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Bochner representable operators on Banach function spaces

  • Marian Nowak
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Abstract

Let \((E,\Vert \cdot \Vert _E)\) be a Banach function space, \(E'\) the Köthe dual of E and \((X,\Vert \cdot \Vert _X)\) be a Banach space. It is shown that every Bochner representable operator \(T:E\rightarrow X\) maps relatively \(\sigma (E,E')\)-compact sets in E onto relatively norm compact sets in X. If, in particular, the associated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\) is order continuous, then every Bochner representable operator \(T:E\rightarrow X\) is \((\gamma _E,\Vert \cdot \Vert _X)\)-compact, where \(\gamma _E\) stands for the natural mixed topology on E. Applications to Bochner representable operators on Orlicz spaces are given.

Keywords

Banach function spaces Orlicz spaces Mixed topologies Bochner representable operators Compact operators 

Mathematics Subject Classification

47B38 46B40 46E30 47B05 

1 Introduction and preliminaries

We assume that \((X,\Vert \cdot \Vert _X)\) is a real Banach space with the Banach dual \((X^*,\Vert \cdot \Vert _{X^*})\). Let \(B_X\) stand for the closed unit ball in X. For terminology concerning Riesz spaces and function spaces, we refer the reader to [1, 2, 9, 10, 20].

We assume that \((\Omega ,\Sigma ,\mu )\) is a complete \(\sigma \)-finite measure space. By \(\Sigma _f\) we denote the \(\delta \)-ring of all sets \(A\in \Sigma \) with \(\mu (A)<\infty \). Let \(L^0\) denote the corresponding space of equivalence classes of all \(\Sigma \)-measurable real functions on \(\Omega \). Then \(L^0\) is a super Dedekind complete Riesz space, equipped with the F-norm topology \(\mathcal{T}_0\) of convergence in measure on sets of finite measure.

Let \((E,\Vert \cdot \Vert _E)\) be a Banach function space, that is, E is an order ideal of \(L^0\) with \(\mathrm{supp}\,E=\Omega \) and \(\Vert \cdot \Vert _E\) is a Riesz norm. By \(\mathcal{T}_E\) we denote the topology of the norm \(\Vert \cdot \Vert _E\). For \(r>0\), let \(B_E(r):=\{u\in E:\Vert u\Vert _E\le r\}.\) The Köthe dual \(E'\) of E is defined by:
$$\begin{aligned} E':=\bigg \{v\in L^0:\int _\Omega |u(\omega )v(\omega )|\,d\mu <\infty \ \text{ for } \text{ all } u\in E\bigg \}. \end{aligned}$$
The associated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\) is defined for \(v\in E'\) by
$$\begin{aligned} \Vert v\Vert _{E'}:=\sup \left\{ \Big |\int _\Omega u(\omega )v(\omega )\,d\mu \Big |:u\in E, \Vert u\Vert _E\le 1\right\} . \end{aligned}$$
Then \(\mathrm{supp}\,E'=\Omega \) (see [9, Theorem 6.1.5]). The \(\sigma \)-order continuous dual \(E^\sim _c\) of E separates the points of E and \(E^\sim _c\) can be identified with \(E'\) through the Riesz isomorphism \(E'\ni v\mapsto F_v\in E^\sim _c\), where
$$\begin{aligned} F_v(u)=\int _\Omega u(\omega )v(\omega )\,d\mu \ \text{ for } u\in E \end{aligned}$$
(see [9, Theorem 6.1.1]). The Mackey topology \(\tau (E,E')\) is a locally convex-solid Hausdorff topology with the \(\sigma \)-Lebesgue property (see [1, Exercise 18, p. 178]). Note that \(\tau (E,E')=\mathcal{T}_E\) if and only if \(\Vert \cdot \Vert _E\) is \(\sigma \)-order continuous.
In addition to these facts, the following assumptions will be in force without futher mention throughout this paper:
  1. (A)

    \(E\subset L^1_{loc}\), i.e., \(A\in \Sigma _f\) implies \(u\mathbb {1}_A\in L^1\) for all \(u\in E\).

     
  2. (B)

    E is perfect, i.e., \(E=E''\) (equivalently, \(\Vert \cdot \Vert _E\) satisfies both the \(\sigma \)-Fatou property and the \(\sigma \)-Levy property (see [1, Definition 3.14], [20, § 110 and Theorem 112.2]).

     
Note that from (A) it follows that for every \(A\in \Sigma _f\), \(\mathbb {1}_A\in E'\). Moreover, from (B) it follows that E is \(\sigma (E,E')\)-sequentially complete (see [13, Proposition 1.2]). Note that if a subset H of E is \(\sigma (E,E')\)-bounded, then \(\sup _{u\in H}\Vert u\Vert _E<\infty \) (see [10, Lemma 1, p. 20], [9, Theorem 6.1.6]). By \((E')_a\) we denote the ideal in \(E'\) of all elements of order continuous norms, that is,
$$\begin{aligned} (E')_a:=\left\{ v\in E':\Vert v_n\Vert _{E'}\rightarrow 0 \ \text{ if } \ |v(\omega )|\ge v_n(\omega ) \downarrow 0 \ \mu -a.e.\right\} . \end{aligned}$$
Then \(\Vert \cdot \Vert _{E'}\) is order continuous if and only if \((E')_a=E'\).
Let \(L^0(X)\) stand for the linear space of \(\mu \)-equivalence classes of all strongly \(\Sigma \)-measurable functions \(g:\Omega \rightarrow X\). Let
$$\begin{aligned} E'(X)=\left\{ g\in L^0(X):\Vert g(\cdot )\Vert _X\in E'\right\} . \end{aligned}$$

Definition 1.1

A bounded linear operator \(T:E\rightarrow X\) is said to be Bochner representable, if there exists \(g\in E'(X)\) such that
$$\begin{aligned} T(u)=\int _\Omega u(\omega )g(\omega )\,d\mu \ \ \text{ for } \ \ u\in E. \end{aligned}$$

2 Bochner representable operators on Banach spaces

It is known that every Bochner representable operator \(T:L^1\rightarrow X\) (where \(\mu (\Omega )<\infty )\) maps relatively \(\sigma (L^1,L^\infty )\)-compact sets onto relatively norm compact sets in X (see [5, Lemma 11, pp. 74–75]). Now we extend this result to Bochner representable operators \(T:E\rightarrow X\).

Theorem 2.1

Assume that \(T:E\rightarrow X\) is a Bochner representable operator. Then the following statements hold:
  1. (i)

    T is \((\tau (E,E'),\Vert \cdot \Vert _X)\)-continuous.

     
  2. (ii)

    T maps relatively \(\sigma (E,E')\)-compact sets in E onto relatively norm compact sets in X.

     

Proof

There exists \(g\in E'(X)\) such that
$$\begin{aligned} T(u)=\int _\Omega u(\omega )g(\omega )\,d\mu \ \ \text{ for } \ \ u\in E. \end{aligned}$$
(i) Assume that \(x^*\in X^*\). Then for \(u\in E\),
$$\begin{aligned} \displaystyle |x^*(T(u))|= & {} \bigg |\int _\Omega u(\omega )x^*(g(\omega ))\,d\mu \bigg |\\\le & {} \int _\Omega |u(\omega )|\cdot \Vert x^*\Vert _{X^*} \Vert g(\omega )\Vert _X\, d\mu = \Vert x^*\Vert _{X^*} F_{\Vert g(\cdot )\Vert _X}(|u|). \end{aligned}$$
It follows that \(x^*\circ T\in E^\sim _c\), so T is \((\sigma (E,E'),\sigma (X,X^*))\)-continuous (see [2, Theorem 9.26]), and hence T is \((\tau (E,E'),\Vert \cdot \Vert _X)\)-continuous (see [8, Theorem 8.6.1]).
(ii) Assume that H is a relatively \(\sigma (E,E')\)-compact subset of E. It follows that H is relatively \(\sigma (E,E')\)-sequentially compact (see [10, Lemma 11, p. 31]). Since E is \(\sigma (E,E')\)-sequentially complete, in view of [13, Proposition 1.1] the set \(\{u\Vert g(\cdot )\Vert _X:u\in H\}\) in \(L^1\) is uniformly integrable. Hence given \(\varepsilon >0\) there exist \(\Omega _0\in \Sigma _f\) and \(\delta >0\) such that
$$\begin{aligned}&\sup _{u\in H}\int _{\Omega \smallsetminus \Omega _0} |u(\omega )|\cdot \Vert g(\omega )\Vert _X \,d\mu \le \frac{\varepsilon }{2}, \end{aligned}$$
(2.1)
$$\begin{aligned}&\sup _{u\in H}\int _A |u(\omega )|\cdot \Vert g(\omega )\Vert _X \,d\mu \le \frac{\varepsilon }{2}, \ \ \text{ whenever } \ \ A\in \Sigma \ \text{ and } \ \mu (A)\le \delta . \end{aligned}$$
(2.2)
Choose a sequence \((s_n)\) of X-valued strongly \(\mu \)-measurable step functions on \(\Omega \) such that \(s_n(\omega )\rightarrow g(\omega )\) \(\mu \)-a.e. and \(\Vert s_n(\omega )\Vert _X\le \Vert g(\omega )\Vert _X\) \(\mu \)-a.e. for all \(n\in \mathbb {N}\) (see [7, Theorem 6, p. 4]). Hence \(s_n\in E'(X)\) for \(n\in \mathbb {N}\). By the Egorov theorem (see [7, Theorem 42, p. 18]), there exists \(A_0\in \Sigma \) with \(A_0\subset \Omega _0\), \(\mu (\Omega _0\smallsetminus A_0)\le \delta \) and \(\sup _{\omega \in A_0}\Vert s_n(\omega )-g(\omega )\Vert _X\rightarrow 0\). For every \(n\in \mathbb {N}\) define an operator \(T_n:E\rightarrow X\) by
$$\begin{aligned} T_n(u):=\int _{A_0} u(\omega )s_n(\omega )\,d\mu \ \ \text{ for } \ \ u\in E. \end{aligned}$$
Note that for every \(n\in \mathbb {N}\), \(T_n\) is a bounded operator of finite dimensional range, so that \(T_n\) is a compact operator. Define an operator \(T_{A_0}:E\rightarrow X\) by
$$\begin{aligned} T_{A_0}(u):=\int _{A_0} u(\omega )g(\omega )\,d\mu \ \ for \ \ u\in E. \end{aligned}$$
Then for \(u\in B_E(1)\), we have
$$\begin{aligned} \displaystyle \bigg \Vert \int _{A_0} u(\omega )(g(\omega )-s_n(\omega ))\,d\mu \bigg \Vert _X\le & {} \int _{A_0}|u(\omega )|\cdot \Vert g(\omega )-s_n(\omega )\Vert _X\,d\mu \\\le & {} \, \bigg (\int _\Omega |u(\omega )|\mathbb {1}_{A_0}(\omega )\,d\mu \bigg ) \sup _{\omega \in A_0}\Vert g(\omega )-s_n(\omega )\Vert _X\\\le & {} \;\big \Vert \mathbb {1}_{A_0}\big \Vert _{E'}\sup _{\omega \in A_0}\Vert g(\omega )-s_n(\omega )\Vert _X. \end{aligned}$$
Hence \(\Vert T_{A_0}-T_n\Vert \le \Vert \mathbb {1}_{A_0}\Vert _{E'}\sup _{\omega \in A_0}\Vert g(\omega )-s_n(\omega )\Vert _X\), and hence \(\Vert T_{A_0}-T_n\Vert \rightarrow 0\). It follows that \(T_{A_0}\) is a compact operator. Since \(\sup _{u\in H}\Vert u\Vert _E<\infty \), we have that \(K_\varepsilon :=\{T(\mathbb {1}_{A_0} u):u\in H\}\) is a relatively norm compact set in X.
Moreover, since \(\mu (\Omega _0\smallsetminus A_0)\le \delta \), for every \(u\in H\), by (2.2) we get
$$\begin{aligned} \big \Vert T\big (\mathbb {1}_{\Omega _0\smallsetminus A_0}u\big )\big \Vert _X\le \int _{\Omega \smallsetminus A_0} |u(\omega )|\cdot \Vert g(\omega )\Vert _X \,d\mu \le \frac{\varepsilon }{2} \end{aligned}$$
i.e., \(\{T(\mathbb {1}_{\Omega _0\smallsetminus A_0}u):u\in H\}\subset \frac{\varepsilon }{2} B_X\). For every \(u\in H\), by (2.1) we have
$$\begin{aligned} \big \Vert T\big (\mathbb {1}_{\Omega \smallsetminus \Omega _0}u\big )\big \Vert _X\le \int _{\Omega \smallsetminus \Omega _0} |u(\omega )|\cdot \Vert g(\omega )\Vert _X \,d\mu \le \frac{\varepsilon }{2}, \end{aligned}$$
i.e., \(\{T(\mathbb {1}_{\Omega \smallsetminus \Omega _0}u):u\in H\}\subset \frac{\varepsilon }{2} B_X\). Hence for every \(u\in H\), we have
$$\begin{aligned} T(u)=T(\mathbb {1}_{\Omega \smallsetminus \Omega _0}u)+T(\mathbb {1}_{\Omega _0\smallsetminus A_0} u)+ T(\mathbb {1}_{A_0} u)\in \varepsilon B_X+K_\varepsilon . \end{aligned}$$
In view of the Grothendieck’s compactness criterion (see [4, Exercise 4(iii)]) T(H) is a relatively norm compact set in X, as desired. \(\square \)
By \(\gamma [\mathcal{T}_E,\mathcal{T}_0]\) (in brief, \(\gamma _E\)) we denote the natural mixed topology on E in the sense of Wiweger (see [3, 11, 12, 19], for more details). Then \(\mathcal{T}_0\big |_E\subset \gamma _E\subset \mathcal{T}_E\) and \(\gamma _E\) is the finest linear topology on E that agrees with \(\mathcal{T}_0\) on every ball \(B_E(r)\), \(r>0\) (see [19, 2.2.2]). \(\mathcal{T}_E\) and \(\gamma _E\) have the same bounded sets in E, and for a sequence \((u_n)\) in E, \(u_n\rightarrow 0\) in \(\gamma _E\) if and only if \(u_n\rightarrow 0\) in \(\mathcal{T}_0\) and \(\sup _n\Vert u_n\Vert _E<\infty \) (see [19, Corollary, p. 56 and Theorem 2.6.1]). If, in particular, \(\mathrm{supp}\,(E')_a=\Omega \), then \(\gamma _E\) is a locally convex-solid Hausdorff topology and \(\gamma _E\) coincides with the mixed topology \(\gamma [\mathcal{T}_E,|\sigma |(E,(E')_a)]\) (see [11, Theorem 3.3]). Note that, then \((E,\gamma _E)\) is a generalized DF-space (see [15] for more details). In view of [11, Theorem 3.1], we have:
$$\begin{aligned} (E,\gamma _E)^*=\{F_v:v\in (E')_a\}. \end{aligned}$$
(2.3)

Theorem 2.2

Assume that \((E,\Vert \cdot \Vert _E)\) is a Banach function space with the order continuous accociated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\). Let \(T:E\rightarrow X\) be a Bochner representable operator. Then the following statements hold:
  1. (i)

    T is \((\gamma _E,\Vert \cdot \Vert _X)\)-continuous and norm-compact operator.

     
  2. (ii)

    T is \((\gamma _E,\Vert \cdot \Vert _X)\)-compact, that is, there exists a \(\gamma _E\)-neighborhood V of 0 in E such that T(V) is a relatively norm compact set in X.

     

Proof

(i) There exists \(g\in (E')_a(X)\) such that
$$\begin{aligned} T(u)=\int _\Omega u(\omega )g(\omega )\,d\mu \ \ \text{ for } \ \ u\in E. \end{aligned}$$
For \(u\in E\), we have
$$\begin{aligned} \Vert T(u)\Vert _X\le \int _\Omega |u(\omega )|\;\Vert g(\omega )\Vert _X \,d\mu =F_{\Vert g(\cdot )\Vert _X}(|u|), \end{aligned}$$
where \(\Vert g(\cdot )\Vert _X\in (E')_a\). Using (2.3) we obtain that T is \((\gamma _E,\Vert \cdot \Vert _X)\)-continuous. Choose a sequence \((s_n)\) of X-valued strongly \(\mu \)-measurable step-functions on \(\Omega \) such that \(\Vert s_n(\omega )-g(\omega )\Vert _X\rightarrow 0\) \(\mu \)-a.e. and \(\Vert s_n(\omega )\Vert _X\le \Vert g(\omega )\Vert _X\) \(\mu \)-a.e. and all \(n\in \mathbb {N}\) (see [7, Theorem 6, p. 4]). Hence \(s_n\in (E')_a(X)\) and \(\Vert s_n(\omega )-g(\omega )\Vert _X\le 2\Vert g(\omega )\Vert _X\) \(\mu \)-a.e. for all \(n\in \mathbb {N}\). Let \(v_n(\omega )=\sup _{k\ge n}\Vert s_n(\omega )-g(\omega )\Vert _X\) for \(\omega \in \Omega \). Then \(2\Vert g(\omega )\Vert _X\ge v_n(\omega )\downarrow 0\) \(\mu \)-a.e., so \(\Vert \;\Vert s_n(\cdot )-g(\cdot )\Vert _X \;\Vert _{E'}\le \Vert v_n\Vert _{E'}\rightarrow 0\).
For each \(n\in \mathbb {N}\), let \(T_n:E\rightarrow X\) be a linear operator defined by
$$\begin{aligned} T_n(u):=\int _\Omega u(\omega )s_n(\omega )\,d\mu \ \ \text{ for } \ \ u\in E. \end{aligned}$$
Note that the range of each \(T_n\) is contained in the span of the finite set of values of \(s_n\). Therefore \(T_n\) is compact for each \(n\in \mathbb {N}\), and for each \(u\in E\),
$$\begin{aligned} \displaystyle \Vert (T_n-T)(u)\Vert _X= & {} \,\bigg \Vert \int _{\Omega }u(\omega )(s_n(\omega )-g(\omega ))\,d\mu \bigg \Vert _X\\\le & {} \int _\Omega |u(\omega )|\cdot \Vert s_n(\omega )-g(\omega )\Vert _X d\,\mu \\\le & {} \,\Vert u\Vert _E \cdot \Big \Vert \,\Vert s_n(\cdot )-g(\cdot )\Vert _X\,\Big \Vert _{E'}. \end{aligned}$$
It follows that \(\Vert T-T_n\Vert \rightarrow 0\), so T is a compact operator.

(ii) Since \((E,\gamma _E)\) is a generalized DF-space, it is quasinormable (see [15, p. 422]). Hence in view of (i) by the Grothendieck’s classical results (see [15, p. 429]), we obtain that T is \((\gamma _E,\Vert \cdot \Vert )\)-compact. \(\square \)

Remark 2.1

A related result to Theorem 2.2 can be found in [17, Corollary 4.8].

An important class of Banach function spaces are Orlicz spaces (see [10, 16] for more details). By a Young function we mean here s convex continuous mapping \(\Phi :[0,\infty )\rightarrow [0,\infty )\) that vanishes only at 0 and \(\Phi (t)/t\rightarrow 0\) as \(t\rightarrow 0\) and \(\Phi (t)/t\rightarrow \infty \) as \(t\rightarrow \infty \). By \(\Phi ^*\) we denote the Young function complementary to \(\Phi \) in the sense of Young.

The Orlicz space
$$\begin{aligned} L^\Phi :=\bigg \{u\in L^0:\int _\Omega \Phi (\lambda |u(\omega )|)\,d\mu <\infty \ \ \text{ for } \text{ some } \ \ \lambda >0\bigg \}, \end{aligned}$$
equipped with the topology \(\mathcal{T}_\Phi \) of the norm:
$$\begin{aligned} \Vert u\Vert _\Phi :=\inf \bigg \{\lambda >0:\int _\Omega \Phi (|u(\omega )|/\lambda )\,d\mu \le 1\bigg \} \end{aligned}$$
is a perfect Banach function space (see [10, 16]). Then \((L^\Phi )'=L^{\Phi ^*}\) and \((L^{\Phi ^*})_a=E^{\Phi ^*}=\{v\in L^{\Phi ^*}: \int _\Omega \Phi (\lambda |v(\omega )|)d\mu <\infty \) for all \(\lambda >0\}\). In particular, \(E^{\Phi ^*}=L^{\Phi ^*}\) if \(\Phi ^*\) satisfies the \(\Delta _2\)-condition, i.e., \(\Phi ^*(2t)\le d\Phi ^*(t)\) for some \(d>1\) and all \(t\ge 0\).

We say that a Young function \(\Psi \) increases more rapidly than \(\Phi \) (in symbols, \(\Phi \prec \Psi \)) if for an arbitrary \(c>0\) there exists \(d>0\) such that \(c\Phi (t)\le \frac{1}{d}\Psi (dt)\) for all \(t\ge 0\).

If \(\Phi \prec \Psi \), then \(L^\Psi \subset L^\Phi \) and by \(i_\Psi :L^\Psi \rightarrow L^\Phi \) we denote the inclusion map.

Proposition 2.3

Let \(T:L^\Phi \rightarrow X\) be a Bochner representable operator. Then for every Young function \(\Psi \) with \(\Phi \prec \Psi \) the operator \(T\circ i_\Psi :L^\Psi \rightarrow X\) is compact.

Proof

Assume that \(\Psi \) is a Young function with \(\Phi \prec \Psi \). Then by [16, Theorem 5.3.3, p. 171] the closed unit ball \(B_\Psi (1)\) in \(L^\Psi \) is relatively \(\sigma (L^\Phi ,L^{\Phi ^*})\)-compact in \(L^\Phi \). Hence according to Theorem 2.1 \(T(B_\Psi (1))\) is relatively norm compact, and this means that \(T\circ i_\Psi :L^\Psi \rightarrow X\) is compact. \(\square \)

We say that a Young function \(\Phi \) increases essentially more rapidly than another \(\Psi \) (in symbols, \(\Psi \ll \Phi \)) if for an arbitrary \(c>0\), \(\Psi (ct)/\Phi (t)\rightarrow 0\) as \(t\rightarrow 0\) and \(t\rightarrow \infty \).

The following characterization of the mixed topology \(\gamma _\Phi (=\gamma [\mathcal{T}_\Phi ,\mathcal{T}_0])\) on \(L^\Phi \) will be useful (see [14, Theorem 2.1]).

Theorem 2.4

Let \(\Phi \) be a Young function. Then the mixed topology \(\gamma _\Phi \) on \(L^\Phi \) is generated by the family of norms \(\{\Vert \cdot \Vert _\Psi \big |_{L^\Phi }:\Psi \ll \Phi \}\).

As a consequence of Theorems 2.2 and 2.4 we have:

Corollary 2.5

Assume that a Young function \(\Phi ^*\) satisfies the \(\Delta _2\)-condition. Let \(T:L^\Phi \rightarrow X\) be a Bochner representable operator. Then there exists a Young function \(\Psi \) with \(\Psi \ll \Phi \) such that \(T(B_\Psi (1)\cap L^\Phi )\) is a relatively norm compact set in X.

Remark

(i) The result of Corollary 2.5 was established in a different way in [14, Theorem 2.3].

(ii) For a bounded linear operator \(T:L^\Phi \rightarrow X\), following [6] one can define its functional norm |||T||| by \(|||T|||=\sup \Sigma \Vert \alpha _iT(\mathbb {1}_{A_i})\Vert _X,\) where the supremum is taken over all finite \(\Sigma \)-partition \((A_i)\) of \(\Omega \) and all \(\alpha _i\in \mathbb {R}\) such that \(\Vert \Sigma \alpha _i\mathbb {1}_{A_i}\Vert _\Phi \le 1\).

Uhl [18, Theorem 1] showed that if X either is reflexive or is a separable dual Banach space and \(\Phi \) obeys the \(\Delta _2\)-condition, then every bounded linear operator \(T:L^\Phi \rightarrow X\) with \(|||T|||<\infty \) is Bochner representable. If, in addition, \(\Phi ^*\) also obeys the \(\Delta _2\)-condition, then every bounded operator \(T:L^\Phi \rightarrow X\) with \(|||T|||<\infty \) is compact (see [18, Corollary 2]).

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Authors and Affiliations

  1. 1.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GoraPoland

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