This paper studies a certain new class of Banach Lie groups in infinite dimensional Lie theory, Kähler homogeneous spaces, and aims to generalize the theory of Hilbert Lie \(L^*\)-algebras.
A pair \(({\mathcal I}_0,{\mathcal I}_1)\) of operator ideals on complex Hilbert space, similar to the Schatten \(p\)-classes and their conjugates, serves as the paper’s framework together with “equivariant monotone operators”. The latter are of the form \(\iota:{\mathfrak g}^\#\to{\mathfrak g}\), where \({\mathfrak g}\) is a real Banach Lie \(*\)-algebra with topological dual \({\mathfrak g}^\#\) such that \(\iota\) intertwines the adjoint and coadjoint action of \({\mathfrak g}\), and is positive in the sense of \(\langle f,\iota(f)\rangle\geq0\) for all \(f\in{\mathfrak g}^\#\) with respect to the canonical pairing of \({\mathfrak g}^\#\) and \({\mathfrak g}\).
For an “admissible” pair \(({\mathcal I}_0,{\mathcal I}_1)\) of operator ideals and a commuting set \(A_1,\dots,A_n\) of bounded Hilbert space operators “pseudo-restricted” Banach Lie group \(U_{{\mathcal I}_0,{\mathcal I}_1}(A_1,\dots,A_n)\) is associated consisting of all unitary operators \(T\in1+{\mathcal I}_0\) with commutation relation \([T,A_i]\in{\mathcal I}_1\) for \(i=1,\ldots,n\). The stronger condition \([T,A_i]=0\) for \(i=1,\dots,n\) yields a “weakly Kähler” homogeneous space.
This paper consists exclusively of definitions and theorems without proofs. The proofs themselves are to be found in the author’s 2002 preprint entitled “Equivariant monotone operators and infinite-dimesional complex homogeneous spaces.” The paper is well readable although largely technical.