Complex homogeneous spaces of pseudo-restricted groups
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.
scientific article; zbMATH DE number 2062123