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SOME HOMOLOGICAL PROPERTIES OF FOURIER ALGEBRAS ON HOMOGENEOUS SPACES

Part of: Selfadjoint operator algebras Abstract harmonic analysis

Published online by Cambridge University Press:  09 November 2020

REZA ESMAILVANDI  and
MEHDI NEMATI
REZA ESMAILVANDI
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan84156-83111, Iran e-mail: [email protected]
MEHDI NEMATI*
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan84156-83111, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, P.O. Box: 19395–5746, Iran
*
e-mail: [email protected]
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Abstract

Let $ H $ be a compact subgroup of a locally compact group $ G $ . We first investigate some (operator) (co)homological properties of the Fourier algebra $A(G/H)$ of the homogeneous space $G/H$ such as (operator) approximate biprojectivity and pseudo-contractibility. In particular, we show that $ A(G/H) $ is operator approximately biprojective if and only if $ G/H $ is discrete. We also show that $A(G/H)^{**}$ is boundedly approximately amenable if and only if G is compact and H is open. Finally, we consider the question of existence of weakly compact multipliers on $A(G/H)$ .

Keywords

Fourier algebrahomogeneous spacehomological propertyweakly compact multiplier

MSC classification

Primary: 43A85: Analysis on homogeneous spaces
Secondary: 43A07: Means on groups, semigroups, etc.; amenable groups 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 46L07: Operator spaces and completely bounded maps
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 1 , August 2021 , pp. 132 - 140
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

This research for the second author was in part supported by a grant from IPM (no. 99170411).

References

Baker, J., Lau, A. T.-M. and Pym, J. S., ‘Module homomorphisms and topological centers associated with weakly sequentially complete Banach algebras’, J. Funct. Anal. 158 (1998), 186–208.CrossRefGoogle Scholar
Choi, Y. and Ghahramani, F., ‘Approximate amenability of Schatten classes, Lipschitz algebras and second duals of Fourier algebras’, Q. J. Math. 62 (2011), 39–58.CrossRefGoogle Scholar
Chu, C-H. and Lau, A. T.-M., ‘Jordan structures in harmonic functions and Fourier algebras on homogeneous spaces’, Math. Ann. 336 (2006), 803–840.CrossRefGoogle Scholar
Dales, H. G., Banach Algebras and Automatic Continuity, London Mathematical Society Monographs. New Series, 24 (Oxford University Press, New York, 2000).Google Scholar
Effros, E. G. and Ruan, Z.-J., Operator Spaces (Oxford University Press, Oxford, 2000).Google Scholar
Eymard, P., ‘L’algèbre de Fourier d’un groupe localement compact’, Bull. Soc. Math. France 92 (1964), 181–236.CrossRefGoogle Scholar
Forrest, B., ‘Fourier analysis on coset spaces’, Rocky Mountain J. Math. 28 (1998), 173–190.CrossRefGoogle Scholar
Forrest, B., Samei, E. and Spronk, N., ‘Convolutions on compact groups and Fourier algebras of coset spaces’, Studia Math. 196(3) (2010), 223–249.CrossRefGoogle Scholar
Ghahramani, F. and Lau, A. T.-M., ‘Multipliers and modulus on Banach algebras related to locally compact groups’, J. Funct. Anal. 150 (1997), 478–497.CrossRefGoogle Scholar
Ghahramani, F. and Zhang, Y., ‘Pseudo-amenable and pseudo-contractible Banach algebras’, Math. Proc. Cambridge Philos. Soc. 142 (2007), 111–123.CrossRefGoogle Scholar
Ilie, M., ‘Dual Banach algebras associated to the coset space of a locally compact group’, J. Math. Anal. Appl. 467 (2018), 550–569.Google Scholar
Kaniuth, K. and Lau, A. T.-M., Fourier and Fourier–Stieljes Algebras on Locally Compact Groups, Mathematical Surveys and Monographs, 231 (American Mathematical Society, Providence, RI, 2018).CrossRefGoogle Scholar
Kumar, N. S., ‘Invariant mean on a class of von Neumann algebras related to ultraspherical hypergroups II’, Canad. Math. Bull. 60 (2017), 402–410.CrossRefGoogle Scholar
Larsen, R., An Introduction to the Theory of Multipliers (Springer-Verlag, Berlin, 1971).CrossRefGoogle Scholar
Lau, A. T.-M., ‘Uniformly continuous functionals on the Fourier algebra of any locally compact group’, Trans. Amer. Math. Soc. 251 (1979), 39–59.CrossRefGoogle Scholar
Lau, A. T.-M., ‘The second conjugate algebra of the Fourier algebra of a locally compact group’, Trans. Amer. Math. Soc. 267 (1981), 53–63.CrossRefGoogle Scholar
Parthasarathy, K. and Kumar, N. S., ‘Homological properties of Fourier algebras on homogeneous spaces’, Arch. Math. 96 (2011), 359–367.CrossRefGoogle Scholar