The Uuniqueness of the Symmetric Structure in Ideals of Compact Operators

Aminov, B. R. , Chilin, V. I.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 1.

Abstract:
Let \(H\) be a separable infinite-dimensional complex Hilbert space, let \(\mathcal L(H)\) be the \(C^*\)-algebra of bounded linear operators acting in \(H\), and let \(\mathcal K(H)\) be the two-sided ideal of compact linear operators in \(\mathcal L(H)\). Let \((E, \|\cdot\|_E)\) be a symmetric sequence space, and let \(\mathcal{C}_E:=\{ x \in \mathcal K(\mathcal H) : \{s_n(x)\}_{n=1}^\infty \in E\}\) be the proper two-sided ideal in \(\mathcal L(H)\), where \(\{s_n(x)\}_{n=1}^{\infty}\) are the singular values of a compact operator \(x\). It is known that \(\mathcal{C}_E\) is a Banach symmetric ideal with respect to the norm \( \|x\|_{\mathcal C_E}=\|\{s_n(x)\}_{n=1}^{\infty}\|_E\).

A symmetric ideal \(\mathcal{C}_E\) is said to have a unique symmetric structure if \(\mathcal{C}_E = \mathcal{C}_F\), that is \(E =F\), modulo norm equivalence, whenever \((\mathcal{C}_E, \|\cdot\|_{\mathcal{C}_E})\) is isomorphic to another symmetric ideal \((\mathcal{C}_F, \|\cdot\|_{\mathcal{C}_F})\). At the Kent international conference on Banach space theory and its applications (Kent, Ohio, August 1979), A. Pelczynsky posted the following problem:

 (P) Does every symmetric ideal have a unique symmetric structure?

This problem has positive solution for Schatten ideals \(\mathcal{C}_p, \ 1\leq p < \infty\) (J. Arazy and J. Lindenstrauss, 1975). For arbitrary symmetric ideals problem (P) has not yet been solved. We consider a version of problem (P) replacing an isomorphism \(U:(\mathcal{C}_E, \|\cdot\|_{\mathcal{C}_E}) \to (\mathcal{C}_F, \|\cdot\|_{\mathcal{C}_F})\) by a positive linear surjective isometry. We show that if \(F\) is a strongly symmetric sequence space, then every positive linear surjective isometry \(U:(\mathcal{C}_E, \|\cdot\|_{\mathcal{C}_E}) \to (\mathcal{C}_F, \|\cdot\|_{\mathcal{C}_F})\) is of the form \(U(x) = u^*xu\), \(x \in \mathcal C_E\), where \(u \in \mathcal L(H)\) is a unitary or antiunitary operator. Using this description of positive linear surjective isometries, it is established that existence of such an isometry \(U:\mathcal{C}_E \to \mathcal{C}_F\) implies that \((E, \|\cdot\|_E)=(F, \|\cdot\|_F)\).

Keywords: symmetric ideal of compact operators, uniqueness of a symmetric structure, positive isometry

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