The Canonical Complex Structure of Flag Manifolds in a C*-algebra

Algebra Environments I.

“…Sections 1 and 2 include new and simplified proofs of results from this article, although many issues such as normalized lifts of continuous curves on structure manifolds to the groups of invertible or unitary elements of the base algebra, or linear connections on structure manifolds have been left out. The two sequels, Martin, Salinas [48,49] analyze flag manifolds of C * -algebras and generalize results from Cowen, Douglas [20] in a C * -algebra framework. Holomorphic mappings of several complex variables with values in Grassmann manifolds, hermitian holomorphic vector bundles of finite or infinite rank, and n-tuples of Hilbert space operators in Cowen-Douglas classes, which all have a role in developing holomorphic spectral theory, are investigated in Martin [46].…”

Similarity Classification and Properties of Some Extended Holomorphic Curves

“…In 1981, Apostol and Martin [3] introduced some concepts and techniques of C * -algebras to the Cowen-Douglas theory. Martin and Salinas [4] proved that extended flag manifold has a natural intrinsic complex structure and gave a criterion for determining the holomorphic maps from Ω to extended flag manifold. They later proved the congruence theorem for the tuples of elements of a C * -algebra in the CowenDouglas class.…”

“…where df is the differential of f , and J Ω denotes the complex structure of Ω [4]. When U is a unital Banach algebra, we can obtain a homeomorphism between ξ n (U) and [11].…”

Integrability of Analytic Almost Complex Structures on Banach Manifolds

“…Our motivation in writing this paper came from the fact that, in the existing literature, the subject of complex coordinates in almost complex Banach manifolds was approached in two very different manners: either one constructed the complex coordinates in a way that seemed to be very specific to the manifolds under consideration (see for instance [Ne04], [BR04], [Be03], [MS98], [MS97], [Wi94] and [Wi90]) or one exhibited examples of almost complex structures that do not come from any complex coordinates at all (see [Pa00]). Our point is to show that there exists a large class of almost complex structures, namely the torsion-free real analytic almost complex structures on real analytic Banach manifolds, that always come from complex coordinates on the corresponding manifold (Theorem 7 below).…”

“…As a matter of fact, our main result on invariant complex structures (Theorem 15) works for homogeneous spaces of not necessarily connected Banach-Lie groups. This point is particularly important for the applications to the adjoint or coadjoint orbits that show up in connection with the operator algebras (see [BR04], [MS98], [MS97] and Corollary 16 below), since the unitary groups of C * -algebras are…”