THE SEPARABLE WEAK BOUNDED APPROXIMATION PROPERTY

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1 Bull. Korea Math. Soc. 52 (2015), No. 1, pp THE SEPARABLE WEAK BOUNDED APPROXIMATION PROPERTY Keu Youg Lee Abstract. I this paper we itroduce ad study the separable weak bouded approximatio properties which is strictly stroger tha the approximatio property ad but weaker tha the bouded approximatio property. It provides ew sufficiet coditios for the metric approximatio property for a dual Baach space. 1. Itroductio Let X ad Y be Baachspaces. We deote by B(X,Y) the spaceofbouded liear operators from X ito Y, ad by F(X,Y), K(X,Y), W(X,Y), ad B S (X,Y) its subspaces of fiite rak operators, compact operators, weakly compact operators, ad separable-valued bouded liear operators. Recall that a Baach space X is said to have the approximatio property (AP) if there exists a et (S α ) F(X,Y) such that S α I X uiformly o compact subsets of X. If (S α ) ca be chose with sup α S α 1, the X is said to have the metric approximatio property (MAP). The followig is a log stadig ope problem [1]. The Metric Approximatio Problem. Does the approximatio property of the dual space X of a Baach space X imply the metric approximatio property of X? Thaks to Grothedieck, he provides a affirmative aswer to the above problem i the case that X is separable. May people i this field have cosider the problem, but ot much progress has bee obtaied (see [3, 8, 9, 12]). The followig theorem is the most far-reachig result o this problem. Theorem 1 ([13, Theorem 5.50]). Let X be a Baach space such that X or X has the Rado-Nikodým property. If X has the approximatio property, the X has the metric approximatio property. Received August 18, 2012; Revised July 10, Mathematics Subject Classificatio. Primary 46B28; Secodary 46B22. Key words ad phrases. the separable weak metric approximatio property, separable weak metric approximatio property with cojugate operators, weak Rado-Nikodým property, metric approximatio property, factorizatio lemma. 69 c 2015 Korea Mathematical Society

2 70 KEUN YOUNG LEE We ca have the followig atural questio from Theorem 1. Questio. What are other o-trivial sufficiet coditios which guaratee the metric approximatio property for a dual Baach space X? Our aim i this paper is to provide aswers to the above questio. To do this, we itroduce the separable weak metric approximatio property ad study first a more geeral cocept so called the separable weak λ-bouded approximatio property. The separable weak metric approximatio property is strictly stroger tha the AP ad weaker tha the MAP. We eed the weak Rado- Nikodým property. Recall that the weak Rado-Nikodým property is a strictly weaker property tha the Rado-Nikodým property (see [10]). Our paper is orgaized as follows. I Sectio 2, we fix otatios ad itroduce basic facts. I Sectio 3, we study the separable weak λ-bouded approximatio property. The we provide characterizatios of the separable weak λ-bouded approximatio property ad several properties about this. I Sectio 4, we give aswers to Questio. 2. Prelimiaries A Baach X is said to have the Rado-Nikodým property if for each fiite measure space (Ω, Σ, µ) ad each µ-cotiuous X-valued coutably additive vector measure F : Σ X of bouded variatio, there exists a Bocher itegrable fuctio f : Ω X such that F(E) = fdµ for all E Σ (see E [13]). A Baach space X is said to have the weak Rado-Nikodým property if for each fiite complete measure space (Ω,Σ,µ) ad each µ-cotiuous X- valued coutably additive vector measure F : Σ X of bouded variatio, there exists a Pettis itegrable fuctio f : Ω X such that F(E) = (P)- E fdµ for all E Σ (see [11]). Clearly if X has the Rado-Nikodým property, the X has the weak Rado-Nikodým property but the coverse is ot true. For example, JT has the weak Rado-Nikodým property but does ot have Rado-Nikodým property where JT is the James tree space (see [10]). Let us fix some otatios. We deote by B w (X,Y ) (B w w(x,y)) its subspace of weak to weak (weak) cotiuous operators. Similarly, we defie K w (X,Y ) (K w w(x,y)) ad F w (X,Y ). Furthermore, we deote by B(X,Y;λ) the space of bouded liear operators T from X to Y satisfyig τ T λ. Similarly, we defie K(X,Y;λ) ad F(X,Y;λ). We deote S α T i B(X,Y) by S α T uiformly o compact subsets of X. 3. The separable weak bouded approximatio property Let X be a Baach space ad let 1 λ <. Let us recall that X is said to have the λ-bouded approximatio property (λ-bap) if there exists a et τ (S α ) F(X,X) such that sup α S α λ ad S α I X. Defiitio 3.1. A Baach space X is said to have the separable weak λ- bouded approximatio property(s-weak λ-bap) if for every separable Baach

3 THE SEPARABLE WEAK BOUNDED APPROXIMATION PROPERTY 71 space Y ad T B(Y,X;1), I X {S F(X,X) : ST λ} τ. If λ = 1, the we say that X has the separable weak metric approximatio property. Also we say that X has the separable weak bouded approximatio property (s-weak-bap) if X has the s-weak λ-bap for some λ. Let us start with followig simple observatios. Give two properties P 1 ad P 2, P 1 P 2 meas that P 1 is stroger tha P 2. Propositio 3.2. We have λ-bap s-weak λ-bap. Furthermore, we obtai that s-weak BAP is strictly stroger tha AP. Proof. First we show that λ-bap s-weak λ-bap. Suppose that X has the λ-bap. Let Y be a separable Baach space ad T B(Y,X;1). The there τ existsaet (S α ) F(X,X;λ)suchthat S α I X. The wehave S α T λ, hece X has the s-weak λ-bap. For s-weak BAP AP, it is clear. Fially, let X be the Figiel ad Johso space (see [4]). The X is a separable Baach space ad has the AP, but it does ot have the BAP. Hece we have I X / F(X,X;λ) τ for all λ 1. Sice X is separable, X does ot have the s-weak BAP. Remark 3.3. We do ot kow whether the s-weak BAP ad the BAP are differet properties. We cojecture that they are. We ow are goig to ivestigate the characterizatio of the s-weak λ-bap. To obtai this characterizatio, we eed followig lemmas. Recall that X K is the Davis-Figiel-Johso-Pelczyski costructio where K is a closed absolutely covex subset of B X (see [8]). Lemma 3.4. Suppose that (V,τ) is a locally covex space ad x V. If C is a balaced covex subset of V, the x C τ if ad oly if for every f (V,τ) f(x) sup f(y). y C Proof. By cotiuity the oly if part is clear. To show the if part, suppose to the cotrary that x / C τ. The, by the separatio theorem, there exist f (V,τ) ad t R such that Ref(x) > t > Ref(y) for all y C. Sice C is balaced, This completes the proof. f(x) > t sup y C Ref(y) = sup f(y). y C Lemma 3.5. Let K be a closed absolutely covex subset of B X ad Z be X K. The for every Baach space Y F(Z,Y) {SJ : S F(X,Y)},

4 72 KEUN YOUNG LEE where J : Z X is the idetity embeddig. Proof. Let T = k=1 ϕ k( )y k F(Z,Y) ad ε > 0. We may assume that k=1 y k = 1. Sice J : Z X is ijective, we have J (X ) = Z. The there exist x 1,...,x X so that ϕ k J (x k ) < ε for k = 1,...,. Cosider S = k=1 x k ( )y k F(X,Y). The we obtai T SJ = ϕ k ( )y k x kj( )y k = ϕ k ( )y k J x k( )y k < ε. k=1 k=1 k=1 k=1 Hece T {SJ : S F(X,Y)}. Lemma 3.6 ([8, Theorem 2.2]). If T : Y X is separably valued, the there exists a separable Baach space Z = X K, R : Y Z, ad J : Z X so that T = JR, Ry = Ty for every y Y, J is the iclusio map, ad T = R, J = 1 where K is T(B Y (0,1/ T )). Theorem 3.7. Let X be a Baach space ad let 1 λ <. The followig are equivalet. (a) X has the s-weak λ-bap. (b) For every separable Baach space Y, we have B(Y,X;1) F(Y,X;λ) τ. (c) For every Baach space Y ad T B S (Y,X;1), we have T {ST : S F(X,X) : ST λ} τ. (d) For every Baach space Y ad T B S (Y,X;1), we have I X {S F(X,X) : ST λ} τ. Proof. (a) (b) ad (d) (a) are clear. (b) (c) Let Y be a Baach space ad let T B S (Y,X;1). By Lemma 3.6, there exist a separable Baach space Z, R : Y Z, ad J : Z X such that T = JR, Ry = Ty for every y Y, J is the iclusio mappig, T = R, ad J = 1. By the assumptio, we have By Lemma 3.5, we have hece we obtai J F(Z,X;λ) τ. F(Z,X;λ) {SJ : S F(Z,X); SJ λ}, J F(Z,X;λ) τ = {SJ : S F(X,X); SJ λ} τ.

5 THE SEPARABLE WEAK BOUNDED APPROXIMATION PROPERTY 73 The there exists a et (S α ) i F(X,X) such that sup α S α J λ ad S α J τ J. For every compact K Y Further, for every α sup S α Ty Ty = sup S α JRy JRy 0. y K y K S α T = sup S α JRy S α J sup Ry λ R = λ T λ. y K y B Y Hece we have T {ST : S F(X,X), ST λ} τ. (c) (d) Let Y be a Baach space ad T be i B S (Y,X;1). Let (x ) ad (x ) be sequeces i X ad X, respectively, so that x x <. We may assume that for every x 1, x 0 ad x <. Now let K = abcov{(x ) T(B Y)}. Sice K is a separable subset of X, by the factorizatio lemma [8], there exists a Baach space Z such that the idetity embeddig J B S (Z,X), J 1, ad K J(B Z ). By the assumptio (c), we have (3.1) J {SJ : S F(X,X), SJ λ} τ B(Z,X). Sice x ( )x (B(Z,X),τ), by (3.1) ad Lemma 3.4, we obtai x (x ) = x Jx { } sup x (SJx ) : S F(X,X), SJ λ { } = sup x (Sx ) : S F(X,X), SJ λ { } sup x (Sx ) : S F(X,X), ST λ because ST SJ. Sice {ST : S F(X,X), ST λ} is a balaced covex subset of B(Z, X), Lemma 3.4 implies I X {S F(X,X) : ST λ} τ. Usig Theorem 3.7, we have the followig corollary. Corollary 3.8. Let X be a Baach space ad let 1 λ <. The followig are equivalet. (a) X has the s-weak λ-bap.

6 74 KEUN YOUNG LEE (b) For every Baach space Y, for every operator T B S (Y,X) with T = 1, ad for all sequece (x ) X, ad (y ) Y with x y <, oe has the iequality x T(y ) λ sup x ST(y ). ST 1,S F(X,X) Proof. (a) (b) Let Y be a Baach space ad T be i B S (Y,X) with T = 1. Take (x ) X, ad (y ) Y with x y < ad let ε > 0. We choosen Nsuchthat >N x y < ε. BytheassumptioadTheorem 3.7, we ca choose S F(X,X) such that ST λ ad x ST(y ) T(y ) < ε/n for all = 1,2,...,N. We obtai x T(y ) N x ST(y N ) + x S(y ) x T(y ) + >N x y x ST(y ) +2ε. Hece we have x T(y ) sup ST λ,s F(X,X) = λ sup ST 1,S F(X,X) x ST(y ) x ST(y ). (b) (a) Let Y be a Baach space ad T be i B S (Y,X) with T = 1. By Theorem 3.7, we are eough to show T {ST : S F(X,X), ST λ} τ. Take (x ) X, ad (y ) Y with x y <. By the assumptio (b), we have x T(y ) sup ST λ,s F(X,X) x ST(y ). Sice x ( )y (B(Y,X),τ) ad {ST : S F(X,X), ST λ} is a balaced covex subset of B(Y,X), by Lemma 3.4, we obtai This completes the proof. T {ST : S F(X,X), ST λ} τ. Let us cosider the trace mappig V from the projective tesor product X ˆ π Y to F(Y,X) defied by V(u)(T) = trace(tu), u X ˆ π Y, T F(Y,X).

7 THE SEPARABLE WEAK BOUNDED APPROXIMATION PROPERTY 75 This meas that if u = x y with x y <, the (Vu)(T) = x (T(y )). It is well kow that Vu u π,u X ˆ π Y. Applyig the trace mappig, Grothedieck [5] proved that a Baach space X has the λ-bap if ad oly if, for every Baach space Y, the trace mappig V : X ˆ π Y F(Y,X) satisfies u π λ Vu for all u X ˆ π Y. O the other had, Lima ad Oja [9] showed that a Baach space X has the AP if ad oly if, for every separable reflexive Baach space Y, the trace mappig V : X ˆ π Y F(Y,X) satisfies u π λ Vu for all u X ˆ π Y. The followig theorem provides some relatio betwee the s-weak λ-bap ad the trace mappig. Theorem 3.9. Suppose that for every separable Baach space Y, the trace mappig V : X ˆ π Y F(Y,X) satisfies u π λ Vu for all u X ˆ π Y. The X has the s-weak λ-bap. Proof. We are eough to verify coditio (b) of Corollary 3.8. Let Y be a Baach space ad T be i B S (Y,X) with T = 1. Take (x ) X, ad (y ) Y with x y <. We may assume that y = 1, N. LetK = T(B Y ). ByLemma3.6,thereexistsaseparableBaachspaceZ = X K so that J : Z X is the iclusio mappig ad K J(B Z ). Let z B Z be such that T(y ) = J(z ), N. Put u = x z X ˆ π Z. Cosider the trace mappig V : X ˆ π Z F(Z,X). Sice Z is separable, by the assumptio, we have u π λ Vu. By Lemma 3.5, we obtai that F(Z,X;1) {SJ : S F(X,X), SJ 1}. Note that ST SJ for all S F(X,X) because T(B Y ) J(B Z ). The we have x T(y ) = x J(z ) = trace(ju) J u π u π λ V(u) = λ λ λ sup SJ 1,S F(X,X) sup ST 1,S F(X,X) sup S F(Z,X;1) trace(su) trace(sju) x SJ(z )

8 76 KEUN YOUNG LEE = λ sup ST 1,S F(X,X) λ V(u). x ST(y ) Corollary Suppose that for every separable Baach space Y, the trace mappig V : X ˆ π Y F(Y,X) is isometric. The X has the s-weak MAP. Also we have the followig propositio. Propositio Let Z be a 1-complemeted subspace of X. If X has the s-weak λ-bap, the Z has the s-weak λ-bap. Proof. Let Y be a separable Baach space ad j Z be the iclusio mappig from Z ito X ad P be the orm 1-projectio from X ito Z ad T be i B(Y,Z). By the assumptio, there exists a et (T α ) F(X,X) such that τ sup α T α j Z T λ ad T α I X. Put S α = PT α j Z. The a et (S α ) is i F(Z,Z) ad we obtai S α T = PT α j Z T T α j Z T λ for all α. Fially we are eough to show that S α τ I Z. Let K be a compact subset of Z ad ε > 0. The there exists β such that sup y K T α j Z (y) j Z (y) < ε wheever α β. Let α β. Sice Pj Z (y) = y for all y Z, we obtai sup S α (y) y = sup PT α j Z (y) Pj Z (y) sup T α j Z (y) j Z (y) < ε. y K y K y K This meas that S α τ I Z. So we have hece Z has the s-weak λ-bap. I Z {S F(Z,Z) : ST λ} τ, Recall that X is a 1-complemeted subspace of X. Thus we have the followig corollary. Corollary If X has the s-weak λ-bap, the X has the s-weak λ- BAP. It is useful to exted the otio of the s-weak λ-bap with cojugate operators as follows. Defiitio A dual space X of a Baach space X has the s-weak λ- BAP with cojugate operators if for every separable Baach space Y ad T B S (Y,X ;1), I X {S : S F(X,X), S T λ} τ, that is, I X {S F w (X,X ) : ST λ} τ. If λ = 1, the we say that X has the separable weak metric approximatio property with cojugate operators.

9 THE SEPARABLE WEAK BOUNDED APPROXIMATION PROPERTY 77 Lemma 3.14 ([2]). For every Baach space X, Furthermore, for each λ > 0, F(X,X ) F w (X,X ) τ. F(X,X ;λ) τ F w (X,X ;λ) τ. Usig the above lemma, we obtai the followig relatio. Propositio We have λ-bap s-weak λ-bap with cojugate operators s-weak λ-bap AP i dual Baach space. Proof. For s-weak λ-bap with cojugate operators s-weak λ-bap, it is clear. We show that λ-bap s-weak λ-bap with cojugate operators. Suppose that X has the λ-bap. Let Y be a separable Baach space ad T B(Y,X ;1). By the assumptio ad Lemma 3.14, we have Sice T 1, we obtai I X F w (X,X ;λ) τ. I X {S F w (X,X ) : ST λ} τ. Also we provide the followig corollary which is similar to Corollary 3.8. Corollary Let X be a Baach space ad let 1 λ <. The followig are equivalet. (a) X has the s-weak λ-bap with cojugate operators. (b) For every Baach space Y ad T B S (Y,X ;1), T {ST : S F w (X,X ), ST λ} τ. (c) For every Baach space Y, for every operator T B S (Y,X ) with T = 1, ad for all sequece (x ) X, ad (y ) Y with x y <, oe has the iequality x T(y ) λ sup x ST(y ). ST 1,S F w (X,X ) Proof. (a) (b) Sice F w (X,X ) is a balaced covex subset of B(X,X ), by the proof of Theorem 3.7((c) (d)), it is clear. (b) (c) It comes from the proof of Corollary 3.8. Remark We do ot kow whether the s-weak λ-bap with cojugate operators, the s-weak λ-bap ad the AP are differet properties i dual Baach spaces. We cojecture that they are. However, if X has the Rado-Nikodým property, the these properties are same.

10 78 KEUN YOUNG LEE Let us cosider a Baach space Y ad take T B S (Y,X ). We defie F T (Y,X ) by F T (Y,X ) = {S T : S F(X,X)}. Clearly, F T (Y,X ) is a subspace of F(Y,X ). The we cosider the trace mappig V from X ˆ π Y to F T (Y,X ), defied by V(u)(S T) = trace(s Tu), u X ˆ π Y, S T F T (Y,X ). This meas that if u = x y with x y <, the (Vu)(S T) = x (S T(y )). We observe that Vu u for all u X ˆ π Y. Also we defie W T from X ˆ π Y to B(X,X) by W T (u)(r) = x (R T(y )), wheever u = x y with x y < ad R B(X,X). The followig theorem provides relatios betwee s-weak λ-bap with cojugate operators ad the trace mappig. Theorem Let X be a Baach space ad let 1 λ <. (a) For every Baach space Y ad T B S (Y,X ;1), the trace mappig V : X ˆ π Y F T (Y,X) satisfies u π λ Vu for all u X ˆ π Y. (b) X has the s-weak λ-bap with cojugate operators. (c) For every Baach space Y ad T B S (Y,X ;1), there exists a oe-tooe operator Φ : F T (Y,X) B(X,X) such that Φ λ, Φ(f)(R) = f(r T) for all f F T (Y,X) ad all R F(X,X), ad Φ V = W T, where V : X ˆ π Y F T (Y,X) is the trace mappig. The we have (a) (b) (c). Proof. (a) (b) The proof is similar to the proof of Theorem 3.9. We are eough to verify coditio (c) of Corollary Let Y be a Baach space ad T be i B S (Y,X ) with T = 1. Take (x ) X, ad (y ) Y with x y <. We may assume that y = 1, N. Let K = T(B Y ). As i the proof of Theorem 3.9, there exists a separable Baach space Z = X K so that J : Z X is the iclusio map ad K J(B Z ). Let z B Z be such that T(y ) = J(z ), N. Put u = x z X ˆ π Z.

11 THE SEPARABLE WEAK BOUNDED APPROXIMATION PROPERTY 79 Cosider the trace mappig V : X ˆ π Z F J (Z,X). The we have u π λ Vu. Note that F J (Z,X ;1) = {S J : S F(X,X), S J 1}. The we have x T(y ) = x J(z ) = trace(ju) J u π u π λ V(u) = λ sup S J F(Z,X ;1) trace(s Ju) λ sup x S J(z ) S T 1,S F(X,X) = λ sup S T 1,S F(X,X) x S T(y ). (b) (c) Let Y be a Baach space ad T be i B S (Y,X ;1). Let us take a et (S α ) F(X,X) such that Sα τ I X ad S T 1. Sice (SαT) B FT(Y,X ), whichisweak compact, afterpassigto asubet, wemayassume that lim α f(sα T) exists for all f F T(Y,X ) (see [6]). For each R B(X,X) ad f F T (Y,X ), we defie f R : F T (Y,X ) F by f R (S T) = f(r S T). It is well defied because SR F(X,X) for all S F(X,X). Also, it is easy to check that f R F T (Y,X ). The we ca defie Φ : F T (Y,X ) B(X,X) by (Φf)(R) = limf R (S α α T) = lim α f(r Sα T) for each R B(X,X) ad f F T (Y,X ) because lim α f(sαt) exists for every f F T (Y,X ). Sice Sα T λ for all α, we obtai Φ λ. Now we claim that Φ(f)(R) = f(r T) for all f F T (Y,X ) ad all R F(X,X). Let f be i F T (Y,X ), R i F(X,X) ad ε > 0. Sice R is a fiite rak operator, R is represeted by R = k=1 x k ( )x k where (x k ) X ad (x k ) X. We may assume that k=1 x k = 1. Sice Sα τ I X, there exists β such that if α β, the S α(x k) x k < ε for all k = 1,2,...,. Let us fix α β. The we have R SαT R T R Sα R = sup Sα (x k )x (x k ) x B X k=1 Sα(x k) x k x k < ε. k=1 x k x (x k ) k=1

12 80 KEUN YOUNG LEE This completes our claim. Furthermore, we observe that Φ is a oe-to-oe operator. Fially, we show that Φ V = W T, where V : X ˆ π Y F T (Y,X) is the trace mappig. Let u = x y X ˆ π Y ad R B(X,X). The we have Φ(Vu)(R) = lim α (x y)(r S αt) = lim α x (R S α T(y)) = x (R T(y)) = W T (u)(r). Remark We do ot kow whether (a), (b) ad (c) i Theorem 3.18 are equivalet. We cojecture that they are. 4. Aswers to Questio I this sectio, we give aswers to Questio as metioed i Itroductio. For this work, we eed the followig represetatio of the dual space of a compact operator space. Theorem 4.1 ([7, Theorem 3.5]). Let X ad Y be Baach spaces such that X or Y has the weak Rado-Nikodým property. The, for all φ K(X,Y), there exist a sequece (((x i ) ) m ) i X ad a sequece (((yi ) ) m ) i Y such that for all T K(X,Y), ad φ,t = lim (x i ) (T ((yi ) )) lim sup (x i ) (yi ) φ. Theorem 4.2 ([7, Corollary4.4]). Suppose that X or Y has the weak Rado- Nikodým property. The, for all φ K w w(x,y), there exist a sequece (((x i ) ) m ) i X ad a sequece (((yi ) ) m ) i Y ad φ,t = lim (yi ) (T((x i ) ) for all T K w w(x,y), ad lim sup (x i ) (yi ) φ. The followig theorem provides oe sufficiet coditio which guaratees the metric approximatio property for a dual Baach space as metioed. We modify the proof i [12, p. 326]. Theorem 4.3. Let X be a Baach space ad let 1 λ <. Suppose that X has the weak Rado-Nikodým property. If X has the s-weak λ-bap, the X has the λ-bap.

13 THE SEPARABLE WEAK BOUNDED APPROXIMATION PROPERTY 81 Proof. Let I X be the idetity operator ad K be a compact subset of X ad let ε > 0. Cosider the semiorm ad put p(a) = sup{ Ax : x K}, A B(X,X ) C = {S : S F(X,X ), p(s I X ) ε/2}. Sice X has the AP, C is a oempty covex subset of F(X,X ). Let us fix δ > 0 such that δ λ+δ (ε/2+p(i X)) ε/2. The we are eough to show that C (λ+δ)b F(X,X ). Suppose to the cotrary that C (1+δ)B F(X,X ) =. By the separatio theorem, there exists φ F(X,X ) such that φ = 1 ad λ+δ = sup{reφ(a) : A λ+δ} if{reφ(a) : A C}. By Theorem 5.1 ad the Hah-Baach theorem, there exist a sequece (((x i ) ) m ) i X ad a sequece (((x i ) ) m ) i X such that for all T F(X,X ) ad φ,t = lim lim sup (x i ) (T((x i ) )) (x i ) (x i ) φ. Deote by C the closed absolutely covex hull i X of the separable set {(( (x i ) (x i ) )m ) }. By the factorizatio lemma [8], there exists a Baach space Z, which is a liear subspace of X, such that the idetity embeddig J : Z X is a separable valued operator ad J 1. Also we have {(( (x i ) (x i ) )m ) } J(B Z ). Sice X has the s-weak λ-bap, there exists S F(X,X ) such that SJ λ ad p(s I X ) ε/2. Sice S C, we obtai λ+δ φ(s). Put ((zi )m ) B Z such that (x i ) = (x i ) J(zi ) for all i,. The we have φ(s) = lim (x i ) (S(x i ) )

14 82 KEUN YOUNG LEE which is a cotradictio. = lim SJ limsup φ < λ+δ, (x i ) (x i ) (SJ(zi )) (x i ) (x i ) We also give aother sufficiet coditio which guaratees the metric approximatio property for a dual Baach space. The proof is similar to Theorem 4.3. We just replace F(X,X ) by F w (X,X ) ad apply Theorem 4.2. Theorem 4.4. Let X be a Baach space ad let 1 λ <. Suppose that X has the weak Rado-Nikodým property. If X has the s-weak λ-bap with cojugate, the X has the λ-bap. Remark 4.5. Recall Theorem 1 i Itroductio, that is, if X has the AP ad the Rado-Nikodým property, the X has the MAP. Hece, to verify that our sufficiet coditios are o-trivial, we eed to check whether our coditios implies coditios of Theorem 1. Fortuately, we ca provide the egative aswer to the above questio. Ideed, we cosider the dual of James tree space JT. As we kow, JT has the s-weak MAP with cojugate operators. O the other had, JT ad JT have both the weak Rado-Nikodým property but JT does ot have the Rado-Nikodým property. Hece, our sufficiet coditios are o-trivial. We fially preset the followig iterestig problems. Problem 1. If X has the s-weak MAP, the does X has s-weak MAP with cojugate operators? Problem 2. If X has the AP ad weak Rado-Nikodým property, the does X has s-weak MAP? If Problem 1 had a affirmative aswer, the we would improve Theorem 4.4. I additio, if Problem 1 ad Problem 2 had a affirmative aswer, the we would improve Theorem 1 i Itroductio. Refereces [1] P. G. Cassaza, Approximate properties, I: W. B. Johso, J. Lidestrauss(eds.), Hadbook of the Geometry of Baach Spaces, Volume 1, , Elsevier, [2] C. Choi ad J. M. Kim, Weak ad quasi approximatio properties i Baach spaces, J. Math. Aal. Appl. 316 (2006), o. 2, [3], Hah-Baach theorem for the compact covergece topology ad applicatios to approximatio properties, Housto J. Math. 37 (2011), o. 4, [4] T. Figiel ad W. B. Johso, The approximatio property does ot imply the bouded approximatio property, Proc. Amer. Math. Soc. 41 (1973),

15 THE SEPARABLE WEAK BOUNDED APPROXIMATION PROPERTY 83 [5] A. Grothedieck, Produits tesoriels topologiques et espaces ucléires, Mem. Amer. Math. Soc. 16 (1955), o. 16, 140 pp. [6] J. Johso, Remarks o Baach spaces of compact operators, J. Fuct. Aal. 32 (1979), o. 3, [7] K. Y. Lee, Dual spaces of compact operators space ad the weak Rado-Nikodým property, Studia Math. 210 (2012), o. 3, [8] Å. Lima, O. Nygraard, ad E. Oja, Isometric factorizatio of weakly compact operators ad the approximatio property, Israel J. Math. 119 (2000), [9] Å. Lima ad E. Oja, The weak metric approximatio property, Math. A. 333 (2005), o. 3, [10] J. Lidestrauss ad C. Stegall, Examples of separable spaces which do ot cotai l 1 ad whose duals are o-separable, Studia Math. 54 (1975), o. 1, [11] K. Musial, The weak Rado-Nikodým property i Baach spaces, Studia Math. 64 (1978), o. 2, [12] E. Oja, The impact of the Rado-Nikodým property o the weak bouded approximatio property, Rev. R. Acad. Ciec. Exactas Fis. Nat. Ser. A Math. 100 (2006), [13] R. A. Rya, Itroductio to Tesor Product of Baach Spaces, Spriger, Lodo, Istitute for Ubiquitous Iformatio Techology ad Applicatios Kokuk Uiversity Seoul , Korea address: orthstar@kaist.ac.kr