Complemented copies of 𝑐₀ in 𝐿^{∞}(𝜇,𝐸)

Díaz, Santiago

doi:10.1090/S0002-9939-1994-1189744-1

Complemented copies of $c_ 0$ in $L^ \infty (\mu ,E)$
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by Santiago Díaz PDF

Proc. Amer. Math. Soc. 120 (1994), 1167-1172 Request permission

Abstract:

Let $E$ be a Banach space, and let $(\Omega ,\Sigma ,\mu )$ be a measure space. We denote by ${L^\infty }(\mu ,E)$ the Banach space of all $E$-valued $\mu$-measurable essentially bounded functions on $\Omega$, two functions being identified if they differ only on a locally $\mu$-null set. We prove that if ${L^\infty }(\mu ,E)$ contains a complemented copy of ${c_0}$, then $E$ contains a copy of ${c_0}$.

References

J. Bourgain, An averaging result for $c_{0}$-sequences, Bull. Soc. Math. Belg. 30 (1978), no. 1, 83–87. MR 549653
S. Kwapień, On Banach spaces containing $c_{0}$, Studia Math. 52 (1974), 187–188. MR 356156
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
G. Emmanuele, On complemented copies of $c_0$ in $L^p_X,\;1\leq p<\infty$, Proc. Amer. Math. Soc. 104 (1988), no. 3, 785–786. MR 930250, DOI 10.1090/S0002-9939-1988-0930250-1
Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121
Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257, DOI 10.1007/978-3-322-90559-8
Denny Leung and Frank Räbiger, Complemented copies of $c_0$ in $l^\infty$ sums of Banach spaces, Illinois J. Math. 34 (1990), no. 1, 52–58. MR 1031883
José Mendoza, Copies of $l_\infty$ in $L^p(\mu ;X)$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 125–127. MR 1012935, DOI 10.1090/S0002-9939-1990-1012935-3
Robert Whitley, Mathematical Notes: Projecting $m$ onto $c_0$, Amer. Math. Monthly 73 (1966), no. 3, 285–286. MR 1533692, DOI 10.2307/2315346