#### Date of Submission

2-28-1981

#### Date of Award

2-28-1982

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Computer Science

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

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], LÃ¢ndenstrauss 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.

#### Control Number

ISILib-TH59

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Rao, T. S.S.R.K. Dr., "The Alfsen-Errors Structure Topology in the Theory of Complex L^{1}-Preduals." (1982). *Doctoral Theses*. 374.

https://digitalcommons.isical.ac.in/doctoral-theses/374

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843727