Ideals of operators and the metric approximation property
A Banach space \(X\) is said to have the metric approximation property (M.A.P. for short) if its identity operator can be uniformly approximated on compact subsets of \(X\) by finite-rank operators whose norms are \(\leq 1\). According to \textit{J. Johnson} [J. Funct. Anal. 32, 304--311 (1979; Zbl 0412.47024)], if \(X\) has the M.A.P., then for all Banach spaces \(Y\), the finite-rank operators \(F(Y,X)\) form an ideal in the Banach space \(L(Y,X)\) of bounded linear operators from \(Y\) to \(X\). This means that there exists a norm one projection on \((L(Y,X))^\ast\) whose kernel equals the annihilator of \(F(Y,X)\). According to \textit{Å. Lima} [Isr. J. Math. 84, No. 3, 451--475 (1993; Zbl 0814.46016)], if \(X\) has the Radon-Nikodým property and \(F(X,X)\) is an ideal in \(L(X,X)\), then \(X\) has the M.A.P. It is not known whether this result holds for arbitrary Banach spaces \(X\). The main result of this paper asserts that a Banach space \(X\) has the M.A.P. whenever \(F(\widehat{X},X)\) is an ideal in \(L(\widehat{X},X)\) for all equivalent renormings \(\widehat{X}\) of \(X\). Interchanging the roles of \(\widehat{X}\) and \(X\) yields the M.A.P. (with conjugate operators) for \(X^\ast\). Similar results hold for the metric compact approximation property. The proofs rely on results from [\textit{Å. Lima, O. Nygaard} and \textit{E. Oja}, Isr. J. Math. 119, 325--348 (2000; Zbl 0983.46024)] and [\textit{Å. Lima} and \textit{E. Oja}, J. Aust. Math. Soc. 77, No. 1, 91--110 (2004; Zbl 1082.46016)]. They develop further the method of [\textit{Å. Lima} and \textit{E. Oja}, Proc. Am. Math. Soc. 130, No. 12, 3631--3640 (2002; Zbl 1006.46004)] of constructing ``good'' Hahn-Banach extension operators. This can be done thanks to the existence of equivalent locally uniformly rotund renormings that are arbitrarily close to the initial norm (see [\textit{R. Deville, G. Godefroy} and \textit{V. Zizler}, Smoothness and renormings in Banach spaces (Pitman Monographs and Surveys in Pure and Applied Mathematics 64, John Wiley \& Sons, New York) (1993; Zbl 0782.46019)]). Relying also on results from [\textit{G. Godefroy, N. J. Kalton} and \textit{P. D. Saphar}, Stud. Math. 104, No. 1, 13--59 (1993; Zbl 0814.46012)], similar characterizations are obtained for the unconditional metric (compact) approximation property in terms of \(u\)-ideals.
scientific article; zbMATH DE number 2091742