Abstract
We define property A for locally compact hypergroups. We also define \(L^p\) versions of property A and show that they are equivalent for \(p=1\) and \(p=2\), and weaker than amenability of the hypergroup.
Similar content being viewed by others
References
Alexandrov, A.D.: A theorem on triangles in a metric space and some of its applications. Trudy Math. Inst. Steks. 38, 5–23 (1951)
Baum, P., Connes, A., Higson, N.: Classifying space for proper actions and K-theory of group C*-algebras. In: C*-algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math., vol. 167, pp. 240–291. American Mathematical Society, Providence, (1994)
Bekka, B., de la Harpe, P., Valette, A.: Kazhdan’s Property (T), New Mathematical Monographs, vol. 11. Cambridge University Press, Cambridge (2008)
Block, J., Weinberger, S.: Aperiodic tilings, positive scalar curvature and amenability of spaces. J. Am. Math. Soc. 5, 907–918 (1992)
Bloom, W.R., Heyer, H.: Harmonic Analysis of Probability Measures on Hypergroups. De Gruyter, Berlin (1995)
Brodzki, J., Niblo, G.A., Spakula, J., Willett, R., Wright, N.: Uniform local amenability. J. Noncommut. Geom. 7, 583–603 (2013)
Deprez, S., Li, K.: Property A and uniform embedding for locally compact groups. J. Noncommut. Geom. 9, 797–819 (2015)
Dunkl, C.F.: The measure algebra of a locally compact hypergroup. Trans. Am. Math. Soc. 179, 331–348 (1973)
Dunkl, C.F., Ramirez, D.E.: A family of countably compact \(P_*\)-hypergroups. Trans. Am. Math. Soc. 202, 339–356 (1975)
Dydak, J., Hoffland, C.S.: An alternative definition of coarse structures. Topol. Appl. 155, 1013–1021 (2008)
Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, New York (1987)
Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, Vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser., vol. 182, pp. 1–295. Cambridge University Press, Cambridge, (1993)
Higson, N.: Bivariant K-theory and the Novikov conjecture. Geom. Funct. Anal. 10, 563–581 (2000)
Higson, N., Roe, J.: Amenable group actions and the Novikov conjecture. J. Reine Angew. Math. 519, 143–153 (2000)
Jewett, R.I.: Spaces with an abstract convolution of measures. Adv. Math. 18, 1–101 (1975)
Roe, J.: Lectures on Coarse Geometry, University Lecture Series, vol. 31. American Mathematical Society, Providence (2003)
Roe, J.: Warped cones and property A. Geom. Topol. 9, 163–178 (2005)
Skandalis, G., Tu, J.L., Yu, G.: The coarse Baum–Connes conjecture and groupoids. Topology 41, 807–834 (2002)
Skantharajah, M.: Amenable hypergroups. Illinois J. Math. 36, 15–46 (1992)
Spector, R.: Aperçu de la théorie des hypergroupes, In: Analyse Harmonique sur les Groups de Lie, Lecture Notes in Math., vol. 497, pp. 643–673 Springer, Berlin, (1975)
Tabatabaie, S.M., Amini, M.: Convolution coarse structures. Topol. Appl. 275, 107–152 (2020)
Voit, M.: Positive characters on commutative hypergroups. Math. Z. 198, 405–421 (1988)
Willson, B.: A fixed point theorem and the existence of a Haar measure for hypergroups satisfying conditions related to amenability. Can. Math. Bull. 58(2), 415–422 (2015)
Yu, G.: The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139, 201–240 (2000)
Acknowledgements
The authors would like to thank the referee for helpful remarks and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Anthony To-Ming Lau.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tabatabaie, S.M., Amini, M. & Amjadi, A.A. Property A for hypergroups. Semigroup Forum 104, 464–479 (2022). https://doi.org/10.1007/s00233-021-10245-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-021-10245-3