**
M. J. Dupre, J. F. Glazebrook**

##
Holomorphic Framings for Projections in a Banach Algebra

**abstract:**

Given a complex Banach algebra, we consider the Stiefel bundle relative to
the similarity class of a fixed projection. In the holomorphic category the
Stiefel bundle is a holomorphic locally trivial principal bundle over a certain
Grassmann manifold. Our main application concerns the holomorphic
parametrization of framings for projections. In the spatial case this amounts to
a holomorphic parametrization of framings for a corresponding complex Banach
space.