Abstract
The present paper contributes to the ongoing programme of quantification of isomorphic Banach space theory focusing on the Hagler–Stegall characterisation of dual spaces containing complemented copies of \(L_{1}\). As a corollary, we obtain the following quantitative version of the Hagler–Stegall theorem asserting that for a Banach space X, the following statements are equivalent:
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X contains almost isometric contains almost isometric copies of \((\bigoplus _{n=1}^{\infty } \ell _{\infty }^{n})_{\ell _1}\);
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for all \(\varepsilon >0\), \(X^{*}\) contains a \((1+\varepsilon )\)-complemented, \((1+\varepsilon )\)-isomorphic copy of \(L_{1}\);
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for all \(\varepsilon >0\), \(X^{*}\) contains a \((1+\varepsilon )\)-complemented, \((1+\varepsilon )\)-isomorphic copy of \(C[0,1]^{*}\).
Moreover, if X is separable, one may add the following assertion:
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for all \(\varepsilon >0\), there exists a \((1+\varepsilon )\)-quotient map \(T:X\rightarrow C(\Delta )\) so that \(T^{*}[C(\Delta )^{*}]\) is \((1+\varepsilon )\)-complemented in \(X^{*}\), where \(\Delta \) is the Cantor set
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1 Introduction
In 1968, Pełczyński [17] showed that if a Banach space X contains an isomorphic copy of \(\ell _{1}\), then the dual space \(X^{*}\) contains an isomorphic copy of \(L_{1}\) and proved that the converse holds as well subject to a mild technical condition that was later removed by Hagler [7]. More precisely, the result stated that the isomorphic containment of \(\ell _1\) is equivalent to the following assertions: \(X^{*}\) contains a subspace isomorphic to \(L_{1}\), \(X^{*}\) contains a subspace isomorphic to \(C[0,1]^{*}\). When X is separable, these are further equivalent to the assertions: \(X^{*}\) contains a subspace isomorphic to \(\ell _1([0,1])\), and C[0, 1] is a quotient of X.
Shortly after, Hagler and Stegall [8] obtained a ‘complemented’ version of Pełczyński’s aforementioned classical work:
Theorem
(Hagler–Stegall). Let X be a Banach space. Then the following assertions are equivalent:
-
(1)
X contains a subspace isomorphic to \((\bigoplus _{n=1}^{\infty } \ell _{\infty }^{n})_{\ell _1}\);
-
(2)
\(X^{*}\) contains a complemented subspace isomorphic to \(L_{1}\);
-
(3)
\(X^{*}\) contains a complemented subspace isomorphic to \(C[0,1]^{*}\);
-
(4)
\(X^{*}\) contains an infinite set K such that K is equivalent to the usual basis of \(\ell _{1}(\Gamma )\) for some \(\Gamma \), [K] is complemented in \(X^{*}\), and K is dense in itself in the weak* topology on \(X^{*}\).
If, in addition, X is separable, then the assertions (1)–(4) are equivalent to
-
(5)
There exists a surjective operator \(T:X\rightarrow C[0,1]\) such that \(T^{*}[C[0,1]^{*}]\) is complemented in \(X^{*}\).
The purpose of this note is to quantify the Hagler–Stegall theorem in the spirit of a large number of recent results on quantitative versions of various theorems on and properties of Banach spaces, such as quantitative versions of Krein’s theorem [6], Gantmacher’s theorem [2], James’ compactness theorem [5], weak sequential completeness and the Schur property [11, 12], the (reciprocal) Dunford–Pettis property [10, 13], the Banach–Saks property [3], etc. More broadly speaking, the present paper contributes to the on-going programme of quantification of Banach space theory.
In the present paper, we quantify the Hagler–Stegall theorem by introducing the following three quantities denoted by lower-case Greek letters and defined as infima of certain sets (when the sets happen to be empty, we use the convention that the corresponding value is \(\infty \)).
Hereinafter X and Y will stand for Banach spaces; is the space of (bounded, linear) operators from X to Y. We then introduce the following quantities:
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\( \alpha _{Y}(X)= \inf \{{\text {d}}(Y,Z):Z \text { is a subspace of}\) \(X\}\), where \({\text {d}}(Y,Z)\) is the Banach–Mazur distance between Y and Z.
The quantity \(\alpha _{Y}(X)\), being directly related to the Banach–Mazur distance, measures how well Y is from being isomorphically embeddable into X. Obviously, \(\alpha _{Y}(X)=1\) if and only if X contains almost isometric copies of Y, that is, for every \(\varepsilon >0\), X contains a subspace \((1+\varepsilon )\)-isomorphic to Y.
-
, \(AB=I_{Y}\}\).
The quantity \(\beta _{Y}(X)\) measures how well Y is from being isomorphic to a complemented subspace of X. It is easy to see that \(\beta _{Y}(X)=1\) if and only if for every \(\varepsilon >0\), there exists a subspace M of X so that M is \((1+\varepsilon )\)-isomorphic to Y and \((1+\varepsilon )\)-complemented in X.
-
.
The quantity \(\theta _{Y}(X)\) measures how well Y is isomorphic to a quotient of X and its dual \(Y^{*}\) is isomorphic to a complemented subspace of \(X^{*}\). We see that \(\theta _{Y}(X)=1\) if and only if, for every \(\varepsilon >0\), there exists a \((1+\varepsilon )\)-quotient map \(T:X\rightarrow Y\) so that \(T^{*}[Y^{*}]\) is \((1+\varepsilon )\)-complemented in \(X^{*}\).
A straightforward argument shows that
By using the aforementioned three quantities, we quantify the Hagler–Stegall theorem as follows:
Theorem A
Let X be a Banach space. Then
If, in addition, X is separable, then
The following \((1+\varepsilon )\)-version of the Hagler–Stegall theorem follows from Theorem A.
Corollary 1.1
Let X be a Banach space. Then the following assertions are equivalent:
-
(1)
X contains almost isometric copies of \((\bigoplus _{n=1}^{\infty }\ell _{\infty }^{n})_{l_{1}}\);
-
(2)
\(X^{*}\) contains a \((1+\varepsilon )\)-complemented subspace that is \((1+\varepsilon )\)-isomorphic to \(L_{1}\) for every \(\varepsilon >0\);
-
(3)
\(X^{*}\) contains a \((1+\varepsilon )\)-complemented subspace that is \((1+\varepsilon )\)-isomorphic to \(C[0,1]^{*}\) for every \(\varepsilon >0\).
If, in addition, X is separable, then
-
(4)
for every \(\varepsilon >0\), there exists a \((1+\varepsilon )\)-quotient map \(T:X\rightarrow C(\Delta )\) so that \(T^{*}[C(\Delta )^{*}]\) is \((1+\varepsilon )\)-complemented in \(X^{*}\).
2 Preliminaries
Our notation and terminology are standard and mostly in-line with [1, 16]. Throughout the paper, all Banach spaces can be considered either real or complex. We work with real scalars but the results can be easily amended to the complex too. By a subspace we understand a closed, linear subspace and by an operator we understand a bounded, linear map. If X is a Banach space, we denote by \(B_{X}\) the closed unit ball of X, by \(I_{X}\) the identity operator on X, and, for a subset \(K \subseteq X\), by [K] the closed linear span of K. For a surjective operator \(T:X\rightarrow Y\), we set
For \(\lambda \geqslant 1\), we say that a surjective operator \(T:X\rightarrow Y\) is a \(\lambda \)-quotient map if \(\Vert T\Vert {\text {co}}(T)\leqslant \lambda \). Quotient maps are 1-quotient maps according to the above terminology. A norm-one surjective operator \(T:X\rightarrow Y\) is a quotient map if and only if T is a \((1+)\)-quotient map, that is, a \((1+\varepsilon )\)-quotient map for every \(\varepsilon >0\).
The Banach–Mazur distance \({\text {d}}(X,Y)\) between two isomorphic Banach spaces X and Y is defined by \(\inf \Vert T\Vert \Vert T^{-1}\Vert \), where the infimum is taken over all isomorphisms T from X onto Y. As defined by Lindenstrauss and Rosenthal [15], for \(\lambda \geqslant 1\), a Banach space X is said to be a \({\mathcal {L}}_{1,\lambda }\)-space whenever for every finite-dimensional subspace E of X, there is a finite-dimensional subspace F of X such that \(F\supseteq E\) and \({\text {d}}(F,l_{1}^{\dim F})\leqslant \lambda \). We say that a Banach space X is an \({\mathcal {L}}_{1,\lambda +}\)-space if it is an \({\mathcal {L}}_{1,\lambda +\varepsilon }\)-space for all \(\varepsilon >0\).
Following the notation from [8], we denote
and, for \((n,i), (m,j)\in {\mathcal {F}}\), we write \((n,i)\geqslant (m,j)\) whenever
-
\(n\geqslant m\),
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\(2^{n-m}j\leqslant i\leqslant 2^{n-m}(j+1)-1\).
Let \(\Delta =\{0,1\}^{\mathbb {N}}\) be the Cantor set endowed with the metric
By Miljutin’s theorem [1, Lemma 4.4.7], C[0, 1] is isomorphic (but not isometric) to \(C(\Delta )\). It is well-known that \(C(\Delta )^{*}\) and \(C[0,1]^{*}\) are linearly isometric, though.
3 Proof of Theorem A
The present section is devoted to the proof of Theorem A and is conveniently split into more digestible parts.
Proof of Theorem A
We split the proof into a number of steps.
Step 1. \(\beta _{C(\Delta )^{*}}(X^{*})\leqslant \alpha _{(\bigoplus _{n=1}^{\infty }\ell _{\infty }^{n})_{\ell _1}}(X)\).
Since \(Z = (\bigoplus _{n=1}^{\infty } \ell _{\infty }^{2^{n}})_{\ell _1}\) embeds isometrically into \((\bigoplus _{n=1}^{\infty } \ell _{\infty }^{n})_{\ell _1}\), it suffices to prove that \(\alpha _{Z}(X)\geqslant \beta _{C(\Delta )^{*}}(X^{*})\). For this, let us fix \(c>\alpha _{Z}(X)\). Then there exists a contractive operator \(R:Z\rightarrow X\) that is bounded below by 1/c.
Let us consider a double-indexed family \((\Delta _{n,i})_{n=0, i=0}^{\infty , 2^{n}-1}\) of clopen subsets of the Cantor set such that
-
(1)
\(\Delta _{0,0}=\Delta \), \(\Delta _{n,i}=\Delta _{n+1,2i}\cup \Delta _{n+1,2i+1}\) (\((n,i)\in {\mathcal {F}}\)), and \(\Delta _{n,i}\cap \Delta _{n,j}=\varnothing \) if \(i\ne j\);
-
(2)
the diameter of \(\Delta _{n,i}\) is \(1/{2^{n}}\) (\(0\leqslant i\leqslant 2^{n}-1\)).
We set \(g_{n,i}=\mathbb {1}_{\Delta _{n,i}}\), which is a continuous function, \([g_{n,i}]_{i=0}^{2^{n}-1}\subseteq [g_{n+1,i}]_{i=0}^{2^{n+1}-1}\), \((g_{n,i})_{i=0}^{2^{n}-1}\) is isometrically equivalent to the unit vector basis of \(\ell _{\infty }^{2^{n}}\) (\(n\in {\mathbb {N}}\)), and \(\bigcup _{n=0}^{\infty }[g_{n,i}]_{i=0}^{2^{n}-1}\) is dense in \(C(\Delta )\). We may then define an operator \(T:Z\rightarrow C(\Delta )\) by the assignment \(Te_{n,i}=g_{n,i}\). For each n, T is an isometry when restricted to \([e_{n,i}:0\leqslant i\leqslant 2^{n}-1]\). Clearly, \(\Vert T\Vert =1\).
Claim 1
If W is a finite-dimensional Banach space and \(S:W\rightarrow C(\Delta )\) is an operator, then for every \(\varepsilon >0\), there exists an operator \({\widehat{S}}:W\rightarrow Z\) so that \(\Vert {\widehat{S}}\Vert \leqslant (1+\varepsilon )\Vert S\Vert \) and \(\Vert S-T{\widehat{S}}\Vert \leqslant \varepsilon \).
Proof of Claim 1
Let us fix an Auerbach basis \((w_{k},w_{k}^{*})_{k=1}^{N}\) for W (\(\dim W=N\)). So if \(w=\sum _{k=1}^{N}a_{k}w_{k}\in W\), then for each \(1\leqslant j\leqslant N\), we get
It follows that \(\sum _{k=1}^{N}|a_{k}|\leqslant N\Vert w\Vert .\) Let \(\delta >0\) be such that \(\delta N\leqslant \varepsilon \Vert S\Vert \) and \(\delta N\leqslant \varepsilon \). Then, there exist a positive integer n and \((f_{k})_{k=1}^{N}\) in \([g_{n,i}]_{i=0}^{2^{n}-1}\) so that \(\Vert Sw_{k}-f_{k}\Vert <\delta \) (\(k=1, 2, \ldots , N\)). Let us write \(f_{k}=\sum _{i=0}^{2^{n}-1}t_{k,i}g_{n,i}\) \((k=1, 2, \ldots , N).\)
Let us define an operator \({\widehat{S}}:W\rightarrow Z\) by
We claim that \({\widehat{S}}\) is the required operator. Indeed, for \(w=\sum _{k=1}^{N}a_{k}w_{k}\in W\), we have
Furthermore,
Let \(\varepsilon >0\). Since \(C(\Delta )\) has the metric approximation property (see, e.g., [4] for the definition), there exists a net \((T_{\alpha })_{\alpha }\) of finite-rank operators on \(C(\Delta )\) such that
-
\(\limsup _{\alpha }\Vert T_{\alpha }\Vert \leqslant 1+\varepsilon \),
-
\(\dim T_{\alpha }(C(\Delta ))\rightarrow \infty \),
-
\(T_{\alpha }\rightarrow I_{C(\Delta )}\) strongly.
For each \(\alpha \), we may apply Claim 1 to the inclusion map \(I_{\alpha }:T_{\alpha }[C(\Delta )]\rightarrow C(\Delta )\) in order to get an operator \(\widehat{I_{\alpha }}:T_{\alpha }[C(\Delta )]\rightarrow Z\) such that
-
\(\Vert \widehat{I_{\alpha }}\Vert \leqslant 1+\varepsilon \),
-
\(\Vert I_{\alpha }-T\widehat{I_{\alpha }}\Vert \leqslant (1+\dim T_{\alpha }[C(\Delta )])^{-2}\).
Hence, for \(f\in C(\Delta )\), we get
Let S be a -cluster point of the net \(((\widehat{I_{\alpha }}T_{\alpha })^{*})_{\alpha }\). We show that \(ST^{*}=I_{C(\Delta )^{*}}\). Indeed, we choose a subnet \(((\widehat{I_{\alpha '}}T_{\alpha '})^{*})_{\alpha '}\) of \(((\widehat{I_{\alpha }}T_{\alpha })^{*})_{\alpha }\) so that \((\widehat{I_{\alpha '}}T_{\alpha '})^{*}\rightarrow S\) in the -topology. Then, for \(f\in C(\Delta )\) and \(\mu \in C(\Delta )^{*}\), we get \(\langle (\widehat{I_{\alpha '}}T_{\alpha '})^{*}T^{*}\mu ,f\rangle \rightarrow \langle ST^{*}\mu ,f\rangle \). On the other hand, we have
Therefore, \(\langle ST^{*}\mu ,f\rangle =\langle \mu ,f\rangle .\)
Claim 2
There exists an operator \({\widetilde{T}}:C(\Delta )^{*}\rightarrow X^{*}\) so that \(R^{*}{\widetilde{T}}=T^{*}\) and \(\Vert {\widetilde{T}}\Vert \leqslant c(1+\varepsilon )\).
The proof of the claim is a variation of the Lindenstrauss’ compactness argument (see [9, Proposition 1] and [14, Lemma 2]). Since certain amendments are required, we present the full reasoning.
Proof of Claim 2
We use the fact that \(C(\Delta )^{*}\) is isometric to \(L_{1}(\mu )\) for some infinite measure \(\mu \), and as such, it is an \({\mathcal {L}}_{1,1+}\)-space. Let \(\Lambda \) be the collection of all finite-dimensional subspaces of \(C(\Delta )^{*}\). Then, for each \(\gamma \in \Lambda \), there exist \(E_{\gamma }\in \Lambda \) with \(\gamma \subseteq E_{\gamma }\) together with an isomorphism \(U_{\gamma }:\ell _{1}^{\dim E_{\gamma }}\rightarrow E_{\gamma }\) so that \(\Vert U_{\gamma }\Vert \Vert U_{\gamma }^{-1}\Vert \leqslant 1+\varepsilon \). Let \(S_{\gamma }:Z\rightarrow E_{\gamma }^{*}\) be an operator such that \(S_{\gamma }^{*}=T^{*}|_{E_{\gamma }}\) (\(\gamma \in \Lambda \)). By the 1-injectivity of \(\ell _{\infty }^{\dim E_{\gamma }}\), there is an operator \(R_{\gamma }:X\rightarrow \ell _{\infty }^{\dim E_{\gamma }}\) so that \(R_{\gamma }R=U_{\gamma }^{*}S_{\gamma }\) and \(\Vert R_{\gamma }\Vert \leqslant \Vert U_{\gamma }^{*}S_{\gamma }\Vert \Vert R^{-1}\Vert \leqslant \Vert U_{\gamma }\Vert \Vert T\Vert \Vert R^{-1}\Vert \). Let \(T_{\gamma }=R_{\gamma }^{*}U_{\gamma }^{-1}:E_{\gamma }\rightarrow X^{*}\). Then \(R^{*}T_{\gamma }= T^{*}|_{E_{\gamma }}\) and \(\Vert T_{\gamma }\Vert \leqslant c(1+\varepsilon )\Vert T\Vert \). For each \(\gamma \), we define a non-linear, discontinuous function from \(C(\Delta )^{*}\) to \(X^{*}\) by
Then \((\widetilde{T_{\gamma }})_{\gamma }\) is a net in the compact space
and as such, it has a cluster point \({\widetilde{T}}\). Standard arguments show that \({\widetilde{T}}\) is linear, \(R^{*}{\widetilde{T}}=T^{*},\) and \(\Vert {\widetilde{T}}\Vert \leqslant c(1+\varepsilon )\Vert T\Vert =c(1+\varepsilon )\).
Finally, we get \(SR^{*}{\widetilde{T}}=ST^{*}=I_{C(\Delta )^{*}}\) and hence
Letting \(\varepsilon \rightarrow 0\), we get \(\beta _{C(\Delta )^{*}}(X^{*})\leqslant c\). As c is arbitrary, we get Step 1.
Step 2. \(\beta _{L_{1}}(X^{*})\leqslant \beta _{C[0,1]^{*}}(X^{*})\).
It is well known that \(L_1\) is isometric to a 1-complemented subspace of \(C[0,1]^{*}\) (see, e.g., [1, p. 85]), which implies Step 2.
Step 3. \(\alpha _{(\bigoplus _{n=1}^{\infty } \ell _{\infty }^{n})_{\ell _1}}(X)\leqslant \beta _{L_{1}}(X^{*})\).
Let \(c>\beta _{L_{1}}(X^{*})\). Then there exist operators \(A:L_{1}\rightarrow X^{*}, B:X^{*}\rightarrow L_{1}\) so that \(BA=I_{L_{1}}, \Vert A\Vert =1\), and \(\Vert B\Vert <c\). Let \(0<\varepsilon <1\) and \(\varepsilon _{n}=\varepsilon /2^{2n+3}\) (\(n=0,1,\ldots \)).
By [8, Lemma 3], we get \((f_{n,i})_{(n,i)\in {\mathcal {F}}}\) in \(L_{\infty }\) and \((x_{n,i})_{(n,i)\in {\mathcal {F}}}\) in X satisfying
-
(1)
\(\Vert f_{n,i}\Vert _{1}=1\) and \(f_{n,i}\geqslant 0\) everywhere for all \((n,i)\in {\mathcal {F}}\);
-
(2)
for each n and \(i\ne j\), \(f_{n,i}(t)\) and \(f_{n,j}(t)\) cannot be both non-zero for the same \(t\in [0,1]\);
-
(3)
$$\begin{aligned}\langle Af_{n,i},x_{m,j}\rangle = \left\{ \begin{array} {rl} 1, &{} (n,i)\geqslant (m,j),\\ 0, &{} \text {otherwise;} \end{array} \right. \end{aligned}$$
-
(4)
\(\max _{0\leqslant i\leqslant 2^{n}-1}|t_{i}|\leqslant \Vert \sum _{i=0}^{2^{n}-1}t_{i}x_{n,i}\Vert \leqslant c(1+\varepsilon _{n}) \max _{0\leqslant i\leqslant 2^{n}-1}|t_{i}|\) (\(n=0,1,\ldots \); \(t_{0},\ldots , t_{2^{n}-1}\in {\mathbb {R}}\)).
We may now define recursively a sequence \((W_{n,i})_{(n,i)\in {\mathcal {F}}}\) of non-empty weak*-closed subsets of \(B_{X^{*}}\) as follows:
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\(W_{0,0}=\{x^{*}\in B_{X^{*}}:|\langle x^{*},x_{0,0}\rangle -1|\leqslant \varepsilon _{0}\},\)
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\(W_{1,0}=W_{0,0}\cap \{x^{*}\in B_{X^{*}}:|\langle x^{*},x_{1,0}\rangle -1|\leqslant \varepsilon _{1}, |\langle x^{*},x_{1,1}\rangle |\leqslant \varepsilon _{1}\},\)
-
\(W_{1,1}=W_{0,0}\cap \{x^{*}\in B_{X^{*}}:|\langle x^{*},x_{1,1}\rangle -1|\leqslant \varepsilon _{1}, |\langle x^{*},x_{1,0}\rangle |\leqslant \varepsilon _{1}\},\)
-
\(W_{2,0}=W_{1,0}\cap \{x^{*}\in B_{X^{*}}:|\langle x^{*},x_{2,0}\rangle -1|\leqslant \varepsilon _{2}, |\langle x^{*},x_{2,j}\rangle |\leqslant \varepsilon _{2},j=1,2,3\},\)
-
\(W_{2,1}=W_{1,0}\cap \{x^{*}\in B_{X^{*}}:|\langle x^{*},x_{2,1}\rangle -1|\leqslant \varepsilon _{2}, |\langle x^{*},x_{2,j}\rangle |\leqslant \varepsilon _{2},j=0,2,3\},\)
-
\(W_{2,2}=W_{1,1}\cap \{x^{*}\in B_{X^{*}}:|\langle x^{*},x_{2,2}\rangle -1|\leqslant \varepsilon _{2}, |\langle x^{*},x_{2,j}\rangle |\leqslant \varepsilon _{2},j=0,1,3\},\)
-
\(W_{2,3}=W_{1,1}\cap \{x^{*}\in B_{X^{*}}:|\langle x^{*},x_{2,3}\rangle -1|\leqslant \varepsilon _{2}, |\langle x^{*},x_{2,j}\rangle |\leqslant \varepsilon _{2},j=0,1,2\}\),
and so on. By (3), each \(W_{n,i}\) is non-empty. By the choice of \(\varepsilon _{n}\), the sets \(W_{n,i}, W_{n,j}\) are disjoint as long as \(i\ne j\). Let
By (3), \(Af_{n,i}\in W_{m,j}\) if \((n,i)\geqslant (m,j)\), which implies that each \(K_{n,i}\) is non-empty. By the construction of the sequence \((W_{n,i})\), we see that \(K_{0,0}=K, K_{n+1,2i}\cup K_{n+1,2i+1}=K_{n,i},\) and \(K_{n,i}\cap K_{n,j}=\varnothing \) if \(i\ne j\).
Let us define an operator \(T:X\rightarrow C(K)\) by \(\langle Tx,x^{*}\rangle =\langle x^{*},x\rangle \) (\(x\in X,x^{*}\in K\)). Then \(|\langle Tx_{n,i},x^{*}\rangle -1|\leqslant \varepsilon _{n}\) if \(x^{*}\in K_{n,i}\), and \(|\langle Tx_{n,i},x^{*}\rangle |\leqslant \varepsilon _{n}\) if \(x^{*}\in \bigcup _{j\ne i}K_{n,j}\). Set \(g_{n,i}=\mathbb {1}_{K_{n,i}}\), which is continuous as \(K_{n,i}\) is clopen. Then \(\Vert Tx_{n,i}-g_{n,i}\Vert \leqslant \varepsilon _{n}\). Moreover, \([g_{n,i}]_{i=0}^{2^{n}-1}\subseteq [g_{n+1,i}]_{i=0}^{2^{n+1}-1}\), \((g_{n,i})_{i=0}^{2^{n}-1}\) is isometrically equivalent to the unit vector basis of \(\ell _{\infty }^{2^{n}}\) for all n, and
is isometric to \(C(\Delta )\). Let Z be a subspace of \(C(\Delta )\) isometric to \((\bigoplus _{n=1}^{\infty } \ell _{\infty }^{n})_{\ell _1}\) and let \((z_{n,j})_{n=1,j=0}^{\infty ,n-1}\) be a basis of Z isometrically equivalent to the unit vector basis of \((\bigoplus _{n=1}^{\infty } \ell _{\infty }^{n})_{\ell _1}\). Fix \(n\geqslant 1\). Then there exist \(m>n\) and unit vectors \(h_{n,j}\in [g_{m,i}]_{i=0}^{2^{m}-1}\) so that \(\Vert z_{n,j}-h_{n,j}\Vert \leqslant \varepsilon /2^{n+3}\) (\(j=0,1,\ldots ,n-1\)). We write \(h_{n,j}=\sum _{i=0}^{2^{m}-1}a_{i,j}g_{m,i}\) and define \(y_{n,j}=\sum _{i=0}^{2^{m}-1}a_{i,j}x_{m,i}\in X.\)
Claim 3
For all \((t_{n,j})_{n=1,j=0}^{\infty , n-1}\in (\bigoplus _{n=1}^{\infty } \ell _{\infty }^{n})_{\ell _1}\), we have
Indeed, by (4), we get
Consequently,
On the other hand, by the choice of m and \(h_{n,j}\), we arrive at
This implies that
Finally, by Claim 3, we get
Letting \(\varepsilon \rightarrow 0\) yields \(\alpha _{(\bigoplus _{n=1}^{\infty } \ell _{\infty }^{n})_{\ell _1}}(X)\leqslant c\). Since c was arbitrary, the proof of Step 3 is completed.
Step 4. \(\beta _{L_{1}}(X^{*})\leqslant \theta _{C(\Delta )}(X)\).
This step follows from (1.1) together with Step 2. We are now ready to establish the final step of the proof.
Step 5. Suppose that X is separable. Then \(\theta _{C(\Delta )}(X)\leqslant \beta _{L_{1}}(X^{*})\).
Let \(c>\beta _{L_{1}}(X^{*})\). Then there exist operators \(A:L_{1}\rightarrow X^{*}, B:X^{*}\rightarrow L_{1}\) so that \(BA=I_{L_{1}}, \Vert A\Vert =1\), and \(\Vert B\Vert <c\).
Let \((f_{n,i})_{(n,i)\in {\mathcal {F}}}\) be a family of functions in \(L_{\infty }\), \((x_{n,i})_{(n,i)\in {\mathcal {F}}}\) in X, and \((W_{n,i})_{(n,i)\in {\mathcal {F}}}\) associated to \(\varepsilon _{n}=1/2^{2n+2}\) (\(n=0,1,\ldots \)) as described in Step 3. Since X is separable, we may assume that the \({\text {d}}\)-diameter of \(W_{n,i}\leqslant 2^{-n}\) for each i, where \({\text {d}}\) is a metric giving the relative \(\sigma (X^{*},X)\)-topology on \(B_{X^{*}}\). Let
Then K is a compact, totally disconnected metric space without isolated points, hence homeomorphic to \(\Delta \). Moreover, \(K_{0,0}=K, K_{n+1,2i}\cup K_{n+1,2i+1}=K_{n,i}\), and \(K_{n,i}\cap K_{n,j}=\varnothing \) if \(i\ne j\). Hence \(K=\bigcup _{i=0}^{2^{n}-1}K_{n,i}\) for all n. As seen in Step 3, the operator \(T:X\rightarrow C(K)\), defined by \(\langle Tx,x^{*}\rangle =\langle x^{*},x\rangle \) (\(x\in X,x^{*}\in K\)), satisfies \(\Vert Tx_{n,i}-g_{n,i}\Vert \leqslant \varepsilon _{n}\), where \(g_{n,i}=\mathbb {1}_{K_{n,i}}\in C(K)\).
An argument analogous to Step 1 yields that, if W is a finite-dimensional Banach space and \(S:W\rightarrow C(K)\) is an operator, then, for every \(\varepsilon >0\), there exists an operator \({\widehat{S}}:W\rightarrow X\) so that \(\Vert {\widehat{S}}\Vert \leqslant c(1+\varepsilon )\Vert S\Vert \) and \(\Vert S-T{\widehat{S}}\Vert \leqslant \varepsilon \).
Fix \(\varepsilon >0\). By an argument analogous to the one from Step 1, we get an operator \(S:X^{*}\rightarrow C(K)^{*}\) with \(\Vert S\Vert \leqslant c(1+\varepsilon )^{2}\) so that \(ST^{*}=I_{C(K)^{*}}\). This means that
Letting \(\varepsilon \rightarrow 0\), we arrive at \(\theta _{C(\Delta )}(X)\leqslant c.\) As c is arbitrary, the proof is complete.\(\square \)
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Acknowledgements
The authors would like to thank Prof. W.B. Johnson for helpful discussions and comments. The first-named author was supported by the National Natural Science Foundation of China (Grant No.11971403) and the Natural Science Foundation of Fujian Province of China (Grant No. 2019J01024). The second-named author acknowledges with thanks funding received from NCN SONATA 15 grant No. 2019/35/D/ST1/01734.
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Chen, D., Kania, T. & Ruan, Y. \((1+)\)-complemented, \((1+)\)-isomorphic copies of \(L_{1}\) in dual Banach spaces. Arch. Math. 119, 495–505 (2022). https://doi.org/10.1007/s00013-022-01778-2
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DOI: https://doi.org/10.1007/s00013-022-01778-2