Renormings and the fixed point property

Renormings and the Fixed Point Property Carlos Alberto Hern´andez Linares Mar´ıa A. Jap´on Pineda Department of Mathematical Analysis Faculty of Mathematics University of Seville Seville, Spain. April 2010 Outline 1Fixed Point Introduction 2Main Result 3Some Examples 4Application to subspaces of L1(µ) 5Application to non-commutative L1(M) Fixed Point Introduction Definitions Definition Let T:C→Cbe a mapping. We say that Thas a fixed point if there exists x∈Csuch that Tx =x. Theorem (Banach contraction) Let Xbe a Banach space and Ca closed subset of X. If T:C→Cis a contraction, i.e. kTx −Tyk ≤ kkx−yk,∀x, y ∈C, with k < 1, then Thas a fixed point. Fixed Point Introduction Definitions Definition A mapping T:C→Cis non-expansive if kTx −Tyk ≤ kx−yk,∀x, y ∈C. Banach’s theorem does not hold for non-expansive mappings. Definition A Banach space Xhas the fixed point property (FPP) if every non-expansive mapping T:C→C, where Cis a closed convex bounded subset of X, has a fixed point. Fixed Point Introduction Fixed Point and Reflexivity Uniformly smooth(⇒Reflexivity) Uniformly Convex(⇒Reflexivity) Normal Structure + Reflexivity Uniformly Kadec Klee + Reflexivity Uniformly Opial Condition + Reflexivity . . . etc + Reflexivity                      ⇒FPP FPP ⇒Reflexivity ? Fixed Point Introduction `1does not have the FPP Theorem `1does not have the FPP. Proof: We consider C={x= (xi)∈`1:∀i∈Nxi≥0,kxk1= 1}. The set Cis a closed convex bounded subset of `1. Let T:C→Cbe the mapping given by T(x1, x2, . . . ) = (0, x1, x2,....) Tis a non-expansive mapping and fixed point free. Fixed Point Introduction The main question. If Xfails to have the FPP, can Xbe renormed to have the FPP? In particular, can `1be renormed to have the FPP? Fixed Point Introduction Some answers Theorem (T. Dom´ınguez Benavides, 2009) Every reflexive Banach space can be renormed to have the FPP. Theorem (P. N. Dowling, C. J. Lennard and B. Turett, 1997-1998) `1(Γ),c0(Γ) and `∞can not be renormed to have the FPP. Fixed Point Introduction Some answers Theorem (P.K. Lin, 2008) The Banach space `1can be renormed to have the FPP. In `1consider the norm given by  ∞ X n=1 anen = sup k γk     ∞ X n=k anen    1 , where {en}nis the canonical basis on `1and γk=8k 1+8k. Then (`1,|||·|||) has the FPP. Some Examples Example Lin’s result can be derived from the Main Theorem defining the seminorms Rk(x) =      ∞ X n=k xnen    1 and taking τas the weak-star topology associated to the duality σ(`1, c0). The condition γk=8k 1+8kcan be dropped. Some Examples Example We can obtain other renormings in `1that have the FPP. For instance, let p > 1 and for k≥2 define for x= (an)∈`1 Rk(x) = ∞ X n=2k |an|+ 2k−1 X n=k |an|p!1 p , and R1(x) = kxk1. Then (`1,|||·|||) has the FPP. Some Examples Corollary Let {Xn}be a sequence of finite dimensional Banach spaces and consider X=⊕1X n Xn=(x= (xn) : xn∈Xn,kxk=X n kxnkXn<∞). Then Xcan be renormed to have the FPP. Proof: Define the seminorms Rk(x) = ∞ X n=k kxnkXn and let τbe the weak star topology where the predual of Xis E=x= (xn) : xn∈Xn,lim kxnkXn= 0,kxk= sup n kxnkXn. Some Examples Example Let 1 <p<∞be and X=⊕1X n `n p. Xcan be renormed to have the FPP. Moreover Xis non-reflexive and it is not isomorphic to any subspace of `1. If Xwere isomorphic to `1then 1 = type(`1) = type(X) = type(`p) = min{2, p} Some Examples Let Gbe a locally compact group. B(G) its Fourier-Stieltjes algebra. Theorem (A.T.-M Lau and M. Leinert, 2008) B(G)has the FPP ⇔Gis finite. Corollary (of the Main Theorem) If Gis a separable compact group, B(G)can be renormed to have the FPP. Proof: B(G) = ⊕1X n T(Hn), where Hnis a finite dimensional Hilbert space and T(Hn) is the trass class operator on Hn. Application to subspaces of L1(µ) Application to subspaces of L1(µ) Consider (Σ,Ω, µ) a σ-finite measure space. Let Ω = ∪nAnwith An⊂ An+1 and µ(An)<+∞for all n∈N. We define for all x∈L1(µ) R1(x) = kxk1=ZΩ |x|dµ, Rk(x) = sup ZE∩Ak |x|dµ :µ(E)<1 k+kxχAc kk1; for k≥2. τ:= the topology of locally convergence in measure (lcm) (≡the topology of convergence a.e., up to subsequences.) dτ(x, y) = ∞ X n=1 1 2n 1 µ(An)ZAn |x−y| 1 + |x−y|dµ;x, y ∈L1(µ). Application to subspaces of L1(µ) Application to subspaces of L1(µ) For a nondecreasing sequence {γk}in (0,1) such that lim kγk= 1 we define a equivalent norm on L1(µ) as |||x||| = sup k γkRk(x). Theorem The seminorms Rk(·)defined above satisfy the properties of the Main Theorem. Thus the following holds: If Xis a subspace of L1(µ)such that BXis lcm-relatively compact then Xcan be renormed to have the FPP. Application to subspaces of L1(µ) Application to subspaces of L1(µ) Remark 1 If µis finite. Consider Ak= Ω, then Rk(x) = sup ZE |x|dµ :µ(E)<1 k; for k≥2. Application to subspaces of L1(µ) Application to subspaces of L1(µ) Remark 2 Assume now that Ω = Nand µis the counting measure defined on the subsets of N. Then the space L1(µ) becomes the sequence space `1. Taken A1=∅and An={1, . . . , n −1}for n≥2 so Rk(x) = kxχAc kk1= ∞ X n=k |x(n)|; for all k∈N lcm =σ(`1, c0) in norm-bounded subsets. In this case we recover the Lin’s renorming taken γk=8k 1+8k. Application to subspaces of L1(µ) Other results Corollary Let Xbe a closed subspace for L1(µ). If Xis a dual space such that the lcm-topology coincides with the w∗-topology on BX, then (X, |||·|||)has the FPP. Application: The Bergman Space La(D) := {f∈L1(D) : fis an analytic function on D}. La(D) is a dual space and τ= topology convergence in measure = weak∗-topology. Then (La(D),|||·|||) has the FPP. Application to non-commutative L1 A little bit of background A little bit of background In a finite von Neumann algebra there always exists a normal faithful finite trace. Example M=L∞(µ), H=L2(µ). For f∈L∞(µ) f:L2(µ)→L2(µ) g7→ fg , τ(f) = Rfdµ and M∗=L1(µ); Application to non-commutative L1 A little bit of background Define for all x∈L1(M) R1(x) := kxk1=τ(|x|) Rk(x) := sup{kxpk1:p∈ P(M), τ(p)<1/k}, k ≥2. The linear topology Assume that Mis a finite von Neumann algebra (τ(1) <+∞). Consider the measure topology defined by the neighborhoods of zero N(, δ) = {x∈ M :∃p∈ P(M) such that kxpk∞≤ε and τ(1 −p)≤δ} for , δ > 0. (E. Nelson, 1974) Application to non-commutative L1 The theorem Theorem (Hern´andez and Jap´on, 2010) Let Mbe a finite von Neumann algebra. If the unit ball is compact for the measure topology, then L1(M)can be renormed to have the FPP. Application to non-commutative L1 Applications Example (The hyperfinite II1factor) Let (R, τ) = Nn≥1(M2, σ2) be the von Neumann algebra tensor product, M2denotes the complex 2 ×2 matrices and σ2a b c d =1 2(a+d). Application to non-commutative L1 Definition A von Neumann algebra is: 1a factor if x∈ M and xy =yx for all y∈ M implies x=λ1for some λ > 0. 2of type II1if it is finite and it does not contain any nonzero abelian projection. 3hyperfinite if there exists a sequence Mn⊂ Mn+1 of finite-dimensional von Neumann algebras such that Mis the closure of ∪nMnwith respect to the WOT. Theorem (F.J. Murray and J. von Neumann, 1943) (R, τ)is the unique, up to isomorphism, hyperfinite II1factor. Application to non-commutative L1 Applications Theorem L1(R)can be renormed to have the FPP. Theorem (U. Haagerup, H. P. Rosenthal & F. A. Sukochev, 2000) If Mis an arbitrary hyperfinite von Neumann algebra of type II1, then L1(M)is isomorphic to L1(R). Corollary If Mis any hyperfinite von Neumann algebra of type II1. Then L1(M)can be renormed to have the FPP Application to non-commutative L1 Applications Carlos A. Hern´andez-Linares and Maria A. Jap´on Pineda, Renormings and Fixed Point Property in non-commutative L1-spaces. Preprint.