Rocky Mountain Journal of Mathematics

Invariantly complemented and amenability in Banach algebras related to locally compact groups

Ali Ghaffari and Somayeh Amirjan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, among other things, we show that there is a close connection between the existence of a bounded projection on some Banach algebras associated to a locally compact group~$G$ and the existence of a left invariant mean on $L^\infty (G)$. A necessary and sufficient condition is found for a locally compact group to possess a left invariant mean.

Article information

Rocky Mountain J. Math., Volume 47, Number 2 (2017), 445-461.

First available in Project Euclid: 18 April 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 43A07: Means on groups, semigroups, etc.; amenable groups
Secondary: 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.

Amenability Banach algebra group algebra homomorphism operator projection weak$^*$ topology


Ghaffari, Ali; Amirjan, Somayeh. Invariantly complemented and amenability in Banach algebras related to locally compact groups. Rocky Mountain J. Math. 47 (2017), no. 2, 445--461. doi:10.1216/RMJ-2017-47-2-445.

Export citation


  • M.E.B. Bekka, Complemented subspace of $L^\infty(G)$, ideals of $L^1(G)$ and amenability, Monatsh. Math. 109 (1990), 195–203.
  • N. Bourbaki, Elements de mathématique, 25, Première partie, Livre VI: Intégration, Chapitre 6: Intégration vectorielle, Act. Sci. Ind. 1281, Hermann, Paris, 1959.
  • H.G. Dales, Banach algebra and automatic continuity, Lond. Math. Soc. Mono. 24, Clarendon Press, Oxford, 2000.
  • B. Forrest, Amenability and bounded approximate identities in ideals of $A(G)$, Illinois J. Math. 34 (1990), 1–25.
  • A. Ghaffari, Projections onto invariant subspaces of some Banach algebras, Acta Math. Sinica 24 (2008), 1089–1096.
  • E. Hewitt and K.A. Ross, Abstract harmonic analysis, Volume I, Springer Verlag, Berlin, 1963; Volume II, Springer Verlag, Berlin, 1970.
  • A.T. Lau, Invariantly complemented subspaces of $L^\infty(G)$ and amenable locally group, Illinois J. Math. 26 (1982), 226–235.
  • ––––, Operators which commute with convolution on subspaces of $L^\infty (G)$, Colloq. Math. 39 (1978), 351–359.
  • A.T. Lau and V. Losert, Complementation of certain subspaces of $L^\infty(G)$ of a locally compact group, Pacific J. Math. 141 (1990), 295–310.
  • ––––, Weak$^\ast$-closed complemented invariant subspaces of $L_\infty(G)$ and amenable locally compact groups, Pacific J. Math. 123 (1986), 149–159.
  • A.T. Lau and J. Pym, Concerning the second dual of the group algebra of a locally compact group, J. Lond. Math. Soc. 41 (1990), 445–460.
  • A.T. Lau and A. Ulger, Characterization of closed ideals with bounded approximate identities in commutative Banach algebras, complemented subspaces of the group von Neumann algebras and applications, Trans. Amer. Math. Soc. 366 (2014), 4151–4171.
  • J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, I, Springer-Verlag, Berlin, 1977.
  • T.S. Lui, A. van Rooij and J.K. Wang, Projections and approximate identities for ideals in group algebras, Trans. Amer. Math. Soc. 175 (1973), 469–482.
  • A.L.T. Paterson, Amenability, Amer. Math. Soc. Math. Surv. Mono. 29, Providence, Rhode Island, 1988.
  • J.P. Pier, Amenable locally compact groups, John Wiley And Sons, New York, 1984.
  • H.P. Rosenthal, Projections onto translation invariant subspaces of $L^p(G)$, Mem. Amer. Math. Soc. 63 (1966), 84 pages.
  • W. Rudin, Functional analysis, McGraw Hill, New York, 1991.
  • ––––, Projections on invariant subspaces, Proc. Amer. Math. Soc. 13 (1962), 429–432.
  • V. Runde, Lectures on amenability, Lect. Notes Math. 1774, Springer-Verlag, Berlin, 2002.
  • M. Takahashi, Remarks on certain complemented subspaces on groups, Hokkaido Math. J. 13 (1984), 260–270.
  • P.J. Wood, Invariant complementation and projectivity in the Fourier algebra, Proc. Amer. Math. Soc. 131 (2002), 1881–1890.
  • ––––, Complemented ideals in the Fourier algebra of a locally compact group, Proc. Amer. Math. Soc. 128 (1999), 445–451.
  • Y. Zhang, Approximate complementation and its applications in studying ideals of Banach algebras, Math. Scand. 92 (2003), 301–308.