On Banach-Lie algebras, spectral decompositions and complex polarizations

Complex K\"ahler polarizations are constructed for a class of real Banach-Lie algebras that are not necessarily $L^*$-algebras but include all the real compact $L^*$-algebras. The approach is based on the theory of spectral decompositions of Banach space operators. The main results are illustrated by means of a family of examples that are constructed starting from the Schatten-von Neumann classes of Hilbert space operators ${\cal C}_p$ with $p\ge2$.

The results of this paper are explored on a global level in the research announcement Complex homogeneous spaces of pseudo-restricted groups.