Abstract
We introduce a geometrical property of norm one complemented subspaces ofC(K) spaces which is useful for computing lower bounds on the norms of projections onto subspaces ofC(K) spaces. Loosely speaking, in the dual of such a space ifx* is a w* limit of a net (x * a ) andx*=x*1+x*2 with ‖x*‖=‖x*1‖ + ‖x*2‖, then we measure how efficiently thex * a 's can be split into two nets converging tox*1 andx*2, respectively. As applications of this idea we prove that if for everyε>0,X is a norm (1+ε) complemented subspace of aC(K) space, then it is norm one complemented in someC(K) space, and we give a simpler proof that a slight modification of anl 1-predual constructed by Benyamini and Lindenstrauss is not complemented in anyC(K) space.
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Research partially supported by a grant of the U.S.-Israel Binational Science Foundation.
Research of the first-named author is supported in part by NSF grant DMS-8602395.
Research of the second-named author was partially supported by the Fund for the Promotion of Research at the Technion, and by the Technion VPR-New York Metropolitan Research Fund.
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Alspach, D.E., Benyamini, Y. A geometrical property ofC(K) spaces. Israel J. Math. 64, 179–194 (1988). https://doi.org/10.1007/BF02787222
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DOI: https://doi.org/10.1007/BF02787222