités Mathématiques, Gauthier-Villars, Paris, 1982, 431–436.
Jean-Paul Pier, Amenable Locally Compact Groups, John Wiley and Sons Inc., New York, 1984.
Walter Rudin, Invariant means on L ∞(G), Studia. Math. T. XLIV, 1972, 219–227.
Chuan Kuan Yuan, The existence of inner invariant means on L ∞(G), J. Math. Anal. Appl., vol. 130, No.2, 1988, 514–524.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag
About this paper
Cite this paper
Yuan, C.K. (1991). On the structures of locally compact groups admitting inner invariant means. In: Cheng, MT., Deng, DG., Zhou, XW. (eds) Harmonic Analysis. Lecture Notes in Mathematics, vol 1494. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087774
Download citation
DOI: https://doi.org/10.1007/BFb0087774
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54901-7
Online ISBN: 978-3-540-46474-7
eBook Packages: Springer Book Archive