An example regarding Kalton's paper "Isomorphisms between spaces of vector-valued continuous functions"
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An example regarding Kalton’s paper “Isomorphismsbetween spaces of vector-valued continuous functions”
F´elix Cabello S´anchez
The paper alluded to in the title contains the following striking result: Let I be theunit interval and ∆ the Cantor set. If X is a quasi Banach space containing no copy of c which is isomorphic to a closed subspace of a space with a basis and C ( I, X ) is linearlyhomeomorphic to C (∆ , X ), then X is locally convex, i.e., a Banach space.Here C ( K, X ) denotes the space of continuous functions F : K −→ X . When K is acompact space and X a quasi Banach space C ( K, X ) is also a quasi Banach space underthe quasinorm k F k = sup {k F ( t ) k : t ∈ K } .When X is a Banach space, the isomorphic theory of the spaces C ( K, X ) is some-how oversimplified by Miljutin theorem (the spaces C ( K ) = C ( K, R ) for K uncountableand metrizable are all mutually isomorphic) and, above all, by Grothendieck’s identity C ( K, X ) = C ( K ) ˇ ⊗ ε X which implies that the isomorphic type of the Banach space C ( K, X )depends only on those of C ( K ) and X . The situation for quasi Banach spaces is morethrilling and actually some seemingly innocent questions remain open: Is C ( I, ℓ p ) isomor-phic to C ( I , ℓ p )? Is C ( I, L p ) isomorphic to C (∆ , L p )? These appear as Problems 7.2 and7.3 at the end of [ ]. Problem 7.1, namely if C ( K ) ⊗ X (the subspace of functions whoserange is contained in some finite dimensional subspace of X ) is always dense in C ( K, X ),was posed by Klee and is connected with quite serious mathematics. While it seems to bewidely open for quasi Banach spaces X the answer is negative for F -spaces (complete linearmetric spaces) as shown by Cauty celebrated example [ ]; see also [ ]. See Waelbroeck [ ,Section 8] for a discussion on Klee’s density problem.The aim of this short note is much more modest: we will show that Kalton result is sharpby exhibiting non locally convex quasi Banach spaces X with a basis for which C ( I, X ) and C (∆ , X ) are isomorphic. Our examples are rather specific and actually in all cases X isisomorphic to C ( K, X ) if K is a metric compactum of finite covering dimension.Recall that the (Lebesgue) covering dimension of a (not necessarily compact) topologicalspace K is the smallest number n ≥ K lies in the intersection of no more than n + 1 sets of the refinement.A quasi Banach space X has the λ -approximation property ( λ -AP) if for every x , . . . , x n ∈ X (or in some dense subset) there is a finite-rank operator T on X suchthat k T k ≤ λ and k x i − T x i k < ε . We say that X has the bounded approximation property(BAP) if it has the λ -AP for some λ ≥ Research supported in part by MICIN Project PID2019-103961GB-C21.2020
Mathematics Subject Classification : 46A16, 46E10.
We end these preliminaires by recalling that a p -norm, where 0 < p ≤
1, is a quasinormsatisfying the inequality k x + y k p ≤ k x k p + k y k p and that every quasinormed space has anequivalent p -norm for some 0 < p ≤
1, so says the Aoki-Rolewicz theorem.
Lemma. If K has finite covering dimension and X has the BAP, then C ( K, X ) has theBAP. Proof.
We first observe that if K has finite covering dimension or X has the BAP,then C ( K ) ⊗ X is dense in C ( K, X ). The part concerning the BAP is obvious; the otherpart is a result by Shuchat [ , Theorem 1].Given f ∈ C ( K ) and x ∈ X we denote by f ⊗ x the function t f ( t ) x . Since everyfunction in C ( K ) ⊗ X can be written as a finite sum P i f i ⊗ x i with f i ∈ C ( K ) , x i ∈ X (whichjustifies our notations, see [ , Proposition 1]) it suffices to see that there is a constant Λsuch that, given f , . . . , f m ∈ C ( K ) , y , . . . , y m ∈ X and ε > T on C ( K, X ) such that k T k ≤ Λ and k f i ⊗ y i − T ( f i ⊗ y i ) k < ε . As ε is arbitrary there isno loss of generality in assuming that k f i k = k y i k = 1 for 1 ≤ i ≤ m .Take an open cover U , . . . , U r of K such that for every i, j one has | f i ( s ) − f i ( t ) | < ε for all s, t ∈ U j . Put n = dim( K ) and take a refinement V , . . . , V k so that each point of K lies in no more than n + 1 of those sets. Finally, let φ , . . . , φ k be a partition of unity of K subordinate to V , . . . , V k .For each j pick t j ∈ V j and define an operator L on C ( K, X ) by letting L ( F ) = P j ≤ k φ j ⊗ F ( t j ), that is, ( LF )( t ) = P j ≤ k φ j ( t ) F ( t j ). Let us estimate k L k assuming X is p -normed:one has k L ( F ) k = sup t ∈ K (cid:13)(cid:13)(cid:13) X j ≤ k φ j ( t ) F ( t j ) (cid:13)(cid:13)(cid:13) , but for each t ∈ K the sum has no more than n + 1 nonzero summands, so (cid:13)(cid:13)(cid:13) X j ≤ k φ j ( t ) F ( t j ) (cid:13)(cid:13)(cid:13) ≤ (cid:16) X j ≤ k φ j ( t ) p k F ( t j ) k p (cid:17) /p ≤ k F k (cid:16) X j ≤ k φ j ( t ) p (cid:17) /p ≤ k F kk I : ℓ n +11 → ℓ n +1 p k = ( n + 1) /p − k F k We claim that k f i ⊗ y i − L ( f i ⊗ y i ) k ≤ ε for all i . We have L ( f i ⊗ y i )( t ) = P j ≤ k f i ( t j ) φ j ( t ) y i ,hence k f i ⊗ y i − L ( f i ⊗ y i ) k = (cid:13)(cid:13)(cid:13) X j ≤ k f i φ j − X j ≤ k f i ( t j ) φ j (cid:13)(cid:13)(cid:13) k y i k = (cid:13)(cid:13)(cid:13) X j ≤ k f i φ j − X j ≤ k f i ( t j ) φ j (cid:13)(cid:13)(cid:13) But, for each j and each t ∈ K one has | f i ( t ) φ j ( t ) − f i ( t j ) φ j ( t ) | ≤ εφ j ( t ): this is obvious if t / ∈ V j since in this case φ j ( t ) = 0, while for t ∈ V j we have | f i ( t ) − f i ( t j ) | ≤ ε by our choiceof V , . . . , V k and thus (cid:12)(cid:12)(cid:12) X j ≤ k f i ( t ) φ j ( t ) − X j ≤ k f i ( t j ) φ j ( t ) (cid:12)(cid:12)(cid:12) ≤ ε X j ≤ k φ j ( t ) = ε = ⇒ (cid:13)(cid:13)(cid:13) X j ≤ k f i φ j − X j ≤ k f i ( t j ) φ j (cid:13)(cid:13)(cid:13) ≤ ε. Let R be a finite-rank operator on X such that k y i − R ( y i ) k < ε , with k R k ≤ λ , where λ is the “approximation constant” of X , and define T on C ( K, X ) by (
T F )( t ) = R (( LF )( t )). EGARDING KALTON 3
Clearly T has finite-rank since for an elementary tensor f ⊗ x one has T ( f ⊗ x ) = X j ≤ k f ( t j ) φ j ⊗ R ( x ) . Finally let us estimate k f i ⊗ y i − T ( f i ⊗ y i ) k . Write f i ⊗ y i − T ( f i ⊗ y i ) = f i ⊗ y i − f i ⊗ R ( y i ) + f i ⊗ R ( y i ) − X j ≤ k f i ( t j ) φ j ⊗ R ( y i )and then k f i ⊗ y i − T ( f i ⊗ y i ) k p ≤ k f i ⊗ y i − f i ⊗ R ( y i ) k p + (cid:13)(cid:13)(cid:13) f i ⊗ R ( y i ) − X j ≤ k f i ( t j ) φ j ⊗ R ( y i ) (cid:13)(cid:13)(cid:13) p = k f i k p k y i − R ( y i ) k p + (cid:13)(cid:13)(cid:13) f i − X j ≤ k f i ( t j ) (cid:13)(cid:13)(cid:13) p k R ( y i ) k p ≤ ε p + ε p λ p , so that C ( K, X ) has the BAP with constant at most λ ( n + 1) /p − . (cid:3) The proof raises the question of whether the lemma is true for, say, the Hilbert cube I ω .The other ingredient we need is a complementably universal space for the BAP. A separa-ble p -Banach space is complementably universal for the BAP if it has the BAP and containsa complemented copy of each separable p -Banach space with the BAP. The existence ofsuch spaces (one for each 0 < p <
1) was first mentioned by Kalton himself in [ , Theorem4.1(b)]. A complete proof appears in the related issues of [ ]. In any case it easily followsfrom the Pe lczy´nski decomposition method that any two separable p -Banach spaces com-plementably universal for the BAP are isomorphic so let us denote by K p the isomorphictype of such specimens and observe that since each separable p -Banach space with the BAPis complemented in one with a basis, it follows that K p does have a basis. Corollary. If K is a (non-empty) metrizable compactum of finite covering dimension,then C ( K, K p ) is linearly homeomorphic to K p . Proof.
This clearly follows from the lemma since C ( K, K p ) is separable, has the BAPand contains K p complemented as the subspace of constant functions. (cid:3) We do not know of any other nonlocally convex quasi Banach space X for which C ( I, X )and C (∆ , X ) are isomorphic, apart from the obvious ones arising as direct sums of K p andBanach spaces lacking the BAP.It’s time to leave. Perhaps the most important question regarding the general topologicalproperties of quasi Banach spaces is to know whether every quotient operator Q : Z −→ X (acting between quasi Banach spaces) admits a continuous section namely a continuous σ : X −→ Z such that Q ◦ σ = I X . More generally, let us say that f ∈ C ( K, X ) liftsthrough Q if there is F ∈ C ( K, Z ) such that f = F ◦ Q . Now, given 0 < p <
1, a quotientoperator between p -Banach spaces Q : Z −→ X and a compactum K , consider the followingstatements:(1) Q admits a continuous section.(2) Every continuous f : K −→ X has a lifting to Z .(3) C ( K ) ⊗ X is dense in C ( K, X ). F´ELIX CABELLO S ´ANCHEZ
Clearly, (1) = ⇒ (2): set F = σ ◦ f , where σ is the hypothesized section of Q ). Besides, if(1) is true for some quotient map ℓ p ( J ) −→ X then so is for every Q . Similarly, if (2) istrue for a given K for some quotient map ℓ p ( J ) −→ X , then it is true for any quotient maponto X and (3) holds.Following (badly) Klee [ , Section 2], let us say that the pair ( K, X ) is admissible if(3) holds, that K is admissible if (3) holds for every quasi Banach space X and that X isadmissible if (3) holds for every compact K .We have mentioned Shuchat’s result that every compactum of finite covering dimensionis admissible. Actually one can prove that (2) holds for any Q if dim( K ) < ∞ . This indeedfollows from Michael’s [ , Theorem 1.2] but a simpler proof can be given using Shuchat’sresult, the argument of the proof of the lemma, and the open mapping theorem. Since everymetrizable compactum is the continuous image of ∆ this implies that every compact subset S ⊂ X there is a compact subset T ⊂ Z such that Q [ T ] = S .Very recently, Caponetti and Lewicki [ ] have shown that the spaces L p are admissiblefor 0 ≤ p <
1; their results are much more general and cover all modular function spaces.We do not know if the quotient map ℓ p −→ L p has a continuous section or satisfies (2) forarbitrary compact K and 0 < p < References [1] F. Cabello S´anchez, J.M.F. Castillo, Y. Moreno,
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Universal spaces and universal bases in metric linear spaces , Studia Math. (1977)161–191.[5] N.J. Kalton, Isomorphisms between spaces of vector-valued continuous functions , Proc. EdinburghMath. Soc. (1983) 29–48.[6] N.J. Kalton, T. Dobrowolski, Cauty’s space enhanced , Topology Appl. (2012) 28–33.[7] V. Klee,
Leray–Schauder theory without local convexity , Math. Ann. (1960) 286–296.[8] E. Michael,
Continuous selections II , Ann. of Math. (2) (1956) 562–580.[9] A.H. Shuchat, Approximation of vector-valued continuous functions , Proc. Amer. Math. Soc. (1972),97–103.[10] L. Waelbroeck, Topological vector spaces , Summer school on topological vector spaces, Bruxelles 1972,Springer Lecture Notes in Math. , Berlin-Heidelberg-New York, 1973, pp. 1–40.
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