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)



Finite-dimensional left ideals in some algebras
associated with a locally compact group

Author: M. Filali
Journal: Proc. Amer. Math. Soc. 127 (1999), 2325-2333
MSC (1991): Primary 43A10, 22D15
Published electronically: April 9, 1999
MathSciNet review: 1487366
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a locally compact group, let $L^{1}(G)$ be its group algebra, let $M(G)$ be its usual measure algebra, let $L^{1}(G)^{**}$ be the second dual of $L^{1}(G)$ with an Arens product, and let $LUC(G)^{*}$ be the conjugate of the space $LUC(G)$ of bounded, left uniformly continuous, complex-valued functions on $G$ with an Arens-type product. We find all the finite-dimensional left ideals of these algebras. We deduce that such ideals exist in $L^{1}(G)$ and $M(G)$ if and only if $G$ is compact, and in $L^{1}(G)^{**}$ (except those generated by right annihilators of $L^{1}(G)^{**}$) and $LUC(G)^{*}$ if and only if $G$ is amenable.

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

  • [1] J. W. Baker and M. Filali, Minimal ideals in group algebras and their biduals, Math. Proc. Cambridge Philos. Soc. 120 (1996), 475-488. CMP 96:12
  • [2] J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on Semigroups: Function Spaces, Compactifications, Representations, Wiley, New York, 1989. MR 91b:43001
  • [3] M. Filali, The ideal structure of some Banach algebras, Math. Proc. Cambridge Philos. Soc. 111 (1992), 567-576. MR 93e:22005
  • [4] M. Filali, Linear equations in $B(\mathbb{Z})^{*}$, Proc. Roy. Soc. Edinburgh Sect. A 123A (1993), 1001-1009. MR 95j:39035
  • [5] M. Filali, Minimal left ideals in the group algebra and its second conjugate, Mathematica Japonica (to appear).
  • [6] E. E. Granirer, The radical of $L^{\infty }(G)^{*}$, Proc. Amer. Math. Soc. 41 (1973), 321-324. MR 48:4646
  • [7] F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand, New York, 1969. MR 40:4776
  • [8] E. Hewitt and K. A. Ross, Abstract harmonic analysis I, Springer-Verlag, Berlin,, 1963, 1970. MR 28:158
  • [9] B. Sz.-Nagy, Uniformly bounded linear transformations in Hilbert space, Acta Math. (Szeged) 11 (1947), 152-157.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 43A10, 22D15

Retrieve articles in all journals with MSC (1991): 43A10, 22D15

Additional Information

M. Filali
Affiliation: Department of Mathematical Sciences, University of Oulu, SF 90570 Finland

Keywords: Locally compact group, Arens product, representation, amenable, $U$-invariant, finite-dimensional left ideal
Received by editor(s): October 28, 1997
Published electronically: April 9, 1999
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society