D029019023

Research Inventy: International Journal Of Engineering And Science Vol.2, Issue 9 (April 2013), Pp 19-23 Issn(e): 2278-4721, Issn(p):2319-6483, Www.Researchinventy.Com

ON SOME ASPECTS OF u -IDEALS DETERMINED BY AND c

1

0

John Wanyonyi Matuya1 , Achiles Simiyu2 and Shem aywa2 1

Department of Mathematics, Maasai Mara University, P.O Box 861-20500, Narok , Kenya 2 Department of Mathematics, Masinde Muliro University of science and Technology, P.O BOX 190-50100, Kakamega, Kenya.

Abstract: The M -ideals defined on a real Banach space are called u -ideals. The u -ideals containing isomorphic copies of  1 are not strict u -ideals. In this paper we show that u -ideals with unconditional basis

 x n  which is shrinking has no isomorphic copy of

u -ideals with unconditional basis

 1 and thus a strict

u -ideal. Finally we show that

 x n  which is boundedly complete is not homeomorphic to copies of



implying that they are weak closed in their biduals X



c0

.

Key Words: u -ideals, strict u -ideals, Shrinking, boundedly complete 2010 Mathematics Subject Classification: 47B10; 46B10; 46A25 This work was supported by National Council of Science and Technology (NCST), Kenya.

I.

INTRODUCTION



A sequence  x n  i  1 is called a basis of a normed space X if for every x  X there exists a unique series



a i x i that converges to x . The basis



 x n  i  1 for a Banach space

X is an unconditional basis if for

i 1 

each

x X

x 

there exists a unique expansion of the form



an xn

where the sum converges

n 1

unconditionally. The basis for which



n k 1

 x  is said to be boundedly complete whenever given a sequence  a  of scalars n

n



ak xk : n  1

is bounded, then lim n



n k 1

a k x K exists. If x n  X is not boundedly

complete then X contains an isomorphic copy of c 0 [1].

Let y  X



n  u ,   i 1 

which belongs to X . We say that  x n  is shrinking, y , u i . xi

     n 

n

y ,  u , xi .u i i 1

  converges to n

y,u



for every u  X .

The notion of u -ideals was introduced and studied thoroughly by Godfrey et al [2]. They generalized M ideals defined on a real Banach space. In their paper on unconditional Ideals they established that u -ideals containing copies of  1 are not strict u -ideals. A Banach space X is said to be a strict u -ideal in its bidual when the canonical decomposition X

u -ideal the u -complement of X





 X



 X



is unconditional. In other words for X to be a strict

must be norming, that is, the range V of the induced projection on X



is a norming subspace of X . Vegard and Asvald  3  characterized Banach spaces which are strict u 

ideals in their biduals and showed that X is a strict u -ideal in a Banach space Y if it contains c 0 . Matuya et al [4] using the approximation properties, hermitian conditions, isometry studied properties of u -ideals and

19

On Some Aspects Of u -Ideals Determined…. their characterization. They showed that u -ideals containing no copies of sequence space  1 are strict u -

 xn 

ideals. In this paper we show that Banach spaces with unconditional basis

that is shrinking is not

homeomorphic to copies of  1 and so it a strict u -ideal. We also show that the u -ideals with unconditional 

basis which are boundedly complete are not bicontinuous to c 0 meaning their u -complement is weak closed.

II. RESULTS ON STRICT u -IDEALS 2.1 Proposition Let X be u -ideal with unconditional basis  x n  . The following statements are equivalent: [1] The sequence  x n  is shrinking [2] X contains no copy of  1 [3] X is a strict u -ideal

 x n  is shrinking then

i  ii it is clear from the definition that if a sequence

Proof:

contains no

X

isomorphic copy of  1 . In fact proving by contradiction it suffices to show that X contains a copy of  1 . Let

 c n  be a sequence of coefficient functional associated with  x n  . Our hypothesis applies that, for some 



c X

,

the



series

does

c, xn .cn

not

converge

X 

in

n 1

     c , x n . c n  cannot be Cauchy in  X  n 1 k k



X

,





,

X



, X  

implying that

, X   . We therefore find a bounded set L   1 and  

for

k

each

we

define

maps

P ,Q : L



 X

P 

by







and

c, xn .cn

n 1

Q 



k



n 1

 c, xn .cn  k

  

L Q



L  P

2 

which  

 holds.

homeomorphically into X . Now L for the subspace P   1  of

implies



together

L  P  Q 

with

P is continuous since

 x n  is bounded, so that

 c n  has a weak limit in

X . In particular

P maps L



and so does  x n  because of

X

n 



where U  p  x



.

Now for each   X

there is a net  x d   B X 

P

that

is dense in  1 . Since  1 is weakly sequentially complete the same is true

ii  iii Considering the map P it is clear from the definition that if x  X



2 

 lim x n , c k ,  k   , this limit has to be x and this is the desired contradiction.

x, ck

xd



converges

to



Thus

u 

 x   u x   so we conclude that 



iii  i Suppose X



converging in the weak -topology of X

x .



and P x  x

we consider the set F   x  X | P   x











lim  d

x   x  



d









then x  U

x  



to u . However, u  x  X so

that

u x



   

x  . 

so that Now



x  F . Hence F  is norming.

is strict u -ideal. We proceed to show that  x n  is shrinking. Let y  X

belongs to X . Since  x n  is shrinking,



then

  

n

u, i 1

20

y , u i . xi

     n 

n

y,  i 1

u , xi .u i

  n



which

On Some Aspects Of u -Ideals Determined…. converges to



n



i 1



for every u  X . We conclude from this that   y , u i . x i  is bounded in X so it 

y,u



has a limit say 

x .



x,u





y , u i . xi , u



y , u , for all u  X

,

i 1

we get y  x  X .

2.2 Proposition Let X be u -ideal with unconditional basis  x n  . Show that if X contains no copy of c 0 then [1]

 x n  is boundedly complete

[2]

 x  is weak 



n

closed in X



 Suppose  x n  is not boundedly complete. Then there exists   i   T with   Cauchy in X . Let now e  

m i  n i  m i 1

and

ai 



and let

 mi 

 jx j  ue

and ,

 ni 

that

is,

n





 i x i  bounded but not

i 1

n

be increasing sequences in  such that

e  ai   1,  i   .

Here

we

set

mi

M i   m i , m i  k , ....., n i  . Choose u i  U e such that 0

e  e M , x   f

such that

By construction of the sets M

   g   ai   g    i    i 

v,  ai

Whence

 x  X , M   and    .

, it follows that

i



 j M

 x,

 1 ,  i   and let f   be

ui , ai

i xi   e g , 

i

 eg  g   , v  U

0 g

.

i 

There exists a constant c  0 such that



v, ai

 g   , vU

 c. e g

0 g

.

i 

Consider now T as a subspace of c 0 and define

 : T  X by   

    n an . n

Then 

g  

 lu b U



0 g

is linear and continuous, because of



  g 

 n

v,   n an

n

 an  

 c.e g . 



,  g   , T .

n

is also open , because given   T

and j  

such that

21

such that 

j

 



, then we can choose   

On Some Aspects Of u -Ideals Determined….  i i u j , a i   i u j , a i



Therefore

 



j



for all i   .



n

u j, an

n

  e 

v j,  n n an



n





n

 n an  

n

   .

 f

Thus  is bicontinuous between T and X . This implies that T is dense in c 0 and since X is sequencially complete,  extends a linear homeomorphic embedding c 0 into X which is a contradiction.

We shall show that when 







xn

n 1

x  

X

n



x   

and

n



X

satisfy

 

xn

and 



n 1



s



x   0.

xn





xn

0

for s  F

X,X ,

then 



n 1

xn



n

We may assume that





x n  0 and M 



x

n 1



  .

Let   0 and choose N   such that



x





for  



 K

Xˆ

. Since X has the weak approximating sequence,

4

nN L



 Xˆ , Z  , there exists a net  s 





F

 Xˆ , Xˆ  such that lu b



 s    1

and

 L s x k    L x k    Lk  x  . 









Therefore s  x  x



weak



for all x  L  k  . In particular s  x  x 



 s   1 , we also have s  L  1 , if 





x





is weak convergent. Hence there is some s  such that





x n  s x n 

, n  1, ...., N

2M

Now we have 







xn



n 1





x









x n  s x n 



xn

n

 s x n

n 1

N







xn







 2





L k n  s L k n

nN

n 1





xn





xn





L  s L









for all x  F . Since 

 1 then x  L k for some k  K . Thus s  x





, so s   x



kn

nN

22



1

On Some Aspects Of u -Ideals Determined…. 





2 Hence



 n 1



 

2 

xn

x   0 

n

III.

CONCLUSION

We have shown that u -ideals can be characterized using sequence spaces. In particular we considered the sequence spaces  1 and c 0 . The u -ideals containing no isomorphic copies of  1 are strict u -ideals 

whereas those that are not homeomorphic to the copies of c 0 their u -complement is weak -closed

IV.

ACKNOWLEDGEMENT

The authors convey their appreciation to the National Council of Science and Technology (NCST), Kenya for funding this research work and Masinde Muliro University of science and Technology for their technical support.

REFERENCES [1] [2] [3] [4]

Diestel. J. “ Sequences and Series in Banach spaces’’. Springer-Verlag Newyork , 1984.

Godefrey, G. Kalton, J. and Saphar,P. “ Unconditional ideals in Banach spaces ’’ .studia Mathematica.104 No.1,(1993) 13-59.

Lima, V. Lima, A. “ Strict u -ideals in Banach spaces ’’.studia Math.3 No.195 (2009) 275-285.

Matuya . J. W. et al “On characterization of u -ideals determined by sequences’’ A.J. Curr.Eng and Math.1( 2012) 247-250.

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