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In the framework of operator theory, we investigate a close Lie theoreticrelationship between all operator ideals and certain classical groups ofinvertible operators that can be described as the solution sets of certainalgebraic equations, hence can be regarded as infinite-dimensional linearalgebraic groups. Historically, this has already been done for only thecomplete-norm ideals; in that case one can work within the framework of thewell-known Lie theory for Banach-Lie groups. That kind of Lie theory is notapplicable for arbitrary operator ideals, so we needed to find a new approachfor dealing with the general situation. The simplest instance of theaforementioned 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 idealin $\Bc(\Hc)$ and $\U(\Hc)$ is the full group of unitary operators. Weinvestigate the Cartan subalgebras (maximal abelian self-adjoint subalgebras)of $\ug_{\Ic}(\Hc)$ for $\{0\}\subsetneqq\Ic\subsetneqq\Bc(\Hc)$, and obtain anuncountably many $\U_{\Ic}(\Hc)$-conjugacy classes of these Cartan subalgebras.The cardinality proof will be given in a follow up paper \cite{BPW13} andstands in contrast to the $\U(\Hc)$-uniqueness work of de la Harpe\cite{dlH72}.

Interplay between Algebraic Groups, Lie Algebras and Operator Ideals