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Operator ideals and the principle of local reflexivity. (English) Zbl 0803.47038
Our aim is, to give necessary and sufficient conditions which allow us to transform the local reflexivity principle of Lindenstrauss and Rosenthal from the canonical operator norm \(\|\cdot\|\) to \(p\)-Banach ideal norms \(\|\cdot\|_{\mathcal A}\), where \(({\mathcal A},\|\cdot\|_{\mathcal A})\) is a given \(p\)-Banach ideal \((0< p\leq 1)\).
We will recognize two important facts:
– By a natural generalization of the weak \({\mathcal A}\)-local reflexivity principle, we can omit the previously assumed maximality of the \(p\)- Banach ideal \(({\mathcal A},\|\cdot\|_{\mathcal A})\). Moreover, we are allowed to consider all \(0< p\leq 1\) and not only the case \(p=1\).
– There are interesting relations between the above-mentioned generalization of weak local reflexivity and structural properties of the ideal \(({\mathcal A},\|\cdot\|_{\mathcal A})\) such as accessibility. Hence, tensor norms are involved.

MSC:
47L20 Operator ideals
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