Complex homogeneous spaces of pseudo-restricted groups

This is a short note where some of the main results in my book Smooth Homogeneous Structures in Operator Theory are announced.

One constructs invariant Kähler structures on homogeneous spaces of certain classes of infinite-dimensional Lie groups. The main new classes of examples which fall under these general constructions are homogeneous spaces of certain Banach-Lie groups associated with admissible pairs of ideals of compact operators on Hilbert spaces.

An important point of this paper is that the existence of pseudo-restricted groups is a consequence of the local enlargibility criterion of V. Pestov [Nova J. Algebra Geom. 1 (1992), no. 4, 371--381]. This enlargibility criterion was discovered and originally proved by means of nonstandard analysis. A ``standard'' proof can be found in my paper Asymptotic products and enlargibility of Banach-Lie algebras.