Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Discontinuous translation invariant functionals


Author: Sadahiro Saeki
Journal: Trans. Amer. Math. Soc. 282 (1984), 403-414
MSC: Primary 43A15; Secondary 43A05
DOI: https://doi.org/10.1090/S0002-9947-1984-0728720-5
MathSciNet review: 728720
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be an infinite $ \sigma$-compact locally compact group. We shall study the existence of many discontinuous translation invariant linear functionals on a variety of translation invariant Fréchet spaces of Radon measures on $ G$. These spaces include the convolution measure algebra $ M(G)$, the Lebesgue spaces $ {L^p}(G)$, where $ 1 \leq p \leq \infty $, and certain combinations thereof. Among other things, it will be shown that $ C(G)$ has many discontinuous translation invariant functionals, provided that $ G$ is amenable. This solves a problem of G. H. Meisters.


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

  • [1] C. Chou, The exact cardinality of the set of invariant means on a group, Proc. Amer. Math. Soc. 55 (1976), 103-106. MR 0394036 (52:14842)
  • [2] P. J. Cohen, Factorization in group algebras, Duke Math. J. 26 (1959), 199-205. MR 0104982 (21:3729)
  • [3] E. Granirer, On left amenable semigroups which admit countable left-invariant means, Bull. Amer. Math. Soc. 69 (1963), 101-105. MR 0147572 (26:5087)
  • [4] -, On the invariant mean on topological semigroups and on topological groups, Pacific J. Math. 15 (1965), 107-140. MR 0209388 (35:286)
  • [5] F. P. Greenleaf, Invariant means on topological groups and their applications, Math. Studies, no 16, Van Nostrand, New York, 1969. MR 0251549 (40:4776)
  • [6] E. Hewitt, The ranges of certain convolution operators, Math. Scand. 15 (1964), 147-155. MR 0187016 (32:4471)
  • [7] E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. I, Springer-Verlag, Berlin, Gottingen and Heidelberg, 1963. MR 551496 (81k:43001)
  • [8] M. Jerison and W. Rudin, Translation-invariant functionals, Proc. Amer. Math. Soc. 13 (1962), 417-423. MR 0146650 (26:4170)
  • [9] C. J. Lester, Continuity of operators on $ {L^2}(G)$ and $ {L^1}(G)$ commuting with translations, J. London Math. Soc. 11 (1975), 144-146. MR 0380269 (52:1169)
  • [10] G. H. Meisters, Translation-invariant linear forms and a formula for the Dirac measure, J. Funct. Anal. 8 (1971), 173-188. MR 0288575 (44:5772)
  • [11] -, Some discontinuous translation-invariant linear forms, J. Funct. Anal. 12 (1973), 199-210. MR 0346418 (49:11143)
  • [12] -, A Guichard theorem on connected monothetic groups, Studia Math. 43 (1973), 161-163. MR 0336228 (49:1004)
  • [13] -, Periodic distributions and non-Liouville numbers, J. Funct. Anal. 26 (1977), 68-88. MR 0448068 (56:6378)
  • [14] -, Some problems and results on translation-invariant linear forms, Preprint, 1982.
  • [15] G. H. Meisters and W. M. Schmidt, Translation-invariant linear forms on $ {L^2}(G)$ for compact abelian groups, J. Funct. Anal. 11 (1972), 407-424. MR 0346417 (49:11142)
  • [16] A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Univ. Press, London and New York, 1973. MR 0350361 (50:2854)
  • [17] W. Roelcke, L. Asam, S. Dierolf and P. Dierolf, Discontinuous translation-invariant linear forms on $ K(G)$, Math. Ann. 239 (1979), 219-222. MR 522779 (80c:43001)
  • [18] J. M. Rosenblatt, Invariant means for the bounded measurable functions on a non-discrete locally compact group, Math. Ann. 220 (1976), 219-228. MR 0397305 (53:1164)
  • [19] -, The number of extensions of an invariant mean, Compositio Math. 33 (1976), 147-159. MR 0435729 (55:8687)
  • [20] J. M. Rosenblatt and M. Talagrand, Different types of invariant means, J. London Math. Soc. 24 (1981), 525-532. MR 635883 (83j:43001)
  • [21] W. Rudin, Invariant means on $ {L^\infty }$, Studia Math. 44 (1972), 219-227. MR 0304975 (46:4105)
  • [22] G. S. Woodward, Translation-invariant linear forms on $ {C_0}(G),\,C(G),\,{L^p}(G)$ for noncompact groups, J. Funct. Anal. 16 (1974), 205-220. MR 0344801 (49:9540)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 43A15, 43A05

Retrieve articles in all journals with MSC: 43A15, 43A05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0728720-5
Keywords: Translation invariant Fréchet space, discontinuous invariant functional, Radon measure
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society