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Local properties of accessible injective operator ideals. (English) Zbl 0926.46054
In contrast to Pisier’s example of a non-accessible maximal Banach ideal, the author exhibits a large class of maximal Banach ideals which are accessible. The construction is based on a characterization of total accessibility of injective hulls of maximal Banach ideals (if \((\mathcal A,\mathbf A)\) is a maximal Banach ideal, then \((\mathcal A^{inj},\mathbf A^{inj})\) is totally accessible if and only if \((\mathcal A\circ \mathcal A^*,\mathbf A\circ \mathbf A^\ast)\subseteq (\mathcal P_1,\mathbf P_1)\), the ideal of absolutely summing operators) and an operator version of Grothendieck’s inequality. As a consequence, it is obtained, for instance, that if \((\mathcal A,\mathbf A)\) is a maximal Banach ideal such that \((\mathcal D_2,\mathbf D_2)\subseteq (\mathcal A,\mathbf A)\subseteq (\mathcal L_1,\mathbf L_1)\) and both \((\mathcal A,\mathbf A)\) and \((\mathcal A^*, \mathbf A^\ast)\) are metrically \(\varepsilon \)-tensorstable, then \((\mathcal A^{inj}, \mathbf A^{inj})\) is totally accessible. Several other criteria for right- and total accessibility are established, and applications to normed products of operator ideals are also given.

MSC:
46M05 Tensor products in functional analysis
47L20 Operator ideals
47A80 Tensor products of linear operators
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References:
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