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A new characterization of the bounded approximation property. (English) Zbl 1360.46013

Denote by \(\mathcal F(Y;X)\) the space of all finite rank operators from \(Y\) to \(X\). Recall that a Banach space \(X\) has the bounded approximation property (BAP) if, for every Banach space \(Y\) and every operator \(T\) from \(Y\) to \(X\), there exists \(\lambda_T>0\) such that \[ T \in \overline{\{S \in \mathcal F(Y;X) : \|S\|\leq \lambda_T\}}^{\tau_c}, \] where \(\tau_c\) denotes the topology of uniform convergence on compact sets.
The authors give the following new characterization of the BAP:
Theorem 1.1. A Banach space \(X\) has the BAP if and only if, for every separable Banach space \(Z\) and every injective operator \(J\) from \(Z\) to \(X\), there exists \(\lambda_J>0\) such that \[ J \in \overline{\{S \in \mathcal F(Z;X) : \|S\|\leq \lambda_J\}}^{\tau_c}. \]
This result should be compared with another characterization of the BAP in [E. Oja, Arch. Math. 107, No. 2, 185–189 (2016; Zbl 1355.46021)].

MSC:

46B28 Spaces of operators; tensor products; approximation properties

Citations:

Zbl 1355.46021