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Weak sequential completeness and related properties of some operator spaces. (Russian) Zbl 0565.47029
Three problems are investigated for Banach ideals of p-nuclear and p- quasi-nuclear operators. [Y. Gordon, D. R. Lewis and J. R. Retherford, J. Funct. Anal. 14, 85-129 (1973; Zbl 0272.47024).] The first is weak completeness for $$1<p<\infty$$. The authors improve previous results by removing the assumption of an unconditional basis on one of the spaces $$X^*$$, Y. In the case of p-quasi-nuclear operators, $$QN_ p(X,Y)$$, they also remove the assumption of an approximation property.
The main result is: Let $$X^*$$, Y be weakly complete, $$1<p<\infty$$, and $$QN_ p(X,Y)=\Pi_ p(X,Y)$$ (the ideal of p-absolutely summing operators). Then the space $$QN_ p(X,Y)$$ is weakly complete. They also observe that if Y contains an unconditional basis, the assumption $$QN_ p=\Pi_ p$$ is necessary. Similar results are obtained for the p-nuclear operators with the assumptions that $$N_ p(X,Y)=I_ p(X,Y)$$ (the ideal of p-integral operators) and that $$Y^*$$ has the approximation property.
Secondly, they present sufficient conditions under which $$N_ p(X,Y)$$ does not contain $$\ell_ 1$$. For $$1<p<\infty$$ they show that this holds if $$X^*$$ does not contain $$\ell_ 1$$, $$Y^*$$ has the Radon-Nikrondym property, and one of $$X^{**}$$, $$Y^{**}$$ has the approximation property. For $$p=1$$ they require Y reflexive and X not containing $$\ell_ 1$$. They also obtain corollaries about p-tensor products as defined by (INVALID INPUT)P. Saphar [Stud. Math. 38, 71-100 (1970; Zbl 0213.142)]. Lastly, they obtain sufficient conditions under which these ideals do not contain $$c_ 0$$. For example, with $$1\leq p<\infty$$, if $$X^*$$, Y do not contain $$c_ 0$$ and $$QN_ p(X,Y)=\Pi_ p(X,Y)$$, then $$QN_ p(X,Y)$$ does not contain $$c_ 0$$.
Reviewer: A.J.Klein
##### MSC:
 47L10 Algebras of operators on Banach spaces and other topological linear spaces 47L05 Linear spaces of operators 46B20 Geometry and structure of normed linear spaces