Banach spaces without approximation properties of type \(p\) (Q650316)

Let \(X\) be a Banach space and let \(1\leq p\leq \infty\). The approximation property (AP) of type \(p\) was recently introduced and studied in [\textit{D. P. Sinha, A. K. Karn}, Math. Nachr. 281, No. 3, 412-423 (2008; Zbl 1144.47017)]. This property can be equivalently described as follows: \(X\) has the AP of type \(p\) if and only if, for all Banach spaces \(Y\), the finite-rank operators \({\mathcal{F}}(Y,X)\) are dense in the Banach space \(QN_p(Y,X)\) of quasi-\(p\)-nuclear operators. The paper [op. cit.] ends with the question whether, for \(p\in[1,2)\), there is a Banach space which fails the AP of type \(p\). The authors answer this question by showing that for every \(p\not= 2\) there is a (reflexive) Banach space that fails to have the AP of type \(p\). They obtain the result relying on the papers [\textit{O. I. Reinov}, ``Approximation properties of order \(p\) and the existence of non-\(p\)-nuclear operators with \(p\)-nuclear second adjoints'', Math. Nachr. 109, 125--134 (1982; Zbl 1231.47018)] and [``Approximation of operators in Banach spaces'', in: Application of Functional Analysis in Approximation Theory (KGU, Kalinin, 1985), 128--142 (1985)]. So, as the authors mention, ``the answer could be obtained, essentially, in 1985''. { Reviewer's remark: In [\textit{J. M. Delgado}, \textit{C. Piñeiro}, \textit{E. Serrano} and the reviewer, J. Math. Anal. Appl. 354, 159--164 (2009; Zbl 1168.46008)], the same answer was obtained relying on Reinov's paper [loc. cit.] from 1982.} For obtaining the result, the authors revisit Reinov's paper [loc. cit.] from 1985. In this paper, the topology \(\tau_p\) of \(\pi_p\)-compact convergence on the space \(\Pi_p(X,Y)\) of absolutely \(p\)-summing operators was introduced and studied. The topology \(\tau_p\) is just the same as the \(\lambda_p\)-topology ``introduced'' in the Sinha--Karn paper [loc. cit.]. The authors prove that \(X\) has Saphar's AP of order \(p'\) (where \(p'\) is the conjugate index of \(p\), i.e., \(1/p+1/p'=1\)) if and only if \({\mathcal{F}}(X,Y)\) is \(\tau_p\)-dense in \(\Pi_p(X,Y)\) for all Banach spaces \(Y\) (or just for all reflexive Banach spaces \(Y\)).