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On Fixed Point Property under Lipschitz and Uniform Embeddings.

1. Introduction

The infinite Rademacher theorem [1, 2] states that every Lipschitz mapping from a separable Banach space to a Banach space with the Radon-Nikodym property (in short, RNP) is almost everywhere Gateaux differentiable, i.e., everywhere Gateaux differentiable off an Aronszajn null set. It is shown in [3, 4] that Aronszajn null sets (introduced by Aronszajn [1]), Gauss null sets (introduced by Phelps [5]), and cube null sets (introduced by Mankiewicz [6]) coincide. In particular, Aronszajn null sets and Lebesgue null sets coincide in a finite dimensional Banach space [3 ]. To establish the localized version of the infinite Rademacher theorem, Cheng and Zhang [7] substituted closed convex subsets of a Banach space X for X and proved that every Lipschitz mapping from a closed convex subset C of a Banach space with nonempty nonsupport point set N(C) to a Banach space with the RNP is almost everywhere Gaateaux differentiable.

As is well known, dual Banach spaces without separability may fail RNP. The Lipschitz mappings from a separable Banach space to a dual Banach space without separability need not be Gaateaux differentiable anywhere. To circumvent this obstruction, a weaker notion of a [[omega].sup.*]-Gateaux differentiability was introduced (see, for instance, [3, 8]). Heinrich and Mankiewicz [9] proved the infinite [[omega].sup.*]-Rademacher theorem: every Lipschitz mapping from a separable Banach space to the dual space of a separable Banach space is almost everywhere [[omega].sup.*]-Gateaux differentiable. Therefore, a question naturally arises.

Question 1. The question is whether the infinite [[omega].sup.*]-Rademacher theorem can be localized to those Lipschitz mappings from a closed convex set C admitting nonempty nonsupport point set N(C) to the dual space of a separable Banach space.

The aim of this paper focuses on the study of the question above. Based on the ideas of the infinite [[omega].sup.*]-Rademacher theorem, we shall prove the following theorem.

Theorem 2. Suppose that C is a closed convex set of a separable Banach space X with N(C) [not equal to] 0 and that Y is a separable Banach space. Let f : C [right arrow] [Y.sup.*] be a Lipschitz mapping. Then f is [[omega].sup.*]-Gateaux differentiable off a Aronszajn null set of N(C).

As an application of Theorem 2, we obtain the following result.

Theorem 3. Every nonempty bounded closed convex subset C of a Banach space X has the fixed point property for isometries if it Lipschitz embeds into a super reflexive space.

With the application of Baudier-Lancien-Schlumprecht's theorem, we finally get the following result.

Theorem 4. Every nonempty bounded closed convex subset C of a Banach space X has the fixed point property for continuous affine mappings if it uniformly embeds into the Tsirelson space [T.sup.*].

The letter X will always be a real Banach space and [X.sup.*] its dual. [S.sub.X] denotes the unit sphere of X. For a subset C [subset] X, we denote by [bar.C] the closure of C. For simplicity, we also denote [X.sub.C] = [bar.span](C) and [C.sub.x] = [union]{[lambda](C - x) : [lambda] > 0} for some x [member of] C.

2. A Proof of Theorem 2

We first recall definitions of support points [5, 7] and Aronszajn null sets [1, 3].

Definition 5. Suppose that C is a convex set in a Banach space X. A point x [member of] C is said to be a support point of C if there exists a non-zero functional [x.sup.*] [member of] [X.sup.*] such that <[x.sup.*], x> = max{<[x.sup.*], y> : y [member of] C}.

We denote by S(C) the set of all support points of C and by N(C) the set of all nonsupport points of C.

Let X be a Banach space. Let [lambda] be the one dimensional Lebesgue measure. For each x [member of] X \ {0}, set

[LAMBDA](x) = {A

[subset] X : A is a Borel set such that [lambda](A [intersection] (z + Rx))

= 0 for every z [member of] X}, (1)

and for a finite or infinite sequence {[x.sub.n]} of nonzero points in X, set

[LAMBDA] ({[x.sub.n]}) = {A = [union] [A.sub.n], [A.sub.n] [member of] [LAMBDA] ([x.sub.n]) for every n [member of] N}. (2)

Definition 6. Suppose that A is a Borel set in a separable Banach space X. The set A is called an Aronszajn null set (or simply, a null set) if

A [member of] [intersection] {[LAMBDA]({[x.sub.n]}) : {[x.sub.]}

are nonzero vectors such that span {[x.sub.]}

is dense in X}. (3)

Note that N(C) is a [G.sub.[delta]] set; hence it is a Borel set (see, for instance, [7,10]).

The following two lemmas are presented in [3, 8].

Lemma 7. Let Y be a Banach space with the RNP. Then every Lipschitz function f from an open subset U of [R.sup.n] into Y is Gateaux differentiable off a null set of U.

Lemma 8. Let F be an n-dimensional subspace of Banach space X, and let [{[x.sub.k]}.sup.n.sub.k=1] be a basis for F. Let [[lambda].sub.n] be the Lebesgue measure on F, and let A be a Borel subset of X such that [[lambda].sub.n](F [intersection] (A + x)) = 0 for every x [member of] X. Then A [member of] [LAMBDA]([{[x.sub.k]}.sup.n.sub.k=1]}).

Definition 9. Suppose that C is a closed convex set of a separable Banach space X with N(C) [not equal to] 0 and that Y is a separable Banach space. Let f : C [right arrow] [Y.sup.*] be a Lipschitz mapping. Then f is said to be [[omega].sup.*]-Gateaux differentiable at x [member of] N(C) if there exists a bounded linear operator [T.sub.x] : C [right arrow] [Y.sup.*] such that for every h [member of] [C.sub.x] and every y [member of] Y the following limit exists:

[mathematical expression not reproducible]. (4)

In this case [T.sub.x] is called the [[omega].sup.*]-derivative of f at x and is denote by [D.sup.*.sub.f](x).

Theorem 10. Suppose that C is a closed convex set of a separable Banach space X with N(C) [not equal to] 0 and that Y is a separable Banach space. Let f : C [right arrow] [Y.sup.*] be a Lipschitz mapping; i.e., there exists K > 0 such that [parallel] f(x) - f(y)[parallel] [less than or equal to] K[parallel]x - y[parallel] for all x, y [member of] C. Let {[x.sub.n]} be a sequence which consists of linearly independent vectors in N(C). For every n [member of] N, set [mathematical expression not reproducible]. Then

(i) [f.sub.n] is [[omega].sup.*]-Gateaux differentiable off a null set of N(C) [intersection] [X.sub.n] for every n [member of] N;

(ii) [mathematical expression not reproducible] exists for every n [member of] N;

(iii) If, in addition, there exists k > 0 such that [parallel]f(x) - f(y) [parallel] [greater than or equal to] k[parallel]x - y[parallel] for all x, y [member of] C, then then [mathematical expression not reproducible] is a linear isomorphism for almost every x [member of] N(C) [intersection] [X.sub.n].

Proof. (i). Without loss of generality, we assume that 0 [member of] N(C). Let {[y.sub.m]} be a dense sequence in [S.sub.Y]. Given n [member of] N, define [[phi].sub.m](x) = [f.sub.n](x)([y.sub.m]) for every m [member of] N and x [member of] C. Then <pm are Lipschitz real-valued functions on C. Since {[x.sub.n]} is a sequence which consists of linearly independent vectors in N(C) and [X.sub.n] = span{[x.sub.k] : k [less than or equal to] n}, we obtain that span(X(C) [intersection] [X.sub.n]) = [X.sub.n], and hence, N(C) [intersection] [X.sub.n] is nonempty open in [X.sub.n] (see also p.p, 9 in [11]).

Applying Lemma 7 to [X.sub.n] and N(C) [intersection] [X.sub.n], then [[phi].sub.m] are Gateaux differentiable off a null set of N(C) [intersection] [X.sub.n]. Let [mathematical expression not reproducible] denote the derivative of [[phi].sub.m] at x.

Consider the set

[mathematical expression not reproducible]. (5)

Then W is not null set in N(C) [intersection] [X.sub.n]. Choose x [member of] W; then the following limits exists:

[mathematical expression not reproducible], (6)

for every m [member of] N, h e [X.sub.n]. Since f is Lipschitz, ([f.sub.n](x + [lambda]h) - [f.sub.n](x))/[lambda] is a bounded [[omega].sup.*]-Cauchy sequence when [lambda] [right arrow] 0. By Alaoglu theorem, [[omega].sup.*]-[lim.sub.[lambda][right arrow]0] (([f.sub.n](x + [lambda]h) - [f.sub.n](x))/[lambda]) exists in [Y.sup.*]. Since [mathematical expression not reproducible] is linear for every m [member of] N, the limit [[omega].sup.*]-[lim.sub.[lambda][right arrow]0] (([f.sub.n](x + [lambda]h) - [f.sub.n](x))/[lambda]) is linear, and hence it is a bounded linear operator from [X.sub.n] into [Y.sup.*] because f is Lipschitz. Therefore, [mathematical expression not reproducible] exists for every n [member of] N and x [member of] W.

(ii) . For every n [member of] N, h [member of] [X.sub.n] and x [member of] W, we have

[mathematical expression not reproducible]. (7)

This completes our assertion (ii).

(iii) . We omit the proof because the proof of assertion (iii) is similar to the proof of assertion (c) of Theorem 14.2.18, p. 385 in [8].

Proof of Theorem 2. Without loss of generality, we assume that 0 [member of] N(C). By Theorem 2.6 in [7], span(C) = span(N(C)) = [[union].sup.[infinity].sub.n=1] nN(C) is a dense subspace of X. Since X is separable, there exists a sequence {[x.sub.n]} of linearly independent vectors in N(C) such that span{[x.sub.n} is dense in X. For each n [member of] N, set [X.sub.n] = span{[x.sub.k] : k [less than or equal to] n}. Let

[mathematical expression not reproducible]. (8)

and let D = [intersection][D.sub.n]. Since [D.sub.n] is a Borel set, so is D. Therefore, for each x [member of] D,

[mathematical expression not reproducible] (9)

defines a bounded linear operator from span{[x.sub.n]} to [Y.sup.*].

Since span{[x.sub.n]} is dense in X, there exists a unique bounded linear extension T of [T.sub.x] from span{[x.sub.n]} to X. Therefore, T = [D.sup.*.sub.f](x) and D is just the set of all [[omega].sup.*]-Gateaux differentiability points of f in N(C). It remains to show that D is not null.

By Theorem 10, (N(C)\[D.sub.n]) [intersection] [X.sub.n] is a null set in N(C) [intersection] [X.sub.n]. Given z [member of] X, we define [f.sub.z] by [f.sub.z](x) = f(x - z), x [member of] (N(C) + z) [intersection] [X.sub.n]. Then [f.sub.z] is a Lipschitz mapping on set (N(C)+z) [intersection] [X.sub.n], and by Lemma 7, (N(C)\ [D.sub.n] + z) [intersection] [X.sub.n] is the set of all non-Gateaux differentiability points of [f.sub.z]. By Lemma 8, N(C) \ [D.sub.n] [member of] [LAMBDA]([x.sub.k] : k [less than or equal to] n). Therefore, N(C) \ D [member of] [LAMBDA]({[x.sub.n]}).

Next, for any sequence {[x'.sub.n]} of non-zero vectors in X whose linear span is dense in X, we shall prove N(C)\ D [member of] [LAMBDA]({[x'.sub.n]}). If span{[x.sub.n]} [subset] span(C), we just repeat the procedure above. Otherwise, for any v [member of] {[x'.sub.n]} such that v [not member of] span(C), if (N(C) \ D) [intersection] (z + Rx) [not equal to] 0 for every z [member of] X, then there exists z' [member of] N(C) \ D such that (N(C) \ D) [intersection] (z + Rx) = (N(C)\D) [intersection] {z + Rx) = {z'}. Therefore, N(C)\D [member of] [LAMBDA](x). This implies that N(C)\D is null set.

3. Fixed Point Property for Isometries

In this section we apply [[omega].sup.*]-Gateaux differentiability of Lipschitz mappings to the fixed point theory. As a result, we obtain Theorem 3.

Let C be a nonempty bounded closed convex subset of a Banach space X. Recall that a mapping T : C [right arrow] C is said to be isometry if

[parallel]T(x) - T(y)[parallel] = [parallel]x - y[parallel], (10)

whenever x, y [member of] C. We say that C has the fixed point property for isometries if every isometry T : C [right arrow] C has a fixed point.

Definition 11. Suppose that C is a subset of a Banach space X and that Y is a Banach space. We say that C Lipschitz embeds into [Y.sup.*] provided that there is a mapping f : C [right arrow] [Y.sup.*] and constants k, K > 0 such that for all x, y [member of] C,

k[parallel]x - y[parallel] [less than or equal to] [parallel]f (x) - f(y)[parallel] [less than or equal to] K [parallel]x - y[parallel]. (11)

In this case, f is said to be Lipschitz embedding and the smallest possible constant K is called the Lipschitz constant of the mapping, in short, Lip(f) = K.

Proposition 12. Suppose that C is a closed convex separable set in a Banach space X. Then N(C) [not equal to] 0 in [X.sub.c].

Proof. We may assume that 0 [member of] C. Since C is separable, [X.sub.c] is a separable space. It follows immediately from Proposition 1 in [10] that N(C) [not equal to] 0.

Theorem 13. Suppose that C is a separable closed convex set of a Banach space X, and that Y is a separable Banach space. Then C Lipschitz embeds into [Y.sup.*] if and only if [X.sub.c] is linearly isomorphic to a subspace of [Y.sup.*].

Proof.

Sufficiency. It is clearly trivial.

Necessity. From Proposition 12, N(C) [not equal to] 0 in [X.sub.c]. We may assume that 0 [member of] N(C). It follows from Theorem 2.6 in [7] that [C.sub.0] = [[union].sup.[infinity].sub.n=1] nC = span(C) is a dense subspace of [X.sub.c]. Note that [X.sub.c] is separable, thus there exists a sequence {[x.sub.n]} of linearly independent vectors in N(C) such that span{[x.sub.n]} is dense in [X.sub.c]. Let f : C [right arrow] [Y.sup.*] be a Lipschitz embedding. For each n [member of] N, set [X.sub.n] = span{x.sub.k] : k [less than or equal to] n}. Consider the set of all points x [member of] N(C) for which there is abounded linear operator [T.sub.n] : [X.sub.n] [right arrow] [Y.sup.*] such that

[mathematical expression not reproducible], (12)

and

k[parallel]h[parallel] [less than or equal to] [parallel][T.sub.n] (h)[parallel], h [member of] [X.sub.n] (13)

From the proof of Theorem 10, the conclusion follows using argument similar to the proof of Theorem 2.

Proposition 14. Suppose that C is a separable closed convex set of a Banach space X, and that Y is a dual Banach space. Then C Lipschitz embeds into Y if and only if [X.sub.c] is linearly isomorphic to a subspace of Y.

Proof. It suffices to show the necessity. Let Y = [Z.sup.*] and let f: C [right arrow] [Z.sup.*] be a Lipschitz embedding. Since C is a separable, [X.sub.f](c) is a separable subspace of [Z.sup.*]. By Corollary 14.2.22 in [8], we can find a separable Banach space W [subset] Z and linearly isometric embeddings [i.sub.1] : [X.sub.f(C)] [right arrow] [W.sup.*] and [i.sub.2] : [W.sup.*] [right arrow] [Z.sup.*]. Hence, [i.sub.1] f : C [right arrow] [W.sup.*] is a Lipschitz embedding. By Theorem 13, [X.sub.c] is linearly isomorphic to a subspace of [W.sup.*]. Note that [W.sup.*] is linearly isometric isomorphic to a subspace of [Z.sup.*]. Therefore, [X.sub.c] is linearly isomorphic to a subspace of Y.

Proof of Theorem 3. Let C be a nonempty bounded closed convex subset of a Banach space X and let C be Lipschitz embedded into a super reflexive space Y. We still assume 0 e C. We can also assume that C is separable because the fixed point property for isometries of a bounded closed convex set in a Banach space is separably determined. Superreflexivity of Y entails Y = [Y.sup.**] and hence [Y.sup.**] is also a super reflexive space. By Proposition 14, Xc is linearly isomorphic to a closed subspace of [Y.sup.**]. This implies [X.sub.C] is a super reflexive space. Therefore, C has the fixed point property for isometries by Maurey's fixed point theorem [12].

Remark 15. The converse Theorem 3 does not hold. In fact, for a nonempty uniformly convexifiable set C of Banach space X, C has the fixed point property for isometries (See [13, Theorem 4.2]). However, Beauzamy [14] proved that there exists a uniformly convexifiable set in a Banach space which can not linearly embedded into a super reflexive space.

4. Fixed Point Property for Continuous Affine Mappings

In this section, with the application of Baudier-Lancien-Schlumprecht's theorem, we obtain Theorem 4.

Let C be a nonempty bounded closed convex subset of a Banach space X. Recall that a mapping T : C [right arrow] C is said to be affine if

T([lambda]x + (1 - [lambda]y)) = [lambda]T(x) + (1 - [lambda])T (y), (14)

whenever x, y [member of] C and X [member of] [0,1]. We say that C has the fixed point property for continuous affine mappings if every continuous affine mapping T : C [right arrow] C has a fixed point.

Let X and Y be two Banach spaces and C be a subset of X, and let f : C [right arrow] Y be a mapping. Set

[[rho].sub.f] (t) = sup{[parallel]f (x) - f (y)[parallel].sub.Y] : [[parallel] x - y [parallel].sub.X] [less than or equal to] t}, (15)

and

[[omega].sub.f] (t) = sup{[parallel]f (x) - f (y)[parallel].sub.Y] : [[parallel] x - y [parallel].sub.X] [less than or equal to] t}. (16)

We say that f is uniformly continuous if [lim.sub.t[right arrow]0] [[omega].sub.f] (t) = 0 and [[rho].sub.f] (t) > 0 for all t > 0. C is said to be uniformly embedded into Y provided that there is a mapping f: C [right arrow] Y such that f is injective and both f and [f.sup.-1] are uniformly continuous.

Proposition 16. Suppose that C is a nonempty bounded closed convex subset of a Banach space X. If C affinely uniformly embeds into a reflexive space Y, then C has the fixed point property for continuous affine mappings.

Proof. Suppose that C affinely uniformly embeds into a reflexive space Y. Then there exist an affinely uniformly continuous mapping T : C [right arrow] Y. We may assume that 0 [member of] C. Defining a mapping [T.sub.1] (x) = T(v) - T(0) for all x [member of] [bar.span](C) is a linearly uniformly continuous mapping from [bar.span](C) to Y. Thus, [bar.span](C) is a reflexive space. It follows from Schauder-Tychonoff theorem [15] that C has the fixed point property for continuous affine mappings.

Remark 17. The affinity in Proposition 16 is necessary. Mazur [16] proved that the unit ball [mathematical expression not reproducible] is uniformly homomorphic onto the unit ball [mathematical expression not reproducible]. It is easy to see that [mathematical expression not reproducible] has the fixed point property for continuous affine mappings by Schauder-Tychonoff theorem [15] and [mathematical expression not reproducible] does not have the fixed point property for continuous affine mappings by Theorems 3.2 in [17]. On the other hand, the Mazur theorem [16] also implies that weak compactness is usually not preserved under uniform embeddings.

Recall the construction of the Tsirelson spaces [T.sup.*] and T originally designed by Tsirelson [18]. Let E, F [subset] N and n [member of] N. We denote E < F if max(F) < min(F) and n [less than or equal to] E if n [less than or equal to] min(E). Here we set max(0) = 0 and min(0) = [infinity]. We say that a sequence [{[E.sub.n]}.sup.n.sub.j=1] [subset] N is admissible if n [less than or equal to] [E.sub.1] < [E.sub.2] < ... < [E.sub.n]. Let {[e.sub.f]} be the canonical basis of [c.sub.00]. For every x = [[summation].sup.[infinity].sub.j=1] [[alpha].sub.j][e.sub.j] [member of] [c.sub.00] we put ||*||[mathematical expression not reproducible] and then define inductively for k = 1, 2, ...

[mathematical expression not reproducible]. (17)

Put

[mathematical expression not reproducible]. (18)

Then [parallel] x [parallel] a norm on [c.sub.00] and T is defined to be the completion of [c.sub.00] with respect to the norm || ? ||. We denote the dual of T by [T.sup.*] which is nowadays usually referred to as [T.sup.*]. For more detail, we refer the reader to [19].

For any infinite subset M of N, let [[M].sup.[omega]] = {A [subset] M : A is infinite}. For each k [member of] N, let [[M].sup.k] = {A [subset] M : [absolute value of A] = k}, where [absolute value of Z] denotes the cardinality of the set A. Elements of [[N].sup.k] will always be listed in an increasing order; i.e., for every [bar.m] = {[m.sub.1], [m.sub.2], ..., [m.sub.k]} [member of] [[M].sup.k], we assume that [m.sub.1] < [m.sub.2] < ... < [m.sub.k]. Recall that the Hamming metric is defined by

[mathematical expression not reproducible] (19)

where [bar.m], [bar.n] [member of] [[N].sup.k] Note that the metric [d.sub.H], can be seen as the graph metric on the Hamming graph over a countable alphabet, denoted [H.sup.[omega].sub.k](N) or simply [H.sup.[omega].sub.k], where two vertices are adjacent if they differ in exactly one coordinate. Let h : N x N [right arrow] N be a bijective. Then the map [[phi].sub.k] : [H.sup.[omega].sub.k] [right arrow] [T.sup.*] defined by [mathematical expression not reproducible] is a Lipschitz mapping with Lip([[phi].sub.k]) = 1 (see, for instant, [20]).

Proof of Theorem 4. By Schauder-Tychonoff theorem [15], it suffices to show that C is weakly compact. Suppose, to the contrary, that C is not weakly compact. Then, by James' theorem [21], there exist [delta] > 0 and a sequences {[X.sub.n]} [subset] C such that for all k [member of] N,

[mathematical expression not reproducible]. (20)

Choose [bar.n] [member of] [[N].sup.2k]. Then

[mathematical expression not reproducible], (21)

which implies that

[mathematical expression not reproducible], (22)

For each [bar.n] [member of] [H.sup.[omega].sub.k] dehne a map [mathematical expression not reproducible]. Then [[phi].sub.k] is a Lipschitz mapping with Lip([[phi].sub.k]) = 2/k.

Let f be a uniform embedding from C into [T.sup.*]. Then Lip(f [omicron] [[phi].sub.k]) [less than or equal to] [[omega].sub.f](2/k). By Theorem 4.4 in [20], for all k [greater than or equal to] 1, there exists M' [member of] [[N].sup.[omega]] such that

[mathematical expression not reproducible], (23)

for all [sub.m], [sub.n] [member of] [[M'].sup.k]. This implies

[mathematical expression not reproducible]. (24)

In particular, for all k [greater than or equal to] 1, choose [bar.m], [bar.n], [member of] [[M'].sup.k] such that [m.sub.1] < [m.sub.1] < ... < [m.sub.k] < [n.sub.1] < [n.sub.2] ... < [n.sub.k]. We obtain

0 < [[rho].sub.f]([delta]) [less than or equal to] 5[[omega].sub.f](2/k). (25)

This is a contradiction when k is sufficiently large and which completes our proof.

https://doi.org/10.1155/2018/4758546

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author was supported by Educational Commission of Hubei Province of China, Grant no. B2018046. The corresponding author was supported by NSFC, Grant no. 11501108, and by NSFF, Grant no. 2015J01579, 1991 Mathematics Subject Classification, 47H10,46B03, and 46G05.

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Jichao Zhang, (1) Lingxin Bao (iD), (2,3) and Lili Su (4)

(1) School of Science, Hubei University of Technology, Wuhan 430068, China

(2) School of Computer and Information, Fujian Agriculture and Forestry University, Fuzhou, 350002, China

(3) Key Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing100190, China

(4) Engineering University of the Chinese People's, Armed Police Force, Xian 710086, China

Correspondence should be addressed to Lingxin Bao; bolingxmu@sina.com

Received 19 June 2018; Accepted 24 September 2018; Published 21 October 2018

Academic Editor: Calogero Vetro
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Title Annotation:Research Article
Author:Zhang, Jichao; Bao, Lingxin; Su, Lili
Publication:Journal of Function Spaces
Date:Jan 1, 2018
Words:4788
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