Cartan subalgebras of operator ideals

TL;DR

This paper studies Cartan subalgebras within operator ideals, revealing multiple conjugacy classes for non-Schatten ideals and describing their geometric structure as smooth manifolds, contrasting with classical single-class results.

Contribution

It constructs uncountably many conjugacy classes of Cartan subalgebras in general ideals and characterizes their geometric structure as smooth Banach manifolds.

Findings

Uncountably many conjugacy classes for nonzero proper ideals.

Conjugacy classes form smooth manifolds modeled on Banach spaces.

Existence of a unique diffeomorphism class of full flag manifolds.

Abstract

Denote by $U_{\mathcal I}({\mathcal H})$ the group of all unitary operators in ${\bf 1}+{\mathcal I}$ where ${\mathcal H}$ is a separable infinite-dimensional complex Hilbert space and ${\mathcal I}$ is any two-sided ideal of ${\mathcal B}({\mathcal H})$. A Cartan subalgebra ${\mathcal C}$ of ${\mathcal I}$ is defined in this paper as a maximal abelian self-adjoint subalgebra of~${\mathcal I}$ and its conjugacy class is defined herein as the set of Cartan subalgebras $\{V{\mathcal C} V^*\mid V\in U_{\mathcal I}({\mathcal H})\}$. For nonzero proper ideals ${\mathcal I}$ we construct an uncountable family of Cartan subalgebras of ${\mathcal I}$ with distinct conjugacy classes. This is in contrast to the by now classical observation of P. de La Harpe who noted that when ${\mathcal I}$ is any of the Schatten ideals, there is precisely one conjugacy class under the action of the full group of unitary operators on~${\mathcal B}$. In the case when ${\mathcal I}$ is a symmetrically normed ideal and is the dual of some Banach space, we show how the conjugacy classes of the Cartan subalgebras of ${\mathcal I}$ become smooth manifolds modeled on suitable Banach spaces. These manifolds are endowed with groups of smooth transformations given by the action of the group $U_{\mathcal I}({\mathcal H})$ 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 $U_{\mathcal I}({\mathcal H})$ and we give its construction.