## Rocky Mountain Journal of Mathematics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 18 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1492502545

Digital Object Identifier
doi:10.1216/RMJ-2017-47-2-445

Mathematical Reviews number (MathSciNet)
MR3635369

Zentralblatt MATH identifier
1371.43002

#### Citation

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. https://projecteuclid.org/euclid.rmjm/1492502545

#### References

• 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.