Interplay between Algebraic Groups, Lie Algebras and Operator Ideals

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}.