Abstract
We prove a fixed point theorem for a family of Banach spaces including notably L 1 and its non-commutative analogues. Several applications are given, e.g. the optimal solution to the “derivation problem” studied since the 1960s.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bader, U., Furman, A., Gelander, T., Monod, N.: Property (T) and rigidity for actions on Banach spaces. Acta Math. 198(1), 57–105 (2007)
Bekka, B., de la Harpe, P., Valette, A.: Kazhdan’s Property (T). New Mathematical Monographs, vol. 11. Cambridge University Press, Cambridge (2008)
Cohen, P.J.: Factorization in group algebras. Duke Math. J. 26, 199–205 (1959)
Connes, A., Weiss, B.: Property T and asymptotically invariant sequences. Israel J. Math. 37(3), 209–210 (1980)
Coronel, D., Navas, A., Ponce, M.: On bounded cocycles of isometries over a minimal dynamics. arXiv:1101.3523
Domínguez Benavides, T., Japón Pineda, M.A., Prus, S.: Weak compactness and fixed point property for affine mappings. J. Funct. Anal. 209(1), 1–15 (2004)
Haagerup, U.: All nuclear C ∗-algebras are amenable. Invent. Math. 74(2), 305–319 (1983)
Kadison, R.V., Pedersen, G.K.: Means and convex combinations of unitary operators. Math. Scand. 57(2), 249–266 (1985)
Kakutani, S.: Concrete representation of abstract (L)-spaces and the mean ergodic theorem. Ann. Math. (2) 42(2), 523–537 (1941)
Konyagin, S.V.: A remark on renormings of nonreflexive spaces and the existence of a Chebyshev center. Vestn. Mosk. Univ., Ser. Filos. 2, 81–82 (1988)
Losert, V.: The derivation problem for group algebras. Ann. Math. (2) 168(1), 221–246 (2008)
Namioka, I., Asplund, E.: A geometric proof of Ryll-Nardzewski’s fixed point theorem. Bull. Am. Math. Soc. 73, 443–445 (1967)
Ringrose, J.R.: Automatic continuity of derivations of operator algebras. J. Lond. Math. Soc. 5, 432–438 (1972)
Runde, V.: Lectures on Amenability. Lectures Notes in Mathematics, vol. 1774. Springer, Berlin (2002)
Takesaki, M.: Theory of Operator Algebras. I. Encyclopaedia of Mathematical Sciences, vol. 124. Springer, Berlin (2002). Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry 5
Veselý, L.: For a dense set of equivalent norms, a non-reflexive Banach space contains a triangle with no Chebyshev center. Comment. Math. Univ. Carol. 42(1), 153–158 (2001)
Yosida, K., Hewitt, E.: Finitely additive measures. Trans. Am. Math. Soc. 72, 46–66 (1952)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bader, U., Gelander, T. & Monod, N. A fixed point theorem for L 1 spaces. Invent. math. 189, 143–148 (2012). https://doi.org/10.1007/s00222-011-0363-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-011-0363-2