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A remark about twisting Schatten classes. (English) Zbl 1328.46015

N. J. Kalton [Trans. Am. Math. Soc. 333, No. 2, 479–529 (1992; Zbl 0776.46033)] proved that the space of Schatten class operators \(\mathcal{S}_p\) can be twisted, i.e., there exists a \(B(H)\)-module, \(\Theta_p\), containing a non-complemented copy of the Schatten class \(\mathcal{S}_p\), such that the quotient is another copy of \(\mathcal{S}_p\). Here it is shown that this quotient mapping is strictly singular, and that the dual of \(\Theta_p\) is \(\Theta_q\), where \(p\) and \(q\) have the usual relationship. While the latter result is to be expected, the author provides a service to the community by presenting a direct proof.

MSC:

46B28 Spaces of operators; tensor products; approximation properties
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)

Citations:

Zbl 0776.46033
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References:

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