Interplay between Algebraic Groups, Lie Algebras and Operator Ideals

TL;DR

This paper explores the deep algebraic and Lie group structures of operator ideals in infinite-dimensional spaces, extending classical Lie theory to a broader class of operator ideals beyond the norm-closed case.

Contribution

It introduces a new approach to Lie theory for arbitrary operator ideals, analyzing their associated Lie algebras and Cartan subalgebras, and classifies conjugacy classes of these subalgebras.

Findings

Uncountably many conjugacy classes of Cartan subalgebras are identified.

Extends Lie algebra and group relationships to general operator ideals.

Contrasts with classical results on the uniqueness of Cartan subalgebras.

Abstract

In the framework of operator theory, we investigate a close Lie theoretic relationship between all operator ideals and certain classical groups of invertible operators that can be described as the solution sets of certain algebraic equations, hence can be regarded as infinite-dimensional linear algebraic groups. Historically, this has already been done for only the complete-norm ideals; in that case one can work within the framework of the well-known Lie theory for Banach-Lie groups. That kind of Lie theory is not applicable for arbitrary operator ideals, so we needed to find a new approach for dealing with the general situation. The simplest instance of the aforementioned relationship is provided by the Lie algebra $\ug_{\Ic}(\Hc)=\{X\in\Ic\mid X^*=-X\}$ associated with the group $\U_{\Ic}(\Hc)=\U(\Hc)\cap(\1+\Ic)$ where $\Ic$ is an arbitrary operator ideal in $\Bc(\Hc)$ and $\U(\Hc)$ is the full group of unitary operators. We investigate the Cartan subalgebras (maximal abelian self-adjoint subalgebras) of $\ug_{\Ic}(\Hc)$ for $\{0\}\subsetneqq\Ic\subsetneqq\Bc(\Hc)$, and obtain an uncountably many $\U_{\Ic}(\Hc)$-conjugacy classes of these Cartan subalgebras. The cardinality proof will be given in a follow up paper \cite{BPW13} and stands in contrast to the $\U(\Hc)$-uniqueness work of de la Harpe \cite{dlH72}.