The isometry group of L^{p}(μ,\X) is SOT-contractible
aa r X i v : . [ m a t h . F A ] A p r THE ISOMETRY GROUP OF L p ( µ, X) IS SOT-CONTRACTIBLE
JARNO TALPONEN
Abstract.
We will show that if (Ω , Σ , µ ) is an atomless positive measurespace, X is a Banach space and 1 ≤ p < ∞ , then the group of isometricautomorphisms on the Bochner space L p ( µ, X) is contractible in the strongoperator topology. We do not require Σ or X above to be separable. Introduction
This article deals with the topological structure of isometry groups of Banachspaces. Recall that an isometry group G X of a Banach space X consists of linearisometric isomorphisms T : X → X.The connectedness of groups of linear automorphisms with respect to the normtopology is a classical topic by now, see e.g. [4, 13]. For example, Kuiper provedalready in 1965 that the linear automorphism group GL ℓ and the isometry group G ℓ of ℓ are operator norm contractible. On the other hand, spaces G ℓ p and G L p ( µ ) are discrete in the operator norm topology for 1 ≤ p ≤ ∞ , p = 2 , ([8, p.112], [1,p.57]) and a fortiori not contractible. Our main result result shows that the situationis surprisingly different when observing G L p ( µ ) in the strong operator topology: Theorem 1.1.
Let X be a Banach space, µ an atomless positive measure and ≤ p < ∞ . Then the isometry group G L p ( µ, X) of the Bochner space L p ( µ, X) endowed with the strong operator topology is contractible. Here X and L p ( µ, X) canbe regarded as real or complex spaces. This result involves the isometric structure and more precisely the L p -structureof Bochner spaces. For a fascinating treatment of the latter topic see [2, 9] whichare implicitly applied in this work. Our study is also closely related to [10], [11]and [12].Recall that for Banach spaces X and Y a projection P : X ⊕ Y → Y is called an L p -projection for a given p ∈ [1 , ∞ ) if || ( x, y ) || p X ⊕ Y = || x || p X + || y || p Y for each x ∈ X , y ∈ Y . Such projections commute and in fact the L p -structure P p (X) of X, i.e. the set ofall L p -projections on X can be regarded as a complete Boolean algebra (see [2]).In order to put our result in the right context, let us mention that our routeto Theorem 1.1 is via analyzing the L p -structure of L p ( µ, X). This structure de-pends on µ and the L p -structure of X, as one might expect. If X is separable,then one can write P p ( L p ( µ, X)) = Σ /µ ⊗ P p (X) in a suitable sense, see [12]. Itmight first seem, bearing the L p -structures in mind, that an effective means to Date : November 21, 2018.1991
Mathematics Subject Classification.
Primary 46B04; Secondary 46B25, 46E40. analyze the connectedness properties of G L p ( µ, X) would be applying suitably nor-malized L p -decompositions of Bochner spaces. For example the p -integral modulerepresentation (see [2]) appears to be a natural tool for such an approach.In fact, the L p -structure of L p ( µ, X) is simple to represent if X is a separableBanach space having only trivial L p -structure and 1 ≤ p < ∞ , p = 2. Then theisometries T ∈ G L p ( µ, X) can be represented, in a suitable sense, as(1.1) T f ( t ) = σ t (cid:18) d ( µ ◦ φ − ) dµ ( t ) (cid:19) p f φ − ( t ) for f ∈ L p (Ω , Σ , µ, X) , where σ : Ω → G X ; t σ t is strongly measurable and φ : Σ /µ ↔ Σ /µ is a Booleanisomorphism (see [9] and also [10]). Moreover, there is in general a close connectionbetween the L p -structure of a given Lebesgue-Bochner space and the correspondingrepresentations of type (1.1).However, there exists an obstruction, that has to be dealt with. Namely, inthe setting of Theorem 1.1 the space X may have a rich L p -structure and be non-separable, so that typically the L p -structure of L p ( µ, X) is very complicated and(1.1) fails. Another substantial difficulty is that if X has a trivial L p -structure butis non-separable , then the L p -structure of L p ( µ, X) is not known explicitly, nor is itknown whether representation (1.1) holds. Thus we note that even though the group G L p ( µ, X) has not been classified in the sense of L p -structures, we are unexpectedlyable to extract enough information of P p ( L p ( µ, X)) in order to establish a verystrong connectedness condition in Theorem 1.1.The way around the described problems is to employ a suitable isometric repre-sentation for L p ( µ, X), which rises from measure-theoretic observations.To introduce the notations applied in this paper, X , Y and Y stand for real
Banach spaces. The closed unit ball and the unit sphere of X are denoted by B X and S X , respectively. We refer to [5] and [6] for necessary background informationin measure theory and isometric theory of classical Banach spaces. It is also usefulto get acquainted with the machinery appearing in [9] regarding Bochner spaces.If F ⊂ P (Ω) and Σ ⊂ Σ is a σ -subring such that F ⊂ Σ and for all A ∈ Σ there is a set { B n | n ∈ N } , where each B n is a countable intersection of suitableelements of F , and S n B n = A , then we say that F σ -generates Σ . Recall that thestrong operator topology (SOT) on L (X) is the topology inherited from X X endowedwith the product topology. Recall that each T ∈ L (X) has a SOT-neighbourhoodbasis consisting of sets of the following type:(1.2) { S ∈ L (X) : || ( S − T ) x || , || ( S − T ) x || , . . . , || ( S − T ) x n || < ǫ } , where x , . . . , x n ∈ X , n ∈ N and ǫ > , Σ , µ ) is a positive measure space, whereΣ is a σ -ring. An arbitrary measure space may be unconveniently rich for ourpurposes, that is, for studying the bands of L p (Ω , Σ , µ ) for p ∈ [1 , ∞ ). Thereforewe wish to extract exactly the information which is ’recoqnized’ by the L p -structure.If Σ is as above, then we will defineΣ L = { f − ( R \ { } ) | f ∈ L p (Ω , Σ , µ ) } /µ. Above { f − ( R \ { } ) | f ∈ L p (Ω , Σ , µ ) } is a σ -subring of Σ and Σ L is the quotient σ -ring formed by identifying µ -null sets with ∅ . By slight abuse of notation we denotethe corresponding measure ring by (Σ L , µ ) and we consider the elements A ∈ Σ L as HE ISOMETRY GROUP OF L p ( µ, X) IS SOT-CONTRACTIBLE 3 contained in Ω in the µ -a.e. sense. Hence, for A, B ∈ Σ L , we will write A S B ∈ Σ L instead of A W B ∈ Σ L , and so on. Note that Σ L does not depend on the value of p ,as long as p < ∞ , since { f − ( R \ { } ) | f ∈ L p (Ω , Σ , µ ) } ⊂ Σ is just the σ -subring of σ -finite sets. The motivation of this concept becomes clear in the subsequent results.Given p ∈ [1 , ∞ ), a set I and Banach spaces X i for i ∈ I , we denote by L pi ∈ I X i the L p -sum of the spaces X i . That is, ( x i ) i ∈ I ∈ Q i ∈ I X i satisfies ( x i ) i ∈ I ∈ L pi ∈ I X i if and only if || ( x i ) i ∈ I || p · = P i ∈ I || x i || p X i < ∞ . The space L pi ∈ I X i is endowed withthe complete norm || · || . We will denote by P j : L pi ∈ I X i → X j the L p -projectiononto X j for each j ∈ I , where X i is regarded as a subspace of L pi ∈ I X i in the naturalway. Hence each x ∈ X j is thought of as an element of L pi ∈ I X i as P j x = x .The Lebesgue measure on [0 ,
1] is denoted by m and if κ is a non-zero cardinal,then we denote the product measure on [0 , κ by m κ : Σ [0 , κ → R , where Σ [0 , κ isthe corresponding product σ -algebra.The orbit of x ∈ S X is G X ( x ) · = { T ( x ) | T ∈ G X } .2. Results
We will require the following auxiliary results.
Lemma 2.1.
For given f ∈ L p ( m, X) and t ∈ [0 , it holds that lim t → t t ∈ [0 , Z t || f ( h t ( s )) − f ( h t ( s )) || p d s = 0 , where h t : [ t , → [ t, h t ( s ) = (cid:16) t − t − t + − t − t s (cid:17) for t, s ∈ [0 , .Proof. The claim reduces to the analogous scalar-valued statement by approximat-ing f with simple functions. This in turn can be obtained by a straightforwardmodification of the proof of the classical fact that lim h → R R | g ( s + h ) − g ( s ) | d s = 0for g ∈ L p ( R ) , p < ∞ , which exploits Lusin’s theorem. (cid:3) Lemma 2.2.
Let X be a Banach space, (Ω , Σ , µ ) be an atomless positive measurespace and p ∈ [1 , ∞ ) . Then there exists a set I such that L p ( µ, X) is isometricallyisomorphic to L pi ∈ I L p ( m, L p ( m λ i , X)) , where λ i are non-zero cardinals for i ∈ I .Proof. The argument is closely related to classical matters discussed in [6] and [12].Recall Lamperti’s classical result [7] that the supports of f, g ∈ L ( µ ) are essentiallydisjoint if and only if || f + g || + || f − g || = 2( || f || + || g || ). The crucial conclusionof this result is that the disjointness of two vectors f, g ∈ L p ( ν ) can be detected bystudying the above norms and hence each linear isometry ψ : L ( ν ) → L ( υ ), where ν and υ are positive measures, preserves disjointness and bands. This also leads tothe fact that a projection P on L ( µ ) is L -projection with a separable range if andonly if there is A ∈ Σ L such that the image of P is { f ∈ L ( µ ) | supp( f ) ⊂ A µ − a . e . } .Since each f ∈ L ( µ ) is σ -finitely supported, it follows that the measure ring(Σ L , µ ) is σ -generated by µ -finite sets. Recall that the Boolean algebra of L -projections on L ( µ ) is complete (see [2, Prop.1.6]). Thus, by recalling Lamperti’sresult we obtain that { A \ S F| A ∈ Σ L } defines an ideal of Σ L for any F ⊂ Σ L .Hence Hausdorff’s maximum principle yields a maximal family { V j } j ∈ J ⊂ Σ L ofpairwise µ -essentially disjoint µ -finite sets. Note that L j ∈ J L ( µ | V j ) is isometricto L ( µ ). JARNO TALPONEN
By using the µ -finiteness of the sets V j we obtain that for each j ∈ J thereis a countable set A j and cardinals λ k for k ∈ A j such that the subspace { f ∈ L ( µ ) | supp( f ) ⊂ V j µ − a . e . } is isometric to L k ∈ A j L ( m λ k ) (see [6, p.127]). Since V j are essentially disjoint, the sets A j can be chosen to be pairwise disjoint.Put I = S j ∈ J A j . Observe that there exists an isometric isomorphims φ : L i ∈ I L ( m λ i ) → L ( µ ). According to Lamperti’s result the map φ preservesbands. Recall that if (Ω , Σ , µ ) and (Ω , Σ , µ ) are positive σ -finite measurespaces such that L ( µ ) and L ( µ ) are isometric, then there is a Boolean isomor-phism τ : Σ /µ ↔ Σ /µ see e.g. [12, p.477]. It follows from the selection of { V j } j ∈ J that there exist ideals B i ⊂ Σ L for i ∈ I such that(a) Σ L is σ -generated by S i ∈ I B i ,(b) B i is Boolean isomorphic to Σ [0 , λi /m λ i for i ∈ I (c) the ideals B i are pairwise essentially disjoint.Indeed, each ideal B i is determined by the isometric embedding φ | L ( m λi ) : L ( m λ i ) ֒ → L ( µ ) for i ∈ A j , namely, B i is the essential support of the functions in the image of φ | L ( m λi ) .Let p ∈ [1 , ∞ ). For each i ∈ I there is a Boolean isomorphism from Σ [0 , λi /m λ i onto the corresponding ideal B i of Σ L . Clearly { f ∈ L p ( µ, X) | supp( f ) ⊂ B i µ − a . e . } ⊂ L p ( µ, X)is a closed subspace isometric to L p ( µ | B i , X). Thus there exists an isometric iso-morphism U i : L p ( m λ i , X) → { f ∈ L p ( µ, X) | supp( f ) ⊂ B i µ − a . e . } for i ∈ I , seee.g. [12, p.476].Define an isometric isomorphism U : L pi ∈ I L p ( m λ i , X) → L p ( µ, X) by ( f i ) i ∈ I P i ∈ I U i ( f i ). Indeed, this mapping is an isometry since the ideals B i are disjoint.Moreover, as Σ L is σ -generated by the ideals B i it is easy to see by analyzing simplefunctions of L p ( µ, X) that U is onto.The following final step finishes the proof. We claim that if κ is a non-zerocardinal, then L p ( m κ , X) is isometrically isomorphic to L p ( m, L p ( m κ , X)). Indeed,recall that each m ⊗ m κ -measurable set A ⊂ [0 , × [0 , κ can be approximatedin measure by countable unions of measurable rectangles (see e.g. [5, p.145]).For the scalar-valued case, see [6, p.127] and [12, p.478-479]. Hence the obviousidentification of the spaces L p ( m ⊗ m κ , X) and L p ( m, L p ( m κ , X)) can be obtained,since each simple function in L p ( m ⊗ m κ , X) can be approximated by a sequenceof simple functions of L p ( m, L p ( m κ , X)) and vice versa. (cid:3)
Now we are ready to prove our main result.
Proof of Theorem. 1.1.
Since µ is atomless we may apply Lemma 2.2 to L p ( µ, X) for1 ≤ p < ∞ . Thus we may write L p ( µ, X) = L pi ∈ I L p ( m, L p ( m κ i , X)) isometrically,where κ i are non-zero cardinals for i ∈ I . In what follows we will denote M · = L pi ∈ I L p ( m, L p ( m κ i , X)) for the sake of brevity. Hence it suffices to show that G M is SOT-contractible.For each i ∈ I we define mappings α i , β i , γ i : [0 , × L p ( m, L p ( m κ i , X)) → L p ( m, L p ( m κ i , X))
HE ISOMETRY GROUP OF L p ( µ, X) IS SOT-CONTRACTIBLE 5 by α i ( t, f i ) = χ [0 ,t ] f i , β i ( t, f i )( s ) = (1 − t ) − p f i ( t + (1 − t ) s ) and γ i ( t, f i ) ◦ β i ( t, f i ) = χ [ t, f i , where f i ∈ L p ( m, L p ( m κ i , X)) and s ∈ [0 , − p = 0. Clearly α i ( t, · ) , β i ( t, · ) and γ i ( t, · ) are contractive linear operators for i ∈ I, t ∈ [0 , α i ( · , f i ) are continuous on [0 ,
1] and according toLemma 2.1 the same is true for β i ( · , f i ) and γ i ( · , f i ).The required homotopy h : [0 , × G M → G M is given by h ( t, T ) (cid:0)P i ∈ I f i (cid:1) = P i ∈ I α i ( t, f i ) + P i ∈ I γ i (cid:0) t, P i T (cid:0)P i ∈ I β i ( t, f i ) (cid:1)(cid:1) for t ∈ [0 , , T ∈ G M and P i ∈ I f i ∈ M . Indeed, it is straightforward to checkthat h ( t, T ) ∈ G M for each t ∈ [0 ,
1] and T ∈ G M . Moreover, h (0 , T ) = T and h (1 , T ) = id for each T ∈ G M . Our next aim is to justify that h is indeed a suitablehomotopy with respect to the strong operator topology.Let t ∈ [0 , T ∈ G M and let V ⊂ G M be a SOT-open neighbourhood of h ( t , T ). In order to justify the | · | × SOT − SOT continuity of h at ( t , T ), we mustfind an open set W ⊂ ([0 , , |·| ) × ( G M , SOT) such that ( t , T ) ∈ W and h ( W ) ⊂ V .Fix ǫ > f = P i ∈ I f i , g = P i ∈ I g i ∈ S M such that || g − h ( t , T )( f ) || < ǫ .Recall that the point h ( t , T ) has a SOT − neighbourhood basis of type (1.2), andthe elementary topological fact that open sets are preserved in finite intersections.Since ǫ , f and g were arbitrary, it suffices to show that there are open sets ∆ ⊂ ([0 , , | · | ) and U ⊂ ( G M , SOT) such that t ∈ ∆ , T ∈ U and(2.1) h (∆ × U ) ⊂ { R ∈ G M : || g − Rf || < ǫ } . Denote δ = ǫ − || g − h ( t , T ) f || >
0. There exists a finite set J ⊂ I such that || P i ∈ I \ J f i || < δ . Put n = | J | . Fix j ∈ J . Similarly as above, let f j = P j f . Since β j ( · , f j ) is continuous we can find an open interval ∆ j ⊂ [0 ,
1] containing t suchthat || β j ( t, f j ) − β j ( t , f j ) || < δ n for t ∈ ∆ j . Define an SOT − open neighbourhood U j of T by U j = { S ∈ G M : || ( S − T ) β j ( t , f j ) || < δ n } . Finally, by recalling the definitions of h , ∆ j and U j we obtain that || h ( t, S ) f j − h ( t , T ) f j || ≤ || h ( t, S ) f j − h ( t , S ) f j || + || h ( t , S ) f j − h ( t , T ) f j || < δ n + || ( S − T ) β j ( t , f j ) || < δ n for t ∈ ∆ j and S ∈ U j , where j ∈ J . Put ∆ = T j ∈ J ∆ j and U = T j ∈ J U j . We get || h ( t, S ) f − h ( t , T ) f ||≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( h ( t, S ) − h ( t , T )) P i ∈ I \ J f i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + P j ∈ J || h ( t, S ) f j − h ( t , T ) f j || < || h ( t, S ) − h ( t , T ) || · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)P I \ J f i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + n δ n < δ for ( t, S ) ∈ ∆ × U . This means that || g − h ( t, S ) f || < || g − h ( t , T ) f || + δ = ǫ for( t, S ) ∈ ∆ × U . Consequently h (∆ × U ) satisfies (2.1) and the proof is complete. (cid:3) Recall that S L p ( m ) , ≤ p < ∞ , consists of (exactly) 2 orbits, namely G L p ( χ [0 , )and G L p (2 p χ [0 , ] ), both of which are dense in S L p ( m ) (see e.g. [14]). Corollary 2.3.
Both the orbits G L p ( χ [0 , ) , G L p (2 p χ [0 , ] ) , ≤ p < ∞ , are path-connected. JARNO TALPONEN
Proof.
Fix 2 points f, g ∈ S X both coming from one of the above orbits. Thenthere exists T ∈ G L p such that T ( f ) = g . According to Theorem 1.1 there is ahomotopy h : [0 , × G L p → G L p such that h (0 , T ) = T and h (1 , T ) = id. Clearly h ([0 , × G L p )( · ) preserves orbits. Note that t h ( t, T ) f defines a continuous pathconnecting g and f in G L p ( f ) ⊂ S L p . (cid:3) The assumptions of µ being atomless or p < ∞ cannot be removed in Theorem1.1 even in the scalar-valued case. If p ∈ [1 , ∞ ) , p = 2 , and µ has some atoms, thenby applying the scalar-valued analogue of representatation (1.1) (see [6]) it can beverified that G L p ( µ ) is not connected in the weak operator topology. However, if ν is the counting measure on N , then L p ( ν, L p ( m )) = L p ( m ) isometrically, whoseisometry group is SOT − contractible according to Theorem 1.1. Proposition 2.4.
Let (Ω , Σ , µ ) be a positive measure space and we will regard L ∞ ( µ ) over the real field. Then G L ∞ ( µ ) is totally separated in the strong operatortopology.Proof. For given
T, S ∈ G L ∞ ( µ ) and A ∈ Σ it holds that T ( χ A ) = S ( χ A ) if and onlyif || T χ A − Sχ A || ≥ T, S ∈ G L ∞ ( µ ) , T = S , if suchexist. It is easy to find a set C ∈ Σ such that
T χ C = Sχ C . Now, { U ∈ G L ∞ ( µ ) : || U χ C − T χ C || < }∪{ V ∈ G L ∞ ( µ ) : || U χ C − T χ C || > } = G L ∞ ( µ ) is SOT − separation completing the claim. (cid:3) To conclude, let us make a few remarks about the homotopy h appearing in theproof of Theorem 1.1. We note that a resembling transformation was applied in [3,p.251] in a different setting. The proof of Theorem 1.1 can easily be modified sothat h becomes a homotopy on the set of linear isometric embeddings L p ( µ, X) → L p ( µ, X), and hence this set is SOT-contractible for atomless µ and p < ∞ . Withequally small modifications it can be verified that the conclusion of Theorem 1.1remains valid, if one investigates the contractibility of GL L p ( µ, X) in place of G L p ( µ, X) . References [1] F. Cabello, 10 Variaciones Sobre un Tema de Mazur, Tesis Doctoral, Universidad de Ex-tremadura, 1996.[2] E. Behrends et al., L p -structure in real Banach spaces. Lecture Notes in Mathematics, Vol.613. Springer-Verlag, Berlin-New York, 1977.[3] J. Dixmier, A. Douady, Champs continus d’espace hilbertiens et de C ∗ -alg`ebres, Bull. Math.Soc. France, 91 (1963), 227-284.[4] N. Kuiper, The homotopy type of the unitary group of Hilbert space. Topology 3 (1965), p.19-30.[5] P.R. Halmos, Measure Theory.
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