Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Topological invariant means on locally compact groups and fixed points.


Author: James C. S. Wong
Journal: Proc. Amer. Math. Soc. 27 (1971), 572-578
MSC: Primary 22.65
DOI: https://doi.org/10.1090/S0002-9939-1971-0272954-X
MathSciNet review: 0272954
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A locally compact group $ G$ is said to have the fixed point property if whenever $ G$ acts affinely on a compact convex subset $ S$ of a separated locally convex space $ E$ with the map $ G \times S \to S$ jointly continuous, there is a fixed point for the action. N. Rickert has proved that $ G$ has this fixed point property if $ G$ is amenable. In this paper, we study the fixed point property for actions of the algebras $ {L_1}(G)$ and $ M(G)$ and prove some fixed point theorems for locally compact groups.


References [Enhancements On Off] (What's this?)

  • [1] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544. MR 19, 1067. MR 0092128 (19:1067c)
  • [2] -, Fixed-point theorems for compact convex sets, Illinois J. Math. 5 (1961), 585-590. MR 25 #1547. MR 0138100 (25:1547)
  • [3] -, Correction to my paper ``Fixed-point theorems for compact convex sets," Illinois J. Math. 8 (1964), 713. MR 29 #6463. MR 0169210 (29:6463)
  • [4] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [5] R. Ellis, Locally compact transformation groups, Duke Math J. 24 (1957), 119-125. MR 19, 561. MR 0088674 (19:561b)
  • [6] E. E. Granirer, Extremely amenable semigroups, Math. Scand. 17 (1965), 177-197. MR 33 #5760. MR 0197595 (33:5760)
  • [7] -, Extremely amenable semigroups. II, Math. Scand. 20 (1967), 93-113. MR 35 #3422; MR 36 #1561. MR 0212551 (35:3422)
  • [8] -, Functional analytic properties of extremely amenable semigroups, Trans. Amer. Math. Soc. 137 (1969), 53-75. MR 39 #765. MR 0239408 (39:765)
  • [9] F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand, Princeton, N. J., 1969. MR 0251549 (40:4776)
  • [10] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #158. MR 551496 (81k:43001)
  • [11] A. Hulanicki, Means and Følner condition on locally compact groups, Studia Math. 27 (1966), 87-104. MR 33 #4178. MR 0195982 (33:4178)
  • [12] T. Mitchell, Constant functions and left invariant means on semigroups, Trans. Amer. Math. Soc. 119 (1965), 244-261. MR 33 #1743. MR 0193523 (33:1743)
  • [13] -, Fixed points and multiplicative left invariant means, Trans. Amer. Math. Soc. 122 (1966), 195-202. MR 32 #7662. MR 0190249 (32:7662)
  • [14] -, Function algebras, means, and fixed points, Trans. Amer. Math. Soc. 130 (1968), 117-126. MR 36 #666. MR 0217577 (36:666)
  • [15] -, Topological semigroups and fixed points (to appear).
  • [16] I. Namioka, On certain actions of semi-groups on $ L$-spaces, Studia Math. 29 (1967), 63-77. MR 36 #6910. MR 0223863 (36:6910)
  • [17] N. W. Rickert, Amenable groups and groups with the fixed point property, Trans. Amer. Math. Soc. 127 (1967), 221-232. MR 36 #5260. MR 0222208 (36:5260)
  • [18] A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Univ. Press, New York, 1964. MR 28 #5318. MR 0162118 (28:5318)
  • [19] R. J. Silverman, Means on semigroups and the Hahn-Banach extension property, Trans. Amer. Math. Soc. 83 (1956), 222-237. MR 18, 910. MR 0084721 (18:910b)
  • [20] James C. S. Wong, Topologically stationary locally compact groups and amenability, Trans. Amer. Math. Soc. 144 (1969), 351-363. MR 0249536 (40:2781)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 22.65

Retrieve articles in all journals with MSC: 22.65


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0272954-X
Keywords: Locally compact groups, fixed point properties, fixed point theorems, amenability, topological left introverted spaces, convolution algebras $ {L_1}(G)$ and $ M(G)$, Silverman's invariant extension property
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society