A characterization of subspaces 𝑋 of 𝑙_{𝑝} for which 𝐾(𝑋) is an 𝑀-ideal in 𝐿(𝑋)

Cho, Chong-Man; Johnson, William B.

doi:10.1090/S0002-9939-1985-0774004-5

A characterization of subspaces $X$ of $l_ p$ for which $K(X)$ is an $M$-ideal in $L(X)$
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by Chong-Man Cho and William B. Johnson PDF

Proc. Amer. Math. Soc. 93 (1985), 466-470 Request permission

Abstract:

Given a subspace $X$ of ${l_p}$, $1 < p < \infty$, the compact operators on $X$ are an $M$-ideal in the bounded linear operators on $X$ if and only if $X$ has the compact approximation property.

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
Ehrhard Behrends, $M$-structure and the Banach-Stone theorem, Lecture Notes in Mathematics, vol. 736, Springer, Berlin, 1979. MR 547509
Moshe Feder, On subspaces of spaces with an unconditional basis and spaces of operators, Illinois J. Math. 24 (1980), no. 2, 196–205. MR 575060
Patrick Flinn, A characterization of $M$-ideals in $B(l_{p})$ for $1<p<\infty$, Pacific J. Math. 98 (1982), no. 1, 73–80. MR 644939
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
W. B. Johnson, On quotients of $L_{p}$ which are quotients of $l_{p}$, Compositio Math. 34 (1977), no. 1, 69–89. MR 454595
Peter Harmand and Åsvald Lima, Banach spaces which are $M$-ideals in their biduals, Trans. Amer. Math. Soc. 283 (1984), no. 1, 253–264. MR 735420, DOI 10.1090/S0002-9947-1984-0735420-4
Åsvald Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1–62. MR 430747, DOI 10.1090/S0002-9947-1977-0430747-4
Åsvald Lima, $M$-ideals of compact operators in classical Banach spaces, Math. Scand. 44 (1979), no. 1, 207–217. MR 544588, DOI 10.7146/math.scand.a-11804
Joram Lindenstrauss, On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 967–970. MR 205040, DOI 10.1090/S0002-9904-1966-11606-3
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
R. R. Smith and J. D. Ward, $M$-ideal structure in Banach algebras, J. Functional Analysis 27 (1978), no. 3, 337–349. MR 0467316, DOI 10.1016/0022-1236(78)90012-5