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)



The weak closure of the set
of left translation operators

Authors: Ching Chou and Guangwu Xu
Journal: Proc. Amer. Math. Soc. 127 (1999), 465-471
MSC (1991): Primary 43A30, 46A50, 46L10; Secondary 43A07, 43A46, 46L05
MathSciNet review: 1468187
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is known that for an amenable locally compact group $G$, $0$ is not in the weak closure of $\{ \lambda (g) : g \in G \}$ of $VN(G)$. In this paper, it is proved that the converse of this is true. In other words, if $G$ is a non-amenable locally compact group, then $0$ is in the weak closure of $\{ \lambda (g) : g \in G \}$. This answers several questions of Ülger. Applications to the algebra $C^{*}_{\delta }(G)$ and the dual of the reduced group $C^{*}$-algebra are obtained.

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

  • 1. E. Bédos, On the $C^{*}-$algebra generated by the left regular representation of a loclly compact group, Proc. Amer. Math. Soc. 120 (1994), 603-608. MR 94d:22004
  • 2. M. Cowling, An application of Littlewood-Paley theory in harmonic analysis, Math. Ann. 241 (1979), 83-96. MR 81f:43003
  • 3. P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. MR 37:4208
  • 4. E. E. Granirer, Some results on $A_{p}(G)$ submodules of $PM_{p}(G)$, Colloq. Math. 51 (1987), 155-163. MR 88f:43008
  • 5. -, 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
  • 6. -, When quotients of the Fourier algebra $A(G)$ are ideals of their bidual and when $A(G)$ has WCHP, Math Japonica 46 (1997), no. 1, 69-72. CMP 97:17
  • 7. F. P. Greenleaf, Invariant Means on Topological Groups, Van Nostrand, New York, 1969. MR 40:4776
  • 8. U. Haagerup, An example of a nonnuclear $C^{*}$-algebra which has the metric approximation property, Invent. Math. 50 (1979), 279-293. MR 80j:46094
  • 9. C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier ( Grenoble ) 23 (1973), 91-123. MR 50:7956
  • 10. M. Leinert, Faltungsoperatoren auf gewissen diskreten gruppen, Studia Mat. 52 (1973), 149-158. MR 50:7954
  • 11. H. Leptin, On locally compact groups with invariant means, Proc. Amer. Math. Soc. 19 (1968), 489-494. MR 39:361
  • 12. J. Pier, Amenable Locally Compact Groups, John Wiley & Sons, New York, 1984. MR 86a:43001
  • 13. A. Ülger, Some results about the spectrum of commutative Banach algebras under the weak topology and applications, Mh Math. 121 (1996), 353-379. MR 98a:46058

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 43A30, 46A50, 46L10, 43A07, 43A46, 46L05

Retrieve articles in all journals with MSC (1991): 43A30, 46A50, 46L10, 43A07, 43A46, 46L05

Additional Information

Ching Chou
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14214

Guangwu Xu
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 1G2

Keywords: Weak closure, von Neumann algebras, Fourier algebras, amenable groups
Received by editor(s): January 20, 1997
Received by editor(s) in revised form: May 21, 1997
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society