IUMJ

Title: Cartan subalgebras of operator ideals

Authors: Daniel Beltita, Sasmita Patnaik and Gary Weiss

Issue: Volume 65 (2016), Issue 1, 1-37

Abstract:

Denote by $U_{Ic}(Hc)$ the group of all unitary operators in $1+Ic$ where $Hc$ is a separable infinite-dimensional complex Hilbert space and $Ic$ is any two-sided ideal of $Bc(Hc)$. A Cartan subalgebra $Cc$ of $Ic$ is defined in this paper as a maximal abelian self-adjoint subalgebra of~$Ic$, and its conjugacy class is defined herein as the set of Cartan subalgebras ${VCc V^*mid Vin U_{Ic}(Hc)}$. For nonzero proper ideals $Ic$, we construct an uncountable family of Cartan subalgebras of $Ic$ with distinct conjugacy classes. This is in contrast to the (by now classical) observation of P. de La Harpe, who noted that when $Ic$ is any of the Schatten ideals, there is precisely one conjugacy class under the action of the full group of unitary operators on~$Hc$. Our perspective is that the action of the full unitary group on Cartan subalgebras of $Ic$ is transitive, while by shrinking to $U_{Ic}(Hc)$, we obtain an action with uncountably many orbits if ${0} eIc eBc(Hc)$. In the case when $Ic$ is a symmetrically normed ideal and is the dual of some Banach space, we show how the conjugacy classes of the Cartan subalgebras of $Ic$ become smooth manifolds modeled on suitable Banach spaces. These manifolds are endowed with groups of smooth transformations given by the action of the group $U_{Ic}(Hc)$ on the orbits, and are equivariantly diffeomorphic to each other. We then find that there exists a unique diffeomorphism class of full flag manifolds of $U_{Ic}(Hc)$, and we give its construction. This resembles the case of compact Lie groups when one has a unique full flag manifold, since all the Cartan subalgebras are conjugated to each other.