The uniqueness of the symmetric structure in ideals of compact operators

Abstract

Let Hbe a separable infinite-dimensional complex Hilbert space, let L(H) be the C∗-algebra of bounded linear operators acting in H, and let K(H) be the two-sided ideal of compact linear operators in L(H). Let (E,∥⋅∥E) be a symmetric sequence space, and let CE:={x∈K(H):{sn(x)}∞n=1∈E} be the proper two-sided ideal in L(H), where {sn(x)}∞n=1 are the singular values of a compact operator x. It is known that CE is a Banach symmetric ideal with respect to the norm ∥x∥CE=∥{sn(x)}∞n=1∥E. A symmetric ideal CE is said to have a unique symmetric structure if CE=CF, that is E=F, modulo norm equivalence, whenever (CE,∥⋅∥CE) is isomorphic to another symmetric ideal (CF,∥⋅∥CF). 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 Cp, 1≤p<∞ (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:(CE,∥⋅∥CE)→(CF,∥⋅∥CF) by a positive linear surjective isometry. We show that if F is a strongly symmetric sequence space, then every positive linear surjective isometry U:(CE,∥⋅∥CE)→(CF,∥⋅∥CF) is of the form U(x)=u∗xu, x∈CE, where u∈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:CE→CF implies that (E,∥⋅∥E)=(F,∥⋅∥F).