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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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