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)

 
 

 

On the set of topologically invariant means on an algebra of convolution operators on $L^{p}(G)$


Author: Edmond E. Granirer
Journal: Proc. Amer. Math. Soc. 124 (1996), 3399-3406
MSC (1991): Primary 43A22, 42B15, 22D15; Secondary 42A45, 43A07, 44A35, 22D25
DOI: https://doi.org/10.1090/S0002-9939-96-03444-2
Erratum: Proc. Amer. Math. Soc. (recently posted)
MathSciNet review: 1340388
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a locally compact group, $A_{p}=A_{p}(G)$ the Banach algebra defined by Herz; thus $A_{2}(G)=A(G)$ is the Fourier algebra of $G$. Let $PM_{p}=A^{*}_{p}$ the dual, $J \subset A_{p}$ a closed ideal, with zero set $F=Z(J)$, and $\mathbb {P} = (A_{p}/J)^{*}$. We consider the set $TIM_{\mathbb {P}}(x) \subset {\mathbb {P}}^{*}$ of topologically invariant means on $\mathbb {P}$ at $x\in F$, where $F$ is ``thin.'' We show that in certain cases card $TIM_{\mathbb {P}}(x) \geq 2^{c}$ and $TIM_{\mathbb {P}}(x)$ does not have the WRNP, i.e. is far from being weakly compact in $\mathbb {P}^{*}$. This implies the non-Arens regularity of the algebra $A_{p}/J$.


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

  • [BL] Y. Benyamini and P.K. Lin, Norm one multipliers on $L^{p}(G)$, Arkiv for Matematik 24 (1986), 159--173. MR 88e:42015
  • [Ch1] Ching Chou, Weakly almost periodic functions and Fourier Stiltjes algebras of locally compact groups, Trans. Amer. Math Soc. 274 (1982), 141--157. MR 84a:43008
  • [Ch2] ------, Topological invariant means on the Von Neumann algebra VN(G), Trans. Amer. Math. Soc. 273 (1982), 207--229. MR 83k:20016
  • [Cow] M. Cowling, An application of Littlewood-Paley theory in Harmonic Analysis, Math. Ann. 241 (1979), 83--96. MR 81f:43003
  • [DU] J. Diestel and J.J. Uhl Jr., Vector Measures, Amer. Math. Soc. Math Surveys No. 15, 1977. MR 56:12216
  • [Ey1] P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181--236. MR 37:4208
  • [Ey2] P. Eymard, Algèbres $A_{p}$ et convoluteurs de $L^{p}$, Seminaire Bourbaki 22e année, 1969/70, no. 367, Lecture Notes in Math., vol. 180, Springer, New York, 1971. MR 42:7461
  • [FG] A. Figa-Talamanca and G.I. Gaudry, Density and representation theorems for multipliers of type $(p,q)$, J. Australian Math. Soc. 7 (1967), 1--6. MR 35:666
  • [GMc] C.C. Graham and O. Caruth McGehee, Essays in commutative harmonic analysis,
    Springer-Verlag, New York, 1979. MR 81d:43001
  • [Gr1] E.E. Granirer, On some spaces of linear functionals on the algebras $A_{p}(G)$ for locally compact groups, Colloq. Math. 52 (1987), 119--132. MR 88k:43006
  • [Gr2] ------, Geometric and topological properties of certain $w^{*}$ compact convex subsets of double duals of Banach spaces which arise from the study of invariant means, Illinois J. Math. 30 (1986), 148--174. MR 87f:43001
  • [Gr3] ------, On convolution operators with small support which are far from being convolution by a bounded measure, Colloq. Math. 67 (1994), 33--60. CMP 94:17
  • [Gr4] ------, Day points for quotients of the Fourier algebra $A(G)$, extreme nonergodicity of their duals and extreme non Arens regularity, Illinois J. Math. (to appear).
  • [He] H. Helson, Fourier transforms on perfect sets, Studia Math. 14 (1954), 209--213. MR 16:817
  • [HR] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis Vols. I,II, Springer-Verlag, 1963, 1970. MR 28:158; MR 41:7378
  • [Hu] Zhiguo Hu, On the set of topologically invariant means on the Von Neumann algebra $VN(G)$, Illinois J. Math. (to appear).
  • [Hz] C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier Grenoble 37 (1973), 91--123. MR 50:7556
  • [LP] Anthony To-Ming Lau and Alan L.T. Paterson, The exact cardinality of topologically invariant means on amenable locally compact groups, Proc. A.M.S. 98 (1986), 75--80. MR 88c:43001
  • [Lo] L.H. Loomis, The spectral characterisation of a class of almost periodic functions, Annals of Math. 72 (1960), 362--368. MR 22:11255
  • [Me] Y. Meyer, Recent advances in spectral synthesis, Conference on Harmonic Analysis. College Park, Maryland. 1971 Springer Lecture Notes in Math., No. 266, pp. 239--253. MR 52:14863
  • [Mc] O. Caruth McGehee, Helson sets in $T^{n}$, Conference on Harmonic Analysis. College Park, Maryland. 1971. Springer Lecture Notes in Math., No. 266, pp. 229--237. MR 52:14862
  • [Pa] A.L.T. Paterson, Amenability, Amer. Math. Soc. Mathematical Surveys and Monographs No. 29, 1988. MR 90e:43001
  • [Ru1] W. Rudin, Fourier Analysis on Groups, John Wiley and Sons, 1960. MR 27:2808
  • [Ru2] ------, Functional Analysis, McGraw-Hill Inc., 1973. MR 51:1315
  • [Ru3] ------, Averages of continuous functions on compact spaces, Duke Math. J. 25 (1958), 197--204. MR 20:4774
  • [Sa1] E. Saab, Some characterizations of weak Radon-Nikodým sets, Proc. A.M.S. 86 (1982), 307--311. MR 84g:46030
  • [Ze] E.I. Zelmanov, On periodic compact groups, Israel J. Math. 77 (1992), 83--95. MR 94e:20055

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 43A22, 42B15, 22D15, 42A45, 43A07, 44A35, 22D25

Retrieve articles in all journals with MSC (1991): 43A22, 42B15, 22D15, 42A45, 43A07, 44A35, 22D25


Additional Information

Edmond E. Granirer
Affiliation: Department of Mathematics, The University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email: granirer@math.ubc.ca

DOI: https://doi.org/10.1090/S0002-9939-96-03444-2
Received by editor(s): March 13, 1995
Received by editor(s) in revised form: May 8, 1995
Communicated by: Dale E. Alspach
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society