Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Roy, Ashoke Kumar (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

A complex Banach space X is said to be an L1-predual if X* is isometric to L1() for some non-negative measure Well known examples of L1-preduals include the space C(X) of complex-valued continuous functions on a compact Hausdorff space and the abstract M-spaces of Kakutani. In [19], Grothendieck introduced a class of L'-preduals, now known as G-spaces, and conjectured that those are all the L'-preduals. In his 1964 memoir [35], Lândenstrauss settled this conjecture by exhibiting a wide class of Banach spaces, other than G-spaces, which are L1-preduals. He also gave several characterizations and interest- ing properties of L1-preduals in terme of intersection properties of balls and extensions of operators. Since that time, the theory of L1;-preduals has attracted wide attention. L1preduals are now sometimes called Lindenstraucs spaces.Let P:X -> X be a linear projection. We call an L-projection if I|x || = ||Px || + ||x - Px || for all x e X. The range of an I-projection is called an L-ideal, As a conse quence of the results of Alfsen-Effros [ and Hirsberg [22], one knows that a norm closed suospace J CA(K) (where A(K), the space of continuous complex-valued affine functions on a compact covex set K is equipped with the supremum norm) is an L-ideal iff J is the linear span of a split face of the image of K in A(K)*under the evaluation map. Through the combined efforts of Lindenstrauss 35), Semadeni [45), Hirsberg and Lazar [21],it is known that a Iindenstrauss space whose unit ball has an extreme point, can be realised isometrically as the space A(E for some compact Choquet simplex K. Now, Ellis [140 proved that a compict convex set is a Choquet simplex iff every closed face of it is split. In view of the one-to-one correspondence bet ween L-ideals and split faces mentioned at the beginning of this paragraph, all these results suggest the possibility of characterising general L1-preduals in terms of L-idealo in their dual opaces. Several such characterizations are obtained in the first three sections of the present thesis.


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