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 1960

Bader, U.

Gelander, T.

Monod, N.

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Invent math (2012) 189:143–148DOI 10.1007/s00222-011-0363-2A fixed point theorem for L1 spacesU. Bader · T. Gelander · N. MonodReceived: 7 December 2010 / Accepted: 23 September 2011 /Published online: 27 October 2011© Springer-Verlag 2011Abstract We prove a fixed point theorem for a family of Banach spaces in-cluding notably L1 and its non-commutative analogues. Several applicationsare given, e.g. the optimal solution to the “derivation problem” studied sincethe 1960s.1 IntroductionAndrés Navas asked us if there is a fixed point theorem for all isometriesof L1 that preserve a given bounded set. Unlike many known cases wherea geometric argument applies, there is a fundamental obstruction in L1: anyinfinite group G admits a fixed-point-free isometric action on a bounded con-vex subset of L1. Example: the G-action on the affine subspace of summablefunctions of sum one on G. This action is fixed-point-free and preserves theclosed convex bounded set of non-negative functions. The obvious (and only)fixed point, zero, is outside the space.Thus, we have to search for fixed points possibly outside the convex set,indeed outside the affine subspace it spans. We shall do this more generallyfor any L-embedded Banach space V , that is, a space whose bidual can beU. BaderTechnion, Haifa, IsraelT. GelanderHebrew University, Jerusalem, IsraelN. Monod ()EPFL, Lausanne, Switzerlande-mail: nicolas.monod@epfl.ch144 U. Bader et al.decomposed as V ∗∗ = V ⊕1 V0 for some V0 ⊆ V ∗∗ (and ⊕1 indicates that thenorm is the sum of the norms on V and V0). Recall that L1 is L-embedded bythe Yosida–Hewitt decomposition [17] and that this holds more generally forthe predual of any von Neumann algebra [15, III.2.14]; in particular, for thedual of any C*-algebra.Theorem A Let A be a non-empty bounded subset of an L-embedded Ba-nach space V .Then there is a point in V fixed by every isometry of V preserving A.Moreover, one can choose a fixed point which minimises supa∈A ‖v − a‖ overall v ∈ V .We recall that an isometric action of a group G on a Banach space V isgiven by a linear part and a cocycle b : G → V . The cocycle is the orbital mapof 0 ∈ V and a fixed point v corresponds to a trivialisation b(g) = v − g.v,where g.v is the linear action. The above norm statement implies that one canarrange ‖v‖ ≤ supg ‖b(g)‖ by considering A = b(G) 	 0.As a special case (the “commutative” case), we recover the main theoremof [11] due to Losert, but with an improved (indeed optimal) norm estimate:Corollary B (cf. [11]) Let G be a group acting by homeomorphisms on a lo-cally compact space X. Then any bounded cocycle b : G → M(X) to thespace of (finite Radon) measures on X is trivial. More precisely, there isa measure μ with ‖μ‖ ≤ supg∈G ‖b(g)‖ such that b(g) = μ − g.μ for allg ∈ G.Indeed, M(X) is the dual of the (commutative) C*-algebra C0(X) andhence the predual of a von Neumann algebra.Numerous consequences of Corollary B are listed in [11]; let us only re-call that it settles the so-called derivation problem whose history began in the1960s: If G is a locally compact group, then any derivation from the con-volution algebra L1(G) to M(G) is inner. This is often phrased in terms ofderivations “of L1(G)” since any derivation L1(G) → M(G) must range inL1(G) by Paul Cohen’s factorisation theorem [3]. It also follows that anyderivation of M(G) is inner. Our norm estimate is stronger and in fact opti-mal by Remark 7.2(a) in [11].As observed by Uffe Haagerup, Theorem A also yields a new proof thatall C*-algebras are weakly amenable, which was proved in [7] using theGrothendieck–Haagerup–Pisier inequality. (Finding such a new proof wasProblem 19 in [14].) In fact, our theorem immediately implies that any con-tinuous derivation from any normed algebra A to a predual M∗ of a von Neu-mann algebra is inner as soon as A is spanned by the elements representedas invertible isometries of M∗ (see the proof of the corollary below). In theparticular case of C*-algebras, we obtain the following general statement.A fixed point theorem for L1 spaces 145Corollary C Let A be a unital C*-algebra. Let M∗ be the predual of a vonNeumann algebra. Assume M∗ is a Banach bimodule over A. Then any arbi-trary derivation D : A → M∗ is inner.Moreover, we can choose v ∈ M∗ with D(a) = v.a − a.v such that‖v‖ ≤ ‖D‖.Haagerup’s weak amenability of A is given by the special case M∗ = A∗.Our definition of Banach bimodule demands ‖a.v.b‖ ≤ ‖a‖ · ‖v‖ · ‖b‖(a, b ∈ A, v ∈ M∗).Proof of Corollary C By Theorem 2 in [13], D is continuous; thus it isbounded (by ‖D‖ < ∞) on the group G of unitaries of A. The map G → M∗given by g → D(g).g−1 is a cocycle for the Banach G-module structure de-fined by the rule v → g.v.g−1. Theorem A thus yields v, with norm boundedby ‖D‖, such that D(g) = v.g − g.v for all g ∈ G. The statement followssince any element of A is a combination of four unitaries (in fact, three [8]). Finally, returning to the case V = L1 of Theorem A, we consider actionswithout a priori boundedness of the orbits and obtain a new characterisationof Kazhdan groups:Corollary D Let  by any measure space. Then any isometric action of aKazhdan group on L1() has a fixed point.Moreover, this fixed point property characterises Kazhdan groups amongstall countable (or locally compact σ -compact) groups.By the Kakutani representation theorem [9], this corollary applies un-changed to abstract L1 spaces, for instance to M(X) for any locally compactspace X.Proof of Corollary D Recall that any isometric action of a Kazhdan group onan L1 space has bounded orbits because of a Fock space argument (see e.g. [1,1.3(2)]). Therefore, Theorem A implies the first part of the statement. Con-versely, let G be a locally compact σ -compact group with this fixed pointproperty. A standard argument shows that G has the L1-version (TL1) ofproperty (T ), see [1, 1.3(1)] (this argument holds in the σ -compact gener-ality). Theorem A in [1] shows that (TL1) implies (T ); although it is claimedthere for Lp with 1 < p < ∞, the proof applies unchanged to L1, using theConnes–Weiss construction [4] (exposed also in Theorem 6.3.4 of [2]). 146 U. Bader et al.2 Proof of the theoremWe first recall the concept of Chebyshev centre. Let A be a non-emptybounded subset of a metric space V . The circumradius of A in V isV (A) = inf{r ≥ 0 : ∃x ∈ V with A ⊆ B(x, r)},where B(x, r) denotes the closed r-ball around x. The Chebyshev centre ofA in V is the (possibly empty) setCV (A) ={c ∈ V : A ⊆ B(c,V (A))}.Notice that CV (A) can be written as an intersection of closed balls as follows:CV (A) =⋂r>V (A)CrV (A) where CrV (A) =⋂a∈AB(a, r).Thus, when V is a normed space, CV (A) is a bounded closed convex set.More importantly, when V is a dual Banach space, we deduce from Alaog˘lu’stheorem that CV (A) is weak-* compact and that it is non-empty becausethe non-empty sets CrV (A) are monotone in r . (For general Banach spaces,CV (A) is very often empty, even when A consists of just three points[10, 16].)Proposition Let A be a non-empty bounded subset of an L-embedded Banachspace V . Then the convex set CV (A) is weakly compact and non-empty.Proof Consider A as a subset of V ∗∗ under the canonical embeddingV ⊆ V ∗∗. In view of the above discussion, CV ∗∗(A) is a non-empty weak-*compact convex set. We claim that it lies in V and coincides with CV (A);the proposition then follows. Let thus c ∈ CV ∗∗(A) and write c = cV + cV0according to the decomposition V ∗∗ = V ⊕1 V0. Then, for any a ∈ A, wehave‖a − c‖ = ‖a − cV ‖ + ‖cV0‖since A ⊆ V . Therefore,V ∗∗(A) = supa∈A‖a − c‖ = supa∈A‖a − cV ‖ + ‖cV0‖ ≥ V (A) + ‖cV0‖.Since V ∗∗(A) ≤ V (A) anyway, we deduce cV0 = 0 and V ∗∗(A) = V (A),whence the claim. Proof of Theorem A Since the definition of CV (A) is metric, it is preservedby any isometry preserving A. By the proposition, we can apply the Ryll-Nardzewski theorem and deduce that there is a point of CV (A) fixed by allA fixed point theorem for L1 spaces 147isometries preserving A. The norm condition follows from the definition ofcentres. We remind the reader that in the present context the Ryll-Nardzewski the-orem has a particularly short geometric proof relying on the dentability ofweakly compact sets [12].3 Commentsa. In marked contrast to classical fixed point theorems, there is no hope tofind a fixed point inside a general bounded closed convex subset of L1,as pointed out in the opening. As a case in point, the weak compactnessof the Ryll-Nardzewski theorem is a strong restriction in L1 since it im-poses equi-integrability, and yet it seems almost unavoidable in light of[6, Theorem 4.2] if one insists on the classical statement.b. For the proposition, a canonical norm one projection V ∗∗ → V is notenough. Indeed, any dual space is canonically 1-complemented in its ownbidual, but the fixed point property in all duals characterises amenability.Specifically, any non-amenable group G has a fixed-point-free action withbounded orbits in (∞(G)/R)∗.c. It would be interesting to find a purely geometric version of the proposi-tion, since we prove compactness out of geometric assumptions. Noticehowever that the compact set CV (A) might still be large. If for instance Aconsists of just the two points 0[0,1],1[0,1] in V = L1([0,1]), then CV (A)is the infinite-dimensional set of functions 0 ≤ f ≤ 1 with ∫ f = 1/2. Itwould further be interesting to study the dynamics of the transformationA → CV (A); for instance, it can have orbits of period 1 or 2, but no otherfinite period.Acknowledgements We are indebted to Andrés Navas for asking the question about L1,motivated by [5]. We thank Uffe Haagerup for pointing out the application of the general resultto weak amenability. Conversations with Narutaka Ozawa were very stimulating. Supported inpart by the Israel Science Foundation, the European Research Council and the Swiss NationalScience Foundation.References1. Bader, U., Furman, A., Gelander, T., Monod, N.: Property (T ) and rigidity for actions onBanach spaces. Acta Math. 198(1), 57–105 (2007)2. Bekka, B., de la Harpe, P., Valette, A.: Kazhdan’s Property (T). New Mathematical Mono-graphs, vol. 11. Cambridge University Press, Cambridge (2008)3. Cohen, P.J.: Factorization in group algebras. Duke Math. J. 26, 199–205 (1959)4. Connes, A., Weiss, B.: Property T and asymptotically invariant sequences. Israel J. Math.37(3), 209–210 (1980)148 U. Bader et al.5. Coronel, D., Navas, A., Ponce, M.: On bounded cocycles of isometries over a minimaldynamics. arXiv:1101.35236. Domínguez Benavides, T., Japón Pineda, M.A., Prus, S.: Weak compactness and fixedpoint property for affine mappings. J. Funct. Anal. 209(1), 1–15 (2004)7. Haagerup, U.: All nuclear C∗-algebras are amenable. Invent. Math. 74(2), 305–319(1983)8. Kadison, R.V., Pedersen, G.K.: Means and convex combinations of unitary operators.Math. Scand. 57(2), 249–266 (1985)9. Kakutani, S.: Concrete representation of abstract (L)-spaces and the mean ergodic theo-rem. Ann. Math. (2) 42(2), 523–537 (1941)10. Konyagin, S.V.: A remark on renormings of nonreflexive spaces and the existence of aChebyshev center. Vestn. Mosk. Univ., Ser. Filos. 2, 81–82 (1988)11. Losert, V.: The derivation problem for group algebras. Ann. Math. (2) 168(1), 221–246(2008)12. Namioka, I., Asplund, E.: A geometric proof of Ryll-Nardzewski’s fixed point theorem.Bull. Am. Math. Soc. 73, 443–445 (1967)13. Ringrose, J.R.: Automatic continuity of derivations of operator algebras. J. Lond. Math.Soc. 5, 432–438 (1972)14. Runde, V.: Lectures on Amenability. Lectures Notes in Mathematics, vol. 1774. Springer,Berlin (2002)15. Takesaki, M.: Theory of Operator Algebras. I. Encyclopaedia of Mathematical Sciences,vol. 124. Springer, Berlin (2002). Reprint of the first (1979) edition, Operator Algebrasand Non-commutative Geometry 516. Veselý, L.: For a dense set of equivalent norms, a non-reflexive Banach space contains atriangle with no Chebyshev center. Comment. Math. Univ. Carol. 42(1), 153–158 (2001)17. Yosida, K., Hewitt, E.: Finitely additive measures. Trans. Am. Math. Soc. 72, 46–66(1952)

A fixed point theorem for L 1 spaces

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A fixed point theorem for L 1 spaces

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