Contractive projections in square Banach spaces

Roy, Nina M.

doi:10.1090/S0002-9939-1976-0428020-8

Contractive projections in square Banach spaces
HTML articles powered by AMS MathViewer

by Nina M. Roy

Proc. Amer. Math. Soc. 59 (1976), 291-296

DOI: https://doi.org/10.1090/S0002-9939-1976-0428020-8

PDF | Request permission

Abstract:

It is proved that if $X$ is a square space and $P$ is a contractive projection in $X$, then $PX$ is square; and if $X$ is regular, then $PX$ is regular. It is also shown that a regular square space is isometric to the image, under a contractive projection, of a regular (square) Kakutani $M$-space. These results are analogous to those obtained for other classes of ${L_1}$-preduals by Lindenstrauss and Wulbert, and in this paper their diagram of ${L_1}$-preduals is enlarged so as to include the classes of square, regular square and regular $M$spaces.

References

Erik M. Alfsen and Edward G. Effros, Structure in real Banach spaces. I, II, Ann. of Math. (2) 96 (1972), 98–128; ibid. (2) 96 (1972), 129–173. MR 352946, DOI 10.2307/1970895
F. Cunningham Jr., $M$-structure in Banach spaces, Proc. Cambridge Philos. Soc. 63 (1967), 613–629. MR 212544, DOI 10.1017/s0305004100041591
F. Cunningham Jr., Square Banach spaces, Proc. Cambridge Philos. Soc. 66 (1969), 553–558. MR 248503, DOI 10.1017/s0305004100045321
Edward G. Effros, On a class of real Banach spaces, Israel J. Math. 9 (1971), 430–458. MR 296658, DOI 10.1007/BF02771459
Edward G. Effros, Structure in simplexes. II, J. Functional Analysis 1 (1967), 379–391. MR 0220041, DOI 10.1016/0022-1236(67)90008-0
A. J. Lazar and J. Lindenstrauss, Banach spaces whose duals are $L_{1}$ spaces and their representing matrices, Acta Math. 126 (1971), 165–193. MR 291771, DOI 10.1007/BF02392030
Joram Lindenstrauss and Daniel E. Wulbert, On the classification of the Banach spaces whose duals are $L_{1}$ spaces, J. Functional Analysis 4 (1969), 332–349. MR 0250033, DOI 10.1016/0022-1236(69)90003-2
Nina M. Roy, A characterization of square Banach spaces, Israel J. Math. 17 (1974), 142–148. MR 344851, DOI 10.1007/BF02882233