The uniqueness of the symmetric structure in ideals of compact operators

B. R. Aminov ; V. I. Chilin

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 1, pp. 30-37

Résumé

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 $(E, \|\cdot\|_E)=(F, \|\cdot\|_F)$.